Maths lessons

276 lessons

• 3D Shapes: Faces, Edges, Vertices, Nets and Views – A 3D shape (solid) has length, width and height, so it takes up space. We describe it by counting its faces (flat or curved surfaces), edges (lines where two faces meet) and vertices (corners). Prisms have two equal ends joined by rectangles; pyramids have one base and triangles that meet at a top point; cylinders, cones and spheres have curved surfaces. A net is a flat pattern that folds into the solid. Drawing a solid from the top, front and side gives its three views. For every solid with flat faces, Faces + Vertices − Edges = 2 (Euler's rule).
• Absolute Value and Intervals – The absolute value |x| is the distance of x from 0 on the number line, so it is never negative. |x - a| is the distance between x and a. Solving |x - a| = b gives x = a + b or x = a - b. An interval is a stretch of the number line, written with round brackets (end not included) or square brackets (end included).
• Advanced Mathematics: Trigonometry, Calculus, Geometry and Probability – Advanced school maths has three big strands. Trigonometry and calculus study waves and change: sin θ from a circle, the derivative as slope and the integral as area. Geometry studies shapes and solids, such as cone : sphere : cylinder = 1 : 2 : 3. Probability and statistics measure chance and data, from dice to the bell curve.
• Algebraic Expressions: Variables, Terms and Simplifying – An algebraic expression uses letters (variables) and numbers joined by +, −, × and ÷, like 3x + 2. A variable stands for a number that can change. An expression is made of terms (3x and 2). In 3x, 3 is the coefficient; a term with no letter, like 2, is the constant. Like terms have exactly the same letter part (3x and 5x) and can be added; unlike terms (3x and 2, or x and x²) cannot. Substitution means putting a number in place of the letter to find the value. Expanding removes brackets: a(b + c) = ab + ac. Factorising is the reverse: take out the common factor. A formula is an expression that gives one quantity from others, and an identity is true for every value of the letter.
• Alligation and Mixture – When two ingredients with prices (or strengths) c and d are mixed, the mixture's mean price m lies between them. The rule of alligation says the quantities must be in the ratio (d − m) : (m − c), cheaper to dearer. It is a short cut for the weighted average m = (q₁c + q₂d) ÷ (q₁ + q₂). For repeated replacement, if x units are drawn from a vessel of V units and replaced with water n times, the original liquid left is V(1 − x/V)ⁿ.
• AM–GM Inequality – For two positive numbers a and b, the arithmetic mean (a+b)/2 is never smaller than the geometric mean √(ab). They are equal only when a = b. This gives the biggest product when the sum is fixed, and the smallest sum when the product is fixed.
• Angle Bisector – An angle bisector is a ray that cuts an angle into two equal parts. Every point on it is the same distance from both arms of the angle, and every point that is the same distance from both arms lies on it. You can draw it with only a compass and a ruler.
• Angles in Space: Dihedral, Trihedral and Polyhedral Angles – A dihedral angle is formed by two half-planes with a common edge; it is measured by its linear angle, the angle between two rays that are perpendicular to the edge. A trihedral angle has three edges, three face (plane) angles α, β, γ and three dihedral angles A, B, C. For it: cos α = cos β cos γ + sin β sin γ cos A, and sin α / sin A = sin β / sin B = sin γ / sin C. A convex polyhedral angle has face angles that add to less than 360°.
• Annuities and Mortgages – An annuity is a series of equal payments made at equal time intervals. A mortgage is a big loan to buy property that you pay back with an annuity. Each payment is split into interest (the cost of borrowing) and principal (the part that reduces the loan). At first most of the payment is interest; later most is principal. An amortization table lists every payment, its interest part, its principal part and the balance left. The payment is PMT = PV × i ÷ (1 − (1 + i)^−n). A higher rate, a longer term or fewer payments per year all raise the total interest you pay.
• Annuities: Equal Payments, Future Value and Present Value – An annuity is a series of equal payments made at equal time gaps, like a monthly saving or a loan instalment. Each payment earns compound interest, so the payments form a geometric series. Future value of an ordinary annuity (payments at the end of each period): FV = P[(1+i)ⁿ − 1]/i. Present value (what the payments are worth today): PV = P[1 − (1+i)⁻ⁿ]/i. Loans use PV: the loan amount equals the present value of all instalments.
• Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
• Application of Integrals: Area Under Curves – The area between a curve y = f(x), the x-axis and the lines x = a and x = b is ∫ₐᵇ |f(x)| dx: we add thin vertical strips of height y and width dx. For curves given as x = g(y) we use horizontal strips. Symmetry saves work: find one part of a circle, parabola or ellipse and multiply. A circle of radius r gives πr² and an ellipse with semi-axes a, b gives πab.
• Area and Perimeter: Heron's Formula, Circles and Sectors – Perimeter is the length of the edge. Area is the space inside. For a triangle with three known sides, Heron's formula gives the area without any height. For a four-sided shape whose corners sit on a circle, Brahmagupta's formula does the same. For circles, the edge is π times the diameter, and a slice (sector) is just a fraction of the whole circle.
• Area of Polygons and Why the Formulas Work – Area is the amount of flat space inside a shape, counted in square units. A rectangle has area base × height. Cutting and moving pieces shows that a parallelogram has the same area (base × height), a triangle is half of a parallelogram (½ × base × height), a rhombus is half of the rectangle drawn on its diagonals (½ × d₁ × d₂), and a trapezium is half of a parallelogram with base (a + b): ½ × (a + b) × h. Units change by squares: 1 m² = 10 000 cm².
• Area of Sector and Segment of a Circle – A sector is a pizza slice of a circle: its area is θ/360 × πr². Its crust is an arc of length θ/360 × 2πr. A segment is the slice minus the triangle inside it: segment = sector − triangle.
• Arithmetic at Work: Unit Price, Total Cost and Change – Shops and workplaces use the four operations every day. Divide the pack price by the number of items to get the unit price (price of one). Multiply the unit price by how many you want to get the total cost. Subtract the cost from the money paid to get the change. To find the best buy, compare unit prices: the smaller one is cheaper.
• Arithmetic Progressions – An arithmetic progression (AP) is a list of numbers where each term is made by adding the same fixed number d (the common difference) to the term before. With first term a: nth term aₙ = a + (n − 1)d; sum of the first n terms Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
• Beauty and Mathematics – Things look beautiful when they are simple, balanced and orderly. Maths describes this order: symmetry (mirror and rotation), periodicity (a pattern that repeats after a fixed distance) and harmony (parts in simple ratios such as 2 : 3). A shape with rotation symmetry of order n turns by 360° ÷ n and looks the same.
• Binomial Distribution – Repeat the same yes/no trial n times, independently, with the same chance of success p each time. The number of successes X follows the binomial distribution B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ with q = 1 − p. Its mean is np and its variance is npq.
• Binomial Theorem for Positive Integers – A binomial is a two-term expression like a + b. The binomial theorem tells us how to expand (a + b)ⁿ for any positive integer n without multiplying again and again: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. There are n + 1 terms. The power of a goes down by 1 and the power of b goes up by 1 in each term, and the two powers always add up to n. The coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle, where each number is the sum of the two above it. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The pattern was known long ago in India (Pingala's Meru Prastara) and in Persia and China, and Pascal later studied the triangle in detail. The theorem is proved by mathematical induction using ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.
• Chi-Square Test for Independence and Homogeneity – A chi-square (χ²) test checks if counts in a table are too far from what we would expect by chance. For a contingency table: E = row total × column total ÷ grand total, χ² = Σ (O − E)² ÷ E, df = (r − 1)(c − 1). If χ² is bigger than the critical value (or p < significance level), reject H0 of no association. For 2×2 tables, Yates' correction uses (|O − E| − 0.5)².
• Circle Theorems – Circle theorems are a small set of angle rules that are always true inside a circle. The angle at the centre is twice the angle at the edge. The angle in a semicircle is 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. A tangent meets the radius at 90°, two tangents from one point are equal, and the tangent–chord angle equals the angle in the alternate segment.
• Circle: Circumference, Pi and Area – The distance around a circle is its circumference: C = π × d = 2πr. The space inside is its area: A = πr². Pi (π) is the number you get when you divide the circumference by the diameter. It is about 3.14 for every circle, big or small.
• Circles and Tangents: How to Draw Them and Why They Work – A tangent touches a circle at one point and is at 90 degrees to the radius. From an outside point P you can draw two equal tangents of length √(D² - r²). Two circles have up to four common tangents. A circle can join two lines smoothly, and arcs with changing centres make spirals. Power of a point says PA x PB = D² - r² for every line through P. The radical axis is the straight line of points that have equal power for two circles.
• Circles: Chords and Angles – A chord joins two points on a circle. Longer chords make bigger angles at the centre, and equal chords make equal angles. The perpendicular from the centre to a chord cuts it in half, and equal chords are the same distance from the centre. The angle an arc makes at the centre is double the angle it makes anywhere on the rest of the circle, so the angle in a semicircle is 90°. In a cyclic 4-gon, opposite angles add to 180°.
• Clock and Calendar: Telling Time by the Sky and by Numbers – Our time units come from the sky. One spin of the Earth relative to the Sun is a solar day (24 h); relative to the stars it is a sidereal day (about 23 h 56 min). The Moon's phases repeat every 29.53 days (a month). The Earth goes round the Sun in about 365.2422 days (a year). Calendars are lunar, solar or lunisolar. The Gregorian leap year rule keeps the calendar in step with the seasons. With numbers we can then solve clock problems (angle = |30H − 5.5M|) and calendar problems (odd days).
• Combined Events, Tree Diagrams and Conditional Probability – When two things happen, list every outcome in a sample space, a table, a Venn diagram or a tree diagram. Multiply along tree branches (the multiplication rule) and add the paths you want. If the first event changes the second, the events are dependent: picking without replacement is the classic case. Conditional probability P(A | B) is the chance of A when we already know B happened: P(A | B) = P(A and B) ÷ P(B).
• Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.
• Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
• Compound Units: Speed, Density, Unit Price and Other Rates – A compound unit joins two units, like metres per second (m/s) or grams per cubic centimetre (g/cm³). "Per" means "in each one", so a rate tells you how much of one amount fits into ONE unit of another. Speed = distance ÷ time, density = mass ÷ volume, unit price = cost ÷ amount. To convert a compound unit, convert the top unit and the bottom unit separately.
• Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).
• Confidence Intervals – A confidence interval is a range of believable values for an unknown population number (a mean μ or a proportion p), worked out from one sample. It has the shape estimate ± margin of error, where margin of error = critical value × standard error. A 95% level means the method catches the true value in about 95% of samples. Higher confidence gives a wider interval; a bigger sample gives a narrower one. If a claimed value lies outside the interval, the data give evidence against the claim.
• Congruence of Triangles – Two triangles are congruent when one can be placed exactly on top of the other: all three sides and all three angles match. You do not need to check all six parts. Any one of SAS, SSS, ASA, AAS or RHS is enough. SSA is not enough. In an isosceles triangle the angles opposite the equal sides are equal, and the converse is also true.
• Congruence: Same Shape, Same Size – Two figures are congruent (≅) if one can be moved exactly onto the other by rigid motions: translations (slides), reflections (flips) and rotations (turns), in any order. Rigid motions keep every length and every angle, so congruent figures have equal corresponding sides and equal corresponding angles. A dilation (enlargement) changes size, so it does not keep congruence; it gives similar figures. To prove two figures congruent, describe a sequence of rigid motions that maps one onto the other, and match vertices in order (A↔A′, B↔B′…). For triangles, SSS, SAS and ASA are shortcuts that follow from rigid motions.
• Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.
• Constructing Sections of Solids: The Trace Method – A section is the flat shape where a plane cuts a solid. To draw it, join points that lie on one face; if lines do not meet inside the solid, extend them to the base plane. The line where the cutting plane meets the base plane is the trace. The trace gives new points of the section. A section has at most one side per face.
• Consumer Mathematics: Earning, Buying and Travelling – Consumer mathematics is everyday money maths. Gross pay is all you earn: hourly wage (hours × rate), salary, commission (a % of sales), piecework, tips, bonus and overtime (often 1.5 × rate). Deductions such as income tax, pension and insurance are taken off, leaving net pay (take-home pay). When buying, take the discount first, then add sales tax (GST or VAT): price × (1 − d) × (1 + t). Unit price = price ÷ quantity, so you can compare packs. Count up to give correct change. A car costs fuel (distance × litres per 100 km ÷ 100 × price), insurance (higher for new drivers), licence, loan, repairs and parking; compare it with bus, train or plane for each trip.
• Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
• Continuity and the Intermediate Value Theorem – A function is continuous at a point if its limit there equals its value. A continuous function on an interval has no jumps: it cannot skip a value between two of its values (intermediate value theorem). If it is also strictly monotone, it takes each such value exactly once.
• Coordinate Geometry: Distance and Section Formula – Every point on a flat plane has an address (x, y). The distance between A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²]; it is just Pythagoras on the x-gap and the y-gap. A point P that cuts AB inside in the ratio m : n is P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). When m = n, P is the midpoint.
• Coordinate Proofs: Using Coordinates to Prove Geometry – In a coordinate proof we place a shape on the grid and use algebra to prove facts about it. The distance formula proves lengths, the midpoint formula proves bisecting, and slopes prove parallel (equal slopes) or perpendicular (slopes multiply to −1). For circles, a point (x, y) lies on the circle with centre (h, k) and radius r exactly when (x − h)² + (y − k)² = r²; completing the square turns a messy equation into this form.
• Critical Path Analysis – A big project is made of many activities. Some must wait for others. We draw them as an activity-on-node network. A forward pass gives each activity its earliest start; a backward pass gives its latest finish. Float = latest start − earliest start tells how much an activity can slip. Activities with zero float form the critical path: the longest route, which fixes the shortest possible project time. A Gantt (cascade) chart turns the network into bars on a time line, and a resource histogram shows workers needed per day. Moving activities inside their float to smooth that histogram is resource levelling.
• Cross-Sections and Solids of Rotation: Linking 2D and 3D Shapes – A cross-section is the flat face you get when a plane cuts a solid. A cube can give a square, rectangle, triangle, pentagon or hexagon; a cylinder, cone or sphere gives a circle when cut across. Spinning a 2D shape around an axis makes a solid of rotation: a rectangle makes a cylinder, a right triangle a cone, a semicircle a sphere. Cross-sections also explain volume: if two solids have equal cross-sections at every height, they have equal volume (Cavalieri's principle).
• Cubes and Cube Roots – The cube of a number n is n × n × n = n³. It is the number of unit blocks in a cube whose edge has n blocks. A perfect cube is a whole number that is the cube of a whole number (1, 8, 27, 64, 125 …). The cube root ∛m undoes cubing: ∛64 = 4 because 4³ = 64. Find it by grouping equal prime factors in threes, or by estimating with the last digit. 1729 is the smallest number that is a sum of two cubes in two ways.
• Curve Sketching Using Derivatives – To sketch y = f(x): find the domain and intercepts, solve f'(x) = 0 for stationary points, use the sign of f'(x) to see where the curve rises or falls, use f''(x) for concavity and points of inflection, check asymptotes and end behaviour, then join everything smoothly. Example: y = x³ − 3x has a maximum at (−1, 2), a minimum at (1, −2) and an inflection at (0, 0).
• Curves and Surfaces: Spinning a Curve into a Surface – A curve is a path of points that follow a rule. Spin a curve around an axis and it sweeps a surface. A straight slanted line makes a cone, a parabola makes a bowl (paraboloid), and a hyperbola makes a cooling tower (hyperboloid). Slicing a surface at a height gives a circle whose radius comes from the curve.
• Cyclic and Tangential Quadrilaterals – A cyclic quadrilateral has all four corners on one circle. Its opposite angles add up to 180°, and each outside angle equals the opposite inside angle. A tangential quadrilateral has a circle inside touching all four sides; there, opposite sides have equal sums: AB + CD = BC + DA. The converse of the intercept (Thales) theorem tells when a line is parallel to a side of a triangle.
• Data Cleaning: Outliers, Missing Values and Normalisation – Real data is messy. Before we analyse it we clean it: find and fix outliers (strange values), deal with missing values (fill or remove them), and normalise (rescale) numbers so that different measurements can be compared fairly. This is called data preprocessing.
• Data Handling: Collect, Organise and Show Data – Data is a set of facts, such as answers, counts or measurements. Data can be qualitative (words) or quantitative (numbers); numbers are discrete (counted) or continuous (measured). We collect data by surveys, observation, experiments or from existing sources, then organise it with tally marks into a frequency table. We show it with the right graph: bar graphs to compare groups, pie charts to show parts of a whole, line graphs for change over time and scatter graphs for links between two variables. Computers store data as structured tables or unstructured text, images and sound.
• De Moivre's Theorem and Roots of Unity – A complex number can be written by its length r and angle θ: z = r(cos θ + i sin θ) = re^{iθ}. When you multiply, the lengths multiply and the angles add. So zⁿ = rⁿ(cos nθ + i sin nθ): this is De Moivre's theorem. It gives quick powers, formulas for cos nθ and sin nθ, and the n roots of any number. The n roots of 1 sit evenly on the unit circle like the corners of a regular polygon.
• Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
• Determinants: Minors, Cofactors, Adjoint, Inverse and Linear Systems – A determinant is one number made from a square matrix, written |A|. For a 2 × 2 matrix it is ad − bc, and it equals the (signed) area made by the columns. For a 3 × 3 matrix we expand along a row using minors and cofactors. |A| = 0 means A is singular and has no inverse. Half of a determinant gives the area of a triangle. The adjoint (transpose of the cofactor matrix) gives A⁻¹ = (adj A)/|A|, and then a system AX = B is solved by X = A⁻¹B. The value of |A| and (adj A)B tell us if a system is consistent.
• Differential Equations – A differential equation connects a function y with its derivatives. Its order is the highest derivative present and its degree is the power of that derivative (when the equation is a polynomial in derivatives). A general solution has arbitrary constants; a condition like y(0) = 1 fixes them to give a particular solution. Class 12 solves first-order equations of three kinds: variables separable, homogeneous (put y = vx) and linear dy/dx + Py = Q (multiply by the integrating factor e^∫P dx).
• Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
• Dynamic Geometry: Drag, Measure and Discover – In dynamic geometry software you build a figure from points, lines and circles, then drag a point and watch the figure follow the rules you built in. What changes shows what is free. What never changes is a geometric rule (an invariant). Testing many positions helps you guess a rule, but a proof is still needed to be sure.
• EMI: Equated Monthly Instalments, Flat Rate and Reducing Balance – An EMI (equated monthly instalment) is the same amount paid every month to clear a loan with interest. In the flat-rate method interest is worked out on the full loan for the whole time: EMI = (P + P×R×T/100) ÷ n. In the reducing-balance method interest is charged only on the amount still owed, so EMI = P·r·(1+r)^n ÷ ((1+r)^n − 1), where r is the monthly rate as a decimal and n the number of months. Each EMI pays some interest and some principal; the interest part falls and the principal part grows until the balance is zero.
• Equation of a Circle – A circle is the set of points at a fixed distance r from a centre (h, k). By the distance formula its equation is (x − h)² + (y − k)² = r²; with centre at the origin, x² + y² = r². Opened up, it becomes x² + y² + Dx + Ey + F = 0, with centre (−D/2, −E/2) and r² = D²/4 + E²/4 − F. A line meets a circle in 2, 1 or 0 points when the distance d from centre to line is less than, equal to or more than r. Two circles are compared by the distance between their centres.
• Equations and Inequalities with a Parameter – A parameter is a letter such as a that stands for one fixed number, but we do not know which. We ask "for which a does the equation have 0, 1, 2 or more solutions?". The graph method moves a, writing the equation as f(x) = a and sliding the horizontal line y = a across the curve y = f(x).
• Estimating Probability Using Simulation – A simulation imitates a chance process with random numbers so we can estimate a probability that is hard to calculate. Plan it: state the question, give each outcome a set of random digits that matches its probability, define one trial and what counts as the event, then repeat many trials. The relative frequency (event count ÷ trials) is the estimate, and it gets closer to the true probability as the number of trials grows (the law of large numbers).
• Estimation and Order of Magnitude – An estimate is a quick, sensible answer made with rounded numbers. Count a small part, multiply up, and give the answer to a sensible precision. The order of magnitude is the nearest power of ten, such as 10² for about 100.
• Ethnomathematics: Maths in Patterns and Buildings – Ethnomathematics studies the maths that people use in their own cultures: weaving, drawing floor patterns, building, measuring and counting. Most patterns come from three moves on one motif: slide (translation), mirror (reflection) and turn (rotation). A shape with n-fold turning symmetry matches itself every 360° ÷ n. Only equilateral triangles, squares and regular hexagons cover a floor alone, because their angles fit 360° at a point.
• Euclid's Geometry: Definitions, Axioms and the Five Postulates – Geometry began as practical measuring of land and altars in Egypt, India and Mesopotamia. Indian Sulbasutras (like Baudhayana's) gave rope rules for making squares, doubling a square and the diagonal rule. Around 300 BCE, Euclid of Alexandria organised geometry as a chain of reasoning: start from a few definitions, common-sense axioms and five geometry postulates, and prove everything else. The fifth postulate is about when two lines meet, and it leads to the idea of parallel lines.
• Exchange Rates: Converting Money Between Currencies – A currency is the money used in a country or group of countries. An exchange rate is the price of one currency in terms of another, for example 1 USD = 83 INR. To convert into the second currency, multiply by the rate; to convert back, divide. Banks and money changers sell foreign currency at a higher rate and buy it at a lower rate, and may charge commission, so each swap costs you a little. Rates change with demand and supply. When a currency appreciates it buys more foreign money: imports get cheaper and exports dearer. When it depreciates, the opposite happens. People and firms gain or lose when rates move between buying and selling.
• Exploring Algebraic Identities – An identity is an equation that is true for every value of the letters. Pictures prove them: a square of side (a + b) splits into a², ab, ab and b², so (a + b)² = a² + 2ab + b². In the same way we get (a − b)², a² − b² = (a + b)(a − b), (x + a)(x + b), (a + b + c)² and (a + b)³. Read backwards, identities help us factorise, calculate fast and simplify rational expressions.
• Exponential Equations and Inequalities – An exponential equation has the unknown in the power, like 2^x = 8. Solve it by making the bases equal, by substitution when it hides a quadratic, or by taking logarithms. For inequalities, keep the sign when the base is bigger than 1 and flip it when the base is between 0 and 1.
• Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
• Exponential, Logarithmic and Irrational Inequalities – Three families of inequalities share one idea: compare two graphs. For powers and logs the base decides whether the sign flips (base above 1 keeps it, base between 0 and 1 flips it). For logs and roots the domain must be checked first, and a root inequality needs care about the sign of the other side.
• Factorising Quadratic Trinomials – A quadratic trinomial like x² + 5x + 6 can be written as a product of two brackets, (x + 2)(x + 3). For x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, find two numbers that multiply to a × c and add to b, split the middle term, then take out common factors. Always check by expanding.
• Fractals and the Sierpinski Triangle – A fractal is a pattern made by repeating the same rule again and again, so a small part looks like the whole. This is called self-similarity. In the Sierpinski triangle, each step removes the middle of every triangle: the number of triangles goes 1, 3, 9, 27 … (3ⁿ) and the area left is multiplied by 3/4 each time.
• Fractions: The Four Operations Made Simple – A fraction is part of a whole cut into equal pieces. The bottom number (denominator) says how many pieces make the whole; the top number (numerator) says how many we take. Equivalent fractions name the same amount (3/4 = 6/8). To add or subtract, first make the denominators the same. To multiply, multiply top by top and bottom by bottom. To divide, keep the first fraction, change ÷ to ×, and flip the second. The same four operations work for decimals and negative numbers, and BIDMAS tells us which operation to do first. An inverse operation (the opposite one) lets us check any answer.
• Function Study: Periodicity, Monotonicity, Extrema, Max and Min on an Interval – To study a function we ask four questions. Does the graph repeat (period T with f(x + T) = f(x))? Where does it go up or down (the sign of the slope)? Where does it turn (slope zero, giving local maximum or minimum)? And on a chosen interval [a, b], which value is the largest and smallest (check the two ends and the turning points inside)?
• Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
• Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
• Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.
• Game Theory: Making the Best Choice When Others Choose Too – Game theory studies decisions where your result depends on what others choose. A pay-off matrix lists each player's gain for every pair of choices. A dominated strategy is always worse and can be removed. A Nash equilibrium is a pair of choices where no player gains by changing alone; the prisoner's dilemma shows it can be worse for everyone than cooperating. In a zero-sum game, the play-safe (maximin/minimax) strategies meet at a saddle point when the game is stable; otherwise players use a mixed strategy, found by drawing expected-pay-off lines and taking the highest point of the lower edge.
• General Binomial Expansion for Any Rational Power – For any rational n, (1 + x)^n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … When n is a positive whole number the series stops. Otherwise it never stops, and it is only true when |x| < 1. For (a + bx)^n, take a^n outside first: a^n(1 + bx/a)^n, valid for |bx/a| < 1, that is |x| < |a/b|. A few terms give good approximations when x is small.
• Geometric Constructions: Drawing Exactly with Compass and Straightedge – A geometric construction draws a figure exactly using only a compass and a straightedge (a ruler used for straight lines). Key constructions: the perpendicular bisector of a segment, the bisector of an angle, a perpendicular from a point to a line, angles of 60°, 30°, 90° and 45°, a triangle from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and regular polygons such as the hexagon. Each works because equal compass arcs make equal lengths, which give congruent triangles.
• Geometric Modelling: Describing Real Objects with Simple Solids – Geometric modelling means replacing a real object with simple shapes (cuboid, cylinder, cone, sphere, prism) so we can calculate with it. The cycle is: look at the real object, simplify it, measure, calculate volume or surface area, then check the answer against reality and improve the model. Scale changes lengths by k, areas by k² and volumes by k³. Geometry also helps us see patterns in nature and art, like hexagons in honeycombs.
• Geometric Probability: Choosing a Random Point – When a point is chosen completely at random from a figure, the chance that it lands in a smaller part is (size of the part) ÷ (size of the whole). Size means length for a segment or arc, and area for a flat figure.
• Geometry Basics: 2D Shapes, 3D Solids, Nets and Scale – Geometry is the maths of shape, size and position. A 2D (flat) shape has length and width only: triangles, squares, hexagons, circles. A 3D (solid) shape also has depth: cubes, prisms, pyramids, cylinders. A solid has faces (flat sides), edges (where two faces meet) and vertices (corners). A net is a flat pattern that folds into a solid. Two shapes are congruent if they have the same shape and size, and similar if they have the same shape but a different size. A tessellation covers a floor with shapes and leaves no gaps. A scale drawing shows a real object smaller or bigger by a fixed scale factor.
• Gradient Descent: Finding the Lowest Point – Gradient descent finds the minimum of a function by taking small steps downhill. At each step, new x = old x − learning rate × slope. A learning rate that is too big overshoots; too small is very slow. It is how machine learning models reduce their error.
• Graph Theory: Dots, Lines and Networks – A graph is a set of vertices (dots) joined by edges (lines). The degree of a vertex is how many edges touch it, and the sum of all degrees is twice the number of edges. An Euler trail uses every edge once and exists only when 0 or 2 vertices have odd degree. A tree is a connected graph with no cycles and n − 1 edges. Weighted graphs model roads and networks; Kruskal’s and Prim’s algorithms find a minimum spanning tree.
• Graphs of Basic Functions and Solving Equations with Graphs – A graph shows every point (x, y) with y = f(x). Five shapes come up again and again: the parabola y = x², the cubic y = x³, the half-parabola y = √x, the V-shape y = |x| and the hyperbola y = k/x. Learn each one's domain, range, symmetry, where it rises or falls, and where it crosses the axes. Then use graphs to solve equations (where graphs cross) and inequalities (where one graph is above the other).
• Graphs of Composite Functions and Geometric Images of Equations – A change inside the bracket moves a graph sideways; a change outside moves it up, down or flips it. |f(x)| flips the part under the x-axis up, and f(|x|) copies the right half to the left. An equation in x and y draws a picture too, such as a circle or a diamond, and it is the graph of a function only if every vertical line cuts it once.
• Group Theory: Binary Operations, Groups, Rings and Fields – A group is a set with one operation that is closed, associative, has an identity and gives every element an inverse. Clock arithmetic Z6 shows all of it: tables, subgroups, the order of an element and Lagrange's theorem. Rings and fields add a second operation.
• Growth Models: Recursive Sequences and y' = ay + b – A growth model says how a quantity changes step by step: u(n+1) = a·u(n) + b. If b = 0 it is geometric, if a = 1 it is arithmetic. For 0 < a < 1 the sequence moves to the fixed point L = b/(1 − a). The continuous version is y' = ay + b with solution y = C·e^(ax) − b/a. Euler's method turns y' = f(y) back into steps, and the logistic model slows growth near a limit.
• Heights and Distances – The line of sight joins your eye to the object. If the object is above you, the angle between the line of sight and the horizontal is the angle of elevation. If it is below you, that angle is the angle of depression. The object, the ground and the line of sight make a right triangle, so tan θ = height ÷ distance. The angle of depression from the top equals the angle of elevation from the bottom. Hard questions use two right triangles that share one side.
• History of Mathematics – Mathematics grew over thousands of years in many places. People first counted with tally marks. Egypt and Babylon used geometry for land and building, and Babylon counted in 60s. Greek thinkers such as Thales, Pythagoras, Euclid and Hypatia turned geometry into proofs. India gave place value with zero, scholars in Baghdad built algebra, and the ideas reached Europe. Women and men from every continent have shaped maths, often against unfair barriers.
• Hyperbolic Functions – Hyperbolic functions are built from eˣ and e⁻ˣ: cosh x = (eˣ + e⁻ˣ)/2, sinh x = (eˣ − e⁻ˣ)/2, tanh x = sinh x/cosh x. The point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, so cosh²x − sinh²x = 1. Their inverses have log forms, e.g. arsinh x = ln(x + √(x² + 1)). They differentiate neatly (d/dx sinh x = cosh x, d/dx cosh x = sinh x) and give standard integrals such as ∫ 1/√(x² + 1) dx = arsinh x + c.
• Hypothesis Testing – A hypothesis test checks a claim about a population using a sample. Start with the null hypothesis H₀ (no change, e.g. p = 0.5) and the alternative H₁ (what we suspect, e.g. p > 0.5). Choose a significance level such as 5%. Work out how likely the sample result (or more extreme) is if H₀ were true: the p-value. If the p-value is below the level, or the result falls in the critical region, reject H₀. Otherwise there is not enough evidence to reject it. Type I error = rejecting a true H₀; Type II error = not rejecting a false H₀.
• Inequalities: Rules, Intervals and Solving Them – An inequality says one amount is bigger or smaller than another, using <, >, ≤ or ≥. On a number line, the smaller number is on the left. You may add or subtract the same number on both sides, and multiply or divide by the same positive number, and the sign stays. If you multiply or divide by a negative number, the sign flips. The answer is usually a whole set of numbers, written as an interval such as (−∞, 4]. A quadratic inequality is solved from its roots and the shape of its graph. |x| < a means −a < x < a. Some inequalities are true for every number, like x² ≥ 0 and the AM–GM inequality.
• Infinite Series – An infinite series adds the terms of a sequence forever: a₁ + a₂ + a₃ + … We study it through its partial sums Sₙ. If Sₙ settles at a number S, the series converges to S; otherwise it diverges. A geometric series a + ar + ar² + … converges to a/(1 − r) when |r| < 1. Terms going to 0 is needed but not enough: the harmonic series 1 + 1/2 + 1/3 + … diverges. Tests (nth-term, p-series, comparison, integral, ratio, alternating) tell us which series converge.
• Inscribed and Circumscribed Solids: When One Solid Sits Inside Another – A solid is inscribed in another when it sits inside and touches it at special points; the outer one is circumscribed. Sphere in a cube: r = a/2. Sphere around a cube: R = a√3/2. Sphere in a cylinder: h = 2r. Cylinder in a sphere: R² = r² + h²/4. Sphere in a cone: ρ = rh/(r + l). Draw a cross-section through the axis and each problem becomes a flat triangle, square or circle problem.
• Integers and Signed Numbers – Integers are the whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. On a number line, numbers grow to the right. Adding a positive moves right, adding a negative moves left. Subtracting a number is the same as adding its opposite. For × and ÷, same signs give a positive answer and different signs give a negative answer. The same sign rules work for signed decimals like −2.5.
• Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
• Integration as Accumulation: Riemann Sums to Volumes – A definite integral adds up a rate. Cut the region into thin rectangles (a Riemann sum), let them get thinner, and the sum becomes the exact area. The same idea gives total change, average value, displacement and distance, and the volume of solids made by spinning or stacking slices.
• Intervals of Real Numbers – An interval is all the real numbers between two ends. A square bracket [ ] means the end is included, a round bracket ( ) means it is left out. [a, b] is closed, (a, b) is open, [a, b) and (a, b] are half-open. Intervals can go on for ever, like [1, ∞), but ∞ always gets a round bracket. On the number line a filled dot means in and a hollow dot means out. The intersection A ∩ B is the part in both intervals; the union A ∪ B is everything in at least one of them.
• Introduction to Polynomials – An algebraic expression is made of terms like 3x², −5x and 7. It is a polynomial when every power of the variable is a whole number (0, 1, 2, …). The degree is the biggest power. Degree 1 polynomials, y = ax + b, are called linear. They model things that grow or shrink by the same amount each step. a is the slope (change per step) and b is the y-intercept (starting value).
• Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.
• Introduction to Three-dimensional Geometry (Class 11) – In space we use three mutually perpendicular axes x, y, z through the origin O. Each pair makes a coordinate plane: XY (z = 0), YZ (x = 0) and ZX (y = 0). The three planes divide space into eight octants, named by the signs of x, y, z. A point P(x, y, z) is reached by moving x along the x-axis, y parallel to the y-axis and z parallel to the z-axis; x, y, z are its distances from the YZ, ZX and XY planes. Points on the x-axis are (x, 0, 0); points on the XY-plane are (x, y, 0). The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is PQ = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
• Inverse Functions – An inverse function undoes what a function does. If f sends a to b, then f⁻¹ sends b back to a. To find it, write y = f(x), swap x and y, and solve for y. The graph of f⁻¹ is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse; if a function is not one-to-one, limit its domain first (for example x² with x ≥ 0 has inverse √x).
• Inverse Trigonometric Functions (Class 12) – sin x, cos x and the other trig functions repeat, so they are not one-one and have no inverse on all of R. We cut each one to a piece (the principal value branch) where it is one-one and onto. On that piece it has an inverse: y = sin⁻¹x means sin y = x with y in [−π/2, π/2]. The graph of an inverse is the mirror image of the branch in the line y = x. Principal ranges: sin⁻¹ [−π/2, π/2], cos⁻¹ [0, π], tan⁻¹ (−π/2, π/2), cot⁻¹ (0, π), sec⁻¹ [0, π] − {π/2}, cosec⁻¹ [−π/2, π/2] − {0}.
• Inverse Variation (Inverse Proportion) – Two quantities vary inversely when their product stays the same: x × y = k, so y = k/x. If x is doubled, y is halved. The graph of y = k/x (k > 0) is a curve called a hyperbola that comes close to both axes but never touches them.
• Irrational Equations (Square Root Equations) – An irrational equation has the unknown inside a root. To solve it, put the root alone on one side, square both sides, solve the new equation, and then check every answer in the original. Squaring can add false roots, so the check is a must. Irrational inequalities need the domain and the sign of the other side.
• Irrational Numbers: √2, √3, the Square Root Spiral and Real Numbers – An irrational number cannot be written as p/q. Its decimal never ends and never repeats, like √2 = 1.41421356… or π. We can still mark √2 exactly on the number line: it is the diagonal of a square of side 1. We prove √2 and √3 are irrational by assuming they are fractions in lowest terms and reaching a contradiction. The square root spiral builds √2, √3, √4, √5 … one right triangle at a time. Rational and irrational numbers together form the real numbers; every repeating decimal can be turned back into p/q.
• Kinematics with Calculus: Velocity, Acceleration, Distance – If position is s(t), then velocity is v = ds/dt and acceleration is a = dv/dt. Going back, displacement is the integral of v dt. Distance travelled uses |v|. In the plane, speed = √(x′² + y′²), and the length of a curve is the integral of that speed.
• Law of Cosines (Cosine Rule) – In any triangle, c² = a² + b² − 2ab cos C, where C is the angle between sides a and b. When C = 90°, cos C = 0 and it becomes Pythagoras. Use it to find the third side when you know two sides and the angle between them (SAS), or to find any angle when you know all three sides (SSS): cos C = (a² + b² − c²) ÷ 2ab.
• Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
• Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
• Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
• Limits of Sequences: Where Do the Terms Go? – A sequence converges to a limit L if its terms get as close to L as we like and stay close from some term onward. If it does not settle on one number, it diverges. Limit laws let us add, multiply and divide limits. The geometric sequence rⁿ converges to 0 when |r| < 1, to 1 when r = 1, and diverges otherwise.
• Line of Best Fit – A line of best fit (trend line) is one straight line drawn through the middle of the dots on a scatter plot. About half the dots sit above it and half below, and the gaps are small. Its equation y = mx + c is a linear model: the slope m tells how much y changes for each 1 unit of x, and the intercept c is the value of y when x = 0. We use it to predict values we did not measure.
• Linear Algebra: Vector Spaces and Linear Maps – A vector space is a set of arrows you can add and stretch. A linear combination a·u + b·w builds new vectors; the span is everything you can reach. A basis is the smallest set that reaches everything, and its size is the dimension. A matrix is a linear map: its columns say where the basis arrows land.
• Linear Equations in One Variable – A linear equation in one variable has one unknown (like x) with power 1, for example 2x + 3 = 11. An equation is like a balance: both sides are equal. To solve it, do the same thing to both sides (add, subtract, multiply or divide by the same non-zero number) until x is alone. Always check by putting the answer back in.
• Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.
• Linear Graphs: Straight Lines and y = mx + c – A linear graph is a straight line. Its equation is y = mx + c. m is the gradient (how steep: rise ÷ run). c is the y-intercept (where the line cuts the y-axis). Lines with the same m are parallel. If m₁ × m₂ = −1 the lines are perpendicular. Where two lines cross, both equations are true, so the crossing point solves them together. In real-life graphs the gradient is a rate (like speed) and the area under the graph can be a total (like distance).
• Linear Inequalities in One Variable – An inequality compares two expressions with <, >, ≤ or ≥. A linear inequality in one variable looks like ax + b < c. Its answer is usually a whole range of numbers, not one number. We solve it like an equation: we may add or subtract the same number on both sides, and multiply or divide by the same positive number. If we multiply or divide by a negative number, the sign must flip. We show the answer on a number line: a hollow dot for < or > (end not included) and a filled dot for ≤ or ≥ (end included), with the shaded part showing all solutions. Double inequalities like −1 ≤ x < 3 give a piece of the line. If x must be a natural number or an integer, only the whole numbers in that range count.
• Linear Programming (Class 12): find the best answer with a graph – Linear programming finds the biggest profit or the smallest cost when you must obey some rules. The rules are straight-line inequalities (constraints). Together they cut out a region of allowed points (the feasible region). The goal, Z = ax + by (the objective function), is always best at a corner of that region. So: draw the lines, shade, find the corners, put each corner in Z, pick the largest or smallest. If the region is open (unbounded), check once more that the answer really holds.
• Linear Regression and the Least Squares Line – Linear regression finds the straight line ŷ = a + bx that best follows paired data (x, y). A residual is the gap between a real point and the line: e = y − ŷ. The least squares line makes the sum of squared residuals as small as possible. Its slope is b = Sxy ÷ Sxx and it always passes through the mean point (x̄, ȳ). We use it to predict y from x, but only inside the data range, and a strong link does not prove that x causes y.
• Linearization, Implicit Curves and Vector-Valued Derivatives – Zoom in on a smooth curve and it looks like a straight line: its tangent. The tangent line L(x) = f(a) + f′(a)(x − a) gives quick estimates near x = a. For curves like x² + y² = 25 we differentiate both sides and solve for dy/dx; a zero numerator gives a horizontal tangent, a zero denominator a vertical one. A moving point r(t) = (x(t), y(t)) has velocity r′(t) = (x′(t), y′(t)) and speed |r′(t)|. The nth term test says: if the terms of a series do not go to 0, the series diverges.
• Lines and Angles: Linear Pair, Vertically Opposite and Parallel Lines – An angle is the turn between two rays that start from the same point. Angles on a straight line add up to 180° (linear pair). When two lines cross, the opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to 180°. The reverse is also true, and we can prove such facts by contradiction: assume the opposite and show it leads to something impossible.
• Lines and Planes in Space – Two points fix a line; three points not on one line fix a plane. Two lines in space meet, are parallel or are skew. A line lies in a plane, is parallel to it or cuts it at one point; two planes are parallel or meet in a line. A line is perpendicular to a plane if it is perpendicular to two crossing lines of the plane. With normal vector n = (a, b, c) the plane is ax + by + cz = d, a line is r = a + t·u, and the distance from (x₀, y₀, z₀) to the plane is |ax₀ + by₀ + cz₀ − d| ÷ √(a² + b² + c²).
• Locus: The Path of a Point That Follows a Rule – A locus is the set of ALL points that obey one rule, and no other points. Fixed distance from one point gives a circle. Equal distance from two points gives the perpendicular bisector. Equal distance from two crossing lines gives the angle bisector. Where two loci meet, you find points that obey both rules, like the centre of the circumcircle. Points that see a segment at a fixed angle lie on an arc, called the capable arc.
• Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
• Logic and Proof: How Mathematicians Show Something Is Always True – A proof is a chain of correct reasons that shows a statement is true in every case. Statements are true or false; 'if P then Q' (P ⇒ Q) means Q must hold whenever P does, and its converse Q ⇒ P may not. P ⇔ Q means both directions hold. Main methods: direct proof (from known facts step by step, often with algebra like 2n + 1 for odd numbers); proof by exhaustion (check every possible case); disproof by counterexample (one failing case kills a claim); proof by contradiction (assume the opposite and reach something impossible). Geometric proofs use given facts, definitions and theorems (parallel lines, congruent and similar triangles), with a reason for every line. Analytic proofs use coordinates; synthetic proofs use pure geometry.
• Logical Reasoning – Logical reasoning means getting from facts (premises) to a conclusion in a way we can check. A statement is either true or false. Statements are joined with NOT, AND, OR, IF…THEN and IF AND ONLY IF, and truth tables show when the result is true. In deduction, if the premises are true and the form is valid, the conclusion must be true (modus ponens, modus tollens, syllogisms). Venn diagrams test syllogisms with 'all', 'no' and 'some'. Induction and analogy go from examples to a general idea: the conclusion is only probable. A direct proof goes step by step from what is known; an indirect proof assumes the opposite and reaches a contradiction. Fallacies are tempting but faulty arguments.
• Maclaurin and Taylor Series – A Maclaurin series writes a function as an endless polynomial: f(x) = f(0) + f′(0)x + f″(0)x²/2! + … . Near x = 0 a few terms copy the curve very well. The standard series for eˣ, sin x, cos x, ln(1+x) and (1+x)ⁿ must be known with their ranges of validity. A Taylor series does the same around any point a. Series also make hard limits easy: replace each function by its first terms.
• Marginal Cost and the Idea of Differentiation – Marginal cost is the extra cost of making one more item. When we make the step smaller and smaller, the extra cost per item becomes the slope of the cost curve. That slope is the derivative. For C = q², the derivative is 2q.
• Marginal Product and Differentiation – Marginal product (MP) is the extra output you get from one more unit of input. When output is a smooth formula Q(L), MP is its derivative: MP = dQ/dL, the slope of the output curve. If MP falls as you add more input, we have diminishing returns. The same idea gives marginal cost MC = dC/dq.
• Mathematical Communication: Saying Maths Clearly – Mathematical communication means sharing maths ideas so that another person can follow and check them. The same situation can be shown in words, symbols (a formula), a table or a graph; each one is good for a different job. Clear working writes one step per line with correct symbols and units. A good communicator also judges graphs critically and presents results with a question, method, results and conclusion.
• Mathematical Inquiry: Asking and Answering Your Own Maths Question – A mathematical inquiry is a small research project where you ask your own maths question and answer it with evidence and reasoning. It follows a cycle: choose a clear question, read what others found (literature research), plan a method (experiment, case study or development research), carry it out and record data, analyse it with maths, then reflect and report. AI tools can help with data and checking, but the thinking, honesty and sources must be yours.
• Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
• Mathematical Processes: How Good Problem Solvers Think – Mathematical processes are the habits that turn facts into real maths skill. Problem solving is a loop: understand, plan, do, look back. Reasoning and proving means giving a reason that works every time, not just for one example. Reflecting means checking that an answer makes sense, often with an estimate. Connecting links ideas to each other and to life (½ = 50% = half price). Representing shows one idea in many forms: objects, pictures, numbers, symbols, tables and graphs. Selecting tools and strategies means choosing mental maths, paper, a calculator or a graph to suit the job. Communicating means writing clear steps, units and words so others can follow you.
• Mathematical Reasoning: Puzzles You Can Solve with Rules – Reasoning questions test whether you can spot a rule and use it. Odd one out: find the rule most items share. Syllogism: draw circles for "all", "some" and "no" and accept only what MUST be true. Blood relations: draw a family tree. Coding-decoding: find how letters or numbers shift. Cryptarithms: digits hide behind letters; use place value, carries and odd/even (parity) to find them.
• Mathematics in Fine Art – Artists use three big ideas from maths. The golden ratio φ ≈ 1.618 gives balanced rectangles and a spiral. Perspective makes a flat drawing look deep: parallel lines meet at a vanishing point on the horizon and size shrinks in proportion to distance. A fractal is a shape made of smaller copies of itself, such as a branching tree.
• Mathematics in Music – A sound is a wave. Its frequency f (in hertz) is the pitch, and its period is T = 1 / f. Strings of length ratio 1 : 2 sound an octave apart, 2 : 3 a fifth. A piano octave has 12 equal steps, each a factor of 2^(1/12) = 1.0595. Rhythm splits a bar into fractions, for example 3 + 3 + 2 = 8 beats.
• Mathematics in Sport – Maths helps athletes and organisers. A ball thrown at speed v and angle θ travels R = v² sin 2θ ÷ g (best at 45° without air drag). Statistics such as mean, median, batting average and strike rate summarise performance. Scheduling counts matches: a round-robin of n teams has n(n−1)/2 matches, a knockout has n−1.
• Maths in Real Life – Maths turns the numbers of daily life into decisions. Put prices on the same measure (price per 100 g), read a graph by checking its axis first, ask what a headline percentage really means, and see how jobs from nursing to farming use the same few tools.
• Maths in the Arts: Music, Pictures, Tiles, Poems and Film – Art is full of maths. Musical notes are ratios of string length (half length = octave, 2/3 length = fifth). Painters use perspective and the golden ratio. Tiles fit when their corner angles add up to 360°. Poems repeat sound patterns. A film is 24 pictures per second.
• Maths Puzzles and Games: Tower of Hanoi, Nim and Magic Squares – Puzzles and games grew with mathematics. The Tower of Hanoi needs at least 2^n - 1 moves for n disks, found by a simple doubling idea. In the game of Nim you win by leaving your opponent a multiple of 4. A 3 by 3 magic square has every row, column and diagonal adding to 15. Euler's 1736 bridge puzzle started graph theory. Play first, then find the rule.
• Matrices as Functions: Linear Transformations – A 2×2 matrix is a function that takes a vector in and gives a vector out. Its first column is where ⟨1, 0⟩ lands and its second column is where ⟨0, 1⟩ lands. The determinant ad − bc tells how areas scale; when it is not zero, the inverse matrix undoes the change. Transition matrices use the same rule to model how shares change step by step.
• Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
• Maximum and Minimum of a Quadratic Function – A parabola has its highest or lowest point at the vertex. If a > 0 the vertex is a minimum; if a < 0 it is a maximum. When x is limited to a domain p ≤ x ≤ q, compare the values at the two ends and at the vertex (only if the vertex is inside). The biggest value is the maximum and the smallest is the minimum.
• Mean Value Theorem and Rolle's Theorem – If a function is continuous on [a, b] and differentiable on (a, b), there is at least one point c between a and b where the tangent slope equals the average slope: f'(c) = (f(b) − f(a)) / (b − a). Rolle's theorem is the special case f(a) = f(b), where f'(c) = 0.
• Measurement Conversions: Within and Between Systems – To change a measurement to another unit, multiply or divide by a conversion factor. Inside the metric system each step on the ladder (km, hm, dam, m, dm, cm, mm) is a factor of 10. Between the metric and imperial systems we use fixed factors such as 1 in = 2.54 cm, 1 mi ≈ 1.609 km and 1 lb ≈ 0.454 kg. Estimate first with benchmarks, then convert and check that the answer makes sense.
• Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.
• Mental Maths: Fast Everyday Calculation Skills – Mental maths means doing routine calculations quickly and correctly in your head or with a few lines. Compare two numbers by difference or by ratio. Write a percent change as a factor: +20% is × 1.2, −20% is × 0.8. Successive changes multiply their factors, so +20% then −20% gives × 0.96, not × 1. To undo +p% use the reciprocal factor. Know powers of ten and unit conversions, the special products (a + b)² = a² + 2ab + b² and (a + b)(a − b) = a² − b², how to make one letter the subject of a formula, and that a product is zero only if a factor is zero.
• Modular Arithmetic: Remainders and Congruences – Euclidean division writes any integer a as a = n × q + r with 0 ≤ r < n. The remainder r is "a mod n". Two numbers are congruent modulo n (a ≡ b mod n) when they leave the same remainder, which means n divides a − b. Congruences can be added, subtracted, multiplied and raised to powers, so we can find remainders of huge numbers using small ones. Remainders of powers repeat in cycles.
• Multiplying Polynomials – To multiply polynomials, multiply every term of the first by every term of the second (the distributive law), then collect like terms. Coefficients multiply; powers of the same letter add (x² · x³ = x⁵). A rectangle of area tiles shows every product: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. To divide a polynomial by a monomial, divide each term by it.
• Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).
• Number Bases: Grouping in Tens, Twos and Sixties – A base is the size of the groups we count in. In base 10 we make bundles of ten, so each place is worth 10 times the place to its right: ones, tens, hundreds. In base b the places are worth 1, b, b², b³ … and only digits 0 to b − 1 are used. Computers use base 2 (binary, digits 0 and 1). Time and angles still use base 60, from ancient Babylon: 60 seconds make a minute and 60 minutes make an hour.
• Number Sense: Big Numbers, Tiny Numbers and Square Roots – Number sense means you can feel how big a number is, compare numbers and estimate quickly. Each place is 10 times the one on its right, so very large and very small numbers are written with powers of 10 (scientific notation: a × 10ⁿ with 1 ≤ a < 10). All real numbers — whole numbers, negatives, fractions, decimals and irrationals like √2 — sit in order on one number line. A square root can be estimated by finding the two perfect squares around the number. Fractions, decimals and percents are three ways to write the same amount.
• Numerical Methods: Finding Roots and Areas Step by Step – Some equations and areas cannot be found with a neat formula. Numerical methods get as close as we like with simple repeated steps: check a change of sign, halve the interval (bisection), slide down tangents (Newton-Raphson), repeat x = g(x) (fixed-point iteration), add up trapeziums for an area, and walk along a slope in small steps (Euler).
• Operations with Integers and Fractions – Work out expressions in a fixed order: brackets, powers (exponents), × and ÷ from left to right, then + and − from left to right. To add integers, cancel zero pairs (+1 and −1); to subtract, add the opposite. For × and ÷, same signs give a positive answer, different signs give a negative answer. Fractions need a common denominator to add or subtract; to multiply, multiply tops and bottoms; to divide, keep, change, flip. In a proportion two ratios are equal, so an unknown is found with the unit rate or by scaling.
• Optimisation: Finding the Best Value with Quadratics – Optimisation means finding the best value: the biggest area, the largest profit or the smallest cost. We write the quantity as a quadratic function, then the vertex of its parabola gives the best value. For y = ax² + bx + c the vertex is at x = −b/(2a); if a < 0 it is a maximum, if a > 0 it is a minimum.
• Organising and Analysing Information – After you collect information, you must make sense of it. Organise it: sort it into groups with one clear rule, count it, and put it in a table or chart. Then analyse it: compare groups, find the mode, mean, median and range, and say what pattern you see and what it does not prove.
• Orthographic Views of Solids – An orthographic view shows a solid as seen straight on from one direction, with no perspective. The three main views are the front view (front elevation), the top view (plan) and the side view (side or end elevation). Each view loses one dimension, so we usually need all three to fix the shape. Views are arranged in a standard layout: first-angle projection (plan below the front view; used in India, Europe and ISO standards) or third-angle projection (plan above; used in the USA). Visible edges are thick continuous lines, hidden edges are dashed, and centre lines are chain lines. A net is the flat pattern that folds into a solid, and a cross-section is the shape you see when you cut through it.
• Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
• Parallel Projection, Skew Lines and Sections of Solids – Two lines in space that never meet and are not parallel are skew. Parallel projection (sun shadows) sends lines to lines, parallel lines to parallel lines, keeps the ratio of segments on a line and midpoints, but changes lengths and angles; so a triangle can be drawn as any triangle and a square as any parallelogram. Central projection (a lamp) loses parallelism. A section of a solid is the polygon where a plane cuts it; its sides lie on the faces the plane crosses, and a plane cuts two parallel faces along parallel lines. A tetrahedron has triangle or quadrilateral sections; a parallelepiped up to hexagons.
• Parametric Equations: Curves, Slopes and Arc Length – In parametric form, x and y are each written using a third variable t, called the parameter. Each value of t gives one point, and as t changes the point traces a curve. You can remove t to get a Cartesian equation, find slopes with dy/dx = (dy/dt)/(dx/dt), and find speed and arc length when t is time.
• Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.
• Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
• Piecewise, Absolute Value and Step Functions – A piecewise function uses different rules on different parts of its domain. To evaluate it, first find which interval x is in, then use only that rule. On the graph, a closed dot ● means the end point is included (≤ or ≥) and an open dot ○ means it is not (< or >). The absolute value function y = |x| is piecewise: −x for x < 0 and x for x ≥ 0, giving a V with vertex (0, 0); y = a|x − h| + k moves the vertex to (h, k). Step functions such as ⌊x⌋, ⌈x⌉ or a parking fee are flat pieces that jump at whole numbers.
• Place Value, Ordering and Negative Numbers – In our number system every place is worth 10 times the place on its right. A digit's value = digit × the value of its place. To compare numbers, line up the places and compare from the left. Negative numbers lie left of zero; the further left, the smaller.
• Poisson Distribution – The Poisson distribution counts how many random events happen in a fixed time or space when they come at a steady average rate λ and independently of each other. P(X = k) = e^(−λ) λ^k / k!. Its mean and its variance are both λ. It is also a good shortcut for a binomial with large n and small p, with λ = np.
• Polar Coordinates – Polar coordinates give the position of a point by its distance r from a fixed point (the pole) and the angle θ turned anticlockwise from a fixed ray (the initial line). Convert with x = r cosθ, y = r sinθ, and back with r² = x² + y², tanθ = y/x (check the quadrant). Equations r = f(θ) draw circles, cardioids, limaçons, roses and spirals. Calculus: slope dy/dx = (dy/dθ)/(dx/dθ), area = ½∫r²dθ.
• Polygons and the Sum of Their Angles – A polygon is a closed shape made of straight sides. A diagonal joins two corners that are not neighbours. From one corner, the diagonals cut a polygon with n sides into (n − 2) triangles. Each triangle has 180°, so the angles of a convex polygon add up to (n − 2) × 180°. A regular polygon has equal angles, so each one is (n − 2) × 180° ÷ n.
• Polyhedra: Prisms, Pyramids and the Platonic Solids – A polyhedron is a closed solid whose surface is made only of flat polygons (faces). Faces meet along edges, and edges meet at vertices. Prisms have two equal parallel bases joined by parallelograms; pyramids have one base and triangles meeting at an apex; a frustum is a pyramid with its top cut off by a plane parallel to the base. For every convex polyhedron, Euler's formula holds: V − E + F = 2. There are exactly five regular (Platonic) polyhedra.
• Polynomial Division, Horner's Scheme and Roots – A polynomial is a sum of terms like a·xⁿ. You can add and multiply polynomials, and divide them with a remainder: p = q·d + r, where r has a smaller degree than d. Dividing by (x − a) is quick with Horner's scheme, and the remainder is exactly p(a) (Bézout's theorem). If p(a) = 0, then a is a root and (x − a) is a factor. Roots and coefficients are tied together by the Viète relations.
• Polynomial Functions and Their Graphs – A polynomial function is y = aₙxⁿ + … + a₁x + a₀ with whole-number powers. The degree n and the leading coefficient aₙ fix the end behaviour. Each real zero r gives a factor (x − r); the graph crosses the x-axis at a zero of odd multiplicity and touches (bounces) at a zero of even multiplicity. A degree-n polynomial has at most n real zeros and at most n − 1 turning points, and exactly n zeros when complex ones are counted. The average rate of change between two points is the slope of the secant line.
• Polynomials: Zeroes and Coefficients – A polynomial like ax² + bx + c has a degree (highest power). A zero is a value of x that makes it 0. On a graph, zeroes are the x-coordinates where the curve y = p(x) meets the x-axis. A polynomial of degree n has at most n zeroes. For ax² + bx + c with zeroes α, β: α + β = −b/a and αβ = c/a. A quadratic with given zeroes is k[x² − (α+β)x + αβ].
• Power Functions: y = a·xⁿ – A power function has the form y = a·xⁿ, where a is a number and n is a fixed exponent. Even whole powers (x², x⁴) make U shapes that are symmetric about the y-axis. Odd whole powers (x³, x⁵) make S shapes that are symmetric about the origin. All y = xⁿ with n > 0 pass through (0, 0) and (1, 1). Negative powers (x⁻¹ = 1/x) have asymptotes and are not defined at x = 0. Fractional powers are roots: x^(1/2) = √x, x^(1/3) = ∛x. The root function is the inverse of the matching power.
• Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.
• Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
• Probability: Events, Algebra of Events and Axioms – An event is any subset of the sample space S. From events A and B we build new events: not A (A′), A and B (A ∩ B), A or B (A ∪ B). Events are mutually exclusive if they share no outcome and exhaustive if together they cover S. The axiomatic approach says every P(E) ≥ 0, P(S) = 1, and for mutually exclusive A, B, P(A ∪ B) = P(A) + P(B). From these follow P(A′) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Problem Solving in Maths – A problem is a question where you do not yet know the method. A good solver follows four steps: understand and analyse the problem, make a plan using a strategy (heuristic) such as drawing a picture, trying small cases, making a table, finding a pattern, guessing and checking, or working backwards, carry out the plan, and look back to check and reflect. The handshake problem, with n people and n(n-1)/2 handshakes, shows all the steps.
• Projection of Regular Plane Figures – A plane figure is drawn as two views: the top view on the floor (HP) and the front view on the wall (VP). Flat on HP, the top view is the true shape. When the figure tilts by an angle θ about an edge on HP, its top view depth becomes depth × cos θ and its front view height becomes depth × sin θ; widths do not change.
• Projections: Central, Parallel and Orthogonal – A projection draws a 3D object on a flat surface using straight rays. If all rays start at one point (a lamp, your eye) it is a central projection: images change size with distance and give perspective. If all rays are parallel (sunlight) it is a parallel projection: sizes do not depend on distance. If the parallel rays are at 90° to the screen it is an orthogonal projection: faces parallel to the screen show their true size. Engineers use orthogonal views, artists use perspective.
• Proof by Mathematical Induction – Mathematical induction proves that a statement P(n) is true for every natural number n. Step 1 (base case): show P(1) is true. Step 2 (inductive step): assume P(k) is true for some k, and use it to show P(k + 1) is true. Then, like a line of dominoes, P(1) makes P(2) true, P(2) makes P(3) true, and so on for ever.
• Properties of Functions: Reading a Graph Like a Story – A function gives exactly one output f(x) for each input x. From its graph we read the domain (allowed x), the range (y values reached), the zeros (where f(x) = 0), the intervals where f is positive or negative, where it increases or decreases, its maximum and minimum, and whether it is even (mirror in the y-axis) or odd (half-turn about the origin).
• Properties of Triangles and Their Centres – The angles of every triangle add up to 180°. A triangle has four famous centres. Medians meet at the centroid G, which cuts each median 2 : 1. Perpendicular bisectors meet at the circumcentre O, the centre of the circle through the corners. Angle bisectors meet at the incentre I, the centre of the circle inside that touches all sides. Altitudes meet at the orthocentre H. O, G and H lie on one line, the Euler line.
• Proportional Relationships – Two quantities are directly proportional if y = k·x (the ratio y/x never changes) and inversely proportional if y = k/x (the product x·y never changes). A straight line is proportional only if it passes through the origin. Many real laws are power relations y = c·xⁿ: if x is multiplied by f, y is multiplied by fⁿ.
• Propositional Logic: From Statements to Valid Arguments – Propositional logic studies statements that are either true or false and the words that join them: not (¬), and (∧), or (∨), if…then (→) and if and only if (↔). A truth table lists every possible combination of truth values. An argument is valid when no row makes all premises true and the conclusion false. Valid forms such as modus ponens and modus tollens become inference rules for natural deduction.
• Propositions and Conditions: The Logic Behind Maths – A proposition is a sentence that is either true or false. We join propositions with NOT, AND, OR, IF…THEN and IF AND ONLY IF. 'If p then q' is false only when p is true and q is false. Its contrapositive 'if not q then not p' always has the same truth value. When p ⇒ q, p is sufficient for q and q is necessary for p. A predicate like 'x > 3' becomes a proposition when we fix x or add 'for all' / 'there exists'.
• Proving √2, √3 and √5 are Irrational – A rational number can be written as p/q (q ≠ 0). An irrational number cannot; its decimal never ends and never repeats. To prove √2 is irrational we assume it is p/q in lowest terms, get p² = 2q², show both p and q are even, and reach a contradiction. The key fact: if a prime divides p², it divides p. Rational + or × irrational (non-zero) is irrational.
• Pythagoras Theorem – In any right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides: a² + b² = c².
• Quadratic Equations – A quadratic equation has x² as its highest power: ax² + bx + c = 0 with a ≠ 0. It has at most two roots. Find them by factorising (split the middle term, then set each bracket to zero) or by the formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells the nature of roots: D > 0 two different real roots, D = 0 two equal roots, D < 0 no real roots.
• Quadratic Functions and Their Graphs – A quadratic function is y = ax² + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola. If a > 0 it opens up and has a lowest point; if a < 0 it opens down and has a highest point. That turning point is the vertex, at x = −b/2a. In vertex form y = a(x − h)² + k the vertex is (h, k).
• Quadratic Inequalities – A quadratic inequality asks where ax² + bx + c is above zero (> 0) or below zero (< 0). Find the roots, picture the parabola, and read the answer from the graph. When a > 0 the curve is below zero between the roots and above zero outside them. If there are no real roots (D < 0), the curve is always on one side of the x-axis.
• Quadrilaterals (4-gons) – A quadrilateral (4-gon) has 4 sides and 4 angles that add to 360°. In a parallelogram, opposite sides are parallel and equal, opposite angles are equal, and the diagonals cut each other in half. Each of these facts also works as a test. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. The three medians of a triangle meet at one point that cuts each median in the ratio 2 : 1.
• Radical Functions (Square Root Functions) – A radical function has the variable under a root sign. The simplest is y = √x. Its graph is a curve that starts at (0, 0) and goes right and up. For y = a√(x − p) + q the start point is (p, q), the domain is x ≥ p, and the range is y ≥ q when a is positive (y ≤ q when a is negative).
• Radicals and nth Roots – The nth root of a number a is the number that, multiplied by itself n times, gives a. We write it ⁿ√a, and it is the same as a^(1/n). Radicals can be multiplied, divided and simplified by pulling out perfect powers, and a root in a denominator can be removed by rationalising.
• Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
• Rate of Change: How Fast Does It Change? – Rate of change tells you how much an output changes for each one-unit change in the input. A graph can rise (increasing), fall (decreasing), or turn at a highest or lowest point. To measure change between two inputs a and b, find the change in input, Δx = b − a, and the change in output, Δy = f(b) − f(a). The average rate of change is Δy ÷ Δx, which is the slope of the straight line (secant) through the two points. A table of differences shows if growth is steady (constant differences, linear) or speeding up (growing differences, for example exponential). If the two points come very close, the average rate becomes the slope of the graph at one point. Sequences are lists of values with a rule: a recursive formula builds each term from the one before, a direct formula gives any term at once.
• Ratio and Proportion – A ratio a : b compares two amounts of the same kind. Divide both parts by the same number to simplify it. To share an amount in a ratio, add the parts, find one part, then multiply. Two quantities are in direct proportion when y = k·x (double one, double the other) and in inverse proportion when x·y = k (double one, halve the other).
• Rational Equations – A rational (fractional) equation has the unknown in the denominator of a fraction. First write the banned values that make any denominator zero. Then multiply every term by the lowest common denominator (LCD) to clear the fractions, and solve the equation that is left (often linear or quadratic). Finally check each root: a root that is a banned value is extraneous and is thrown out. The same method solves rate problems such as boats on rivers and people working together.
• Rational Functions: Graphs, Asymptotes and Sketching – A rational function is one polynomial divided by another, like y = (2x + 1)/(x − 1). It is not defined where the bottom is zero. Near those x-values the graph shoots up or down along a vertical asymptote. Far out, it settles near a horizontal (or slant) asymptote. Find domain, asymptotes, intercepts and holes, then sketch.
• Rational Graphs, Conics and Inequalities – To sketch y = (ax + b)/(cx + d), find where the bottom is zero (vertical asymptote), the value of a/c far away (horizontal asymptote) and the axis crossings. For a quadratic over a quadratic, set y = k and use the discriminant to find which values y can take. The conics y² = 4ax, x²/a² + y²/b² = 1 and x²/a² − y²/b² = 1 move and stretch by simple swaps (x → x − p, x → x/s). Rational and polynomial inequalities are solved with critical values and a sign check, never by multiplying by an unknown sign.
• Rational Numbers: Number Line, Density and Decimals – A rational number is any number that can be written as p/q, where p and q are integers and q is not 0. Examples: 3/4, −5/3, 7 (= 7/1), 0.25 (= 1/4). To place p/q on the number line, cut each unit into q equal parts and hop p parts from 0. Between any two rational numbers there is always another one (their average), so there are endlessly many between them. The decimal of a rational number either ends (terminating) or repeats a block for ever (non-terminating repeating).
• Recurrence Relations – A recurrence relation makes each term of a sequence from the term (or terms) before it, for example u(n+1) = u(n) + 3 with u(0) = 2. You always need a rule and a starting value. Rules like u(n+1) = u(n) + d give arithmetic sequences, u(n+1) = r·u(n) give geometric ones, and u(n+1) = a·u(n) + b settles at the fixed point b ÷ (1 − a) when −1 < a < 1.
• Regular Polygons and Their Circles – A regular polygon has all sides equal and all angles equal. Every regular polygon has a centre O, a circumscribed circle through all its vertices (radius R) and an inscribed circle touching every side (radius r, the apothem). For n sides of length a: central angle = 360°/n, interior angle = (n − 2)·180°/n, R = a / (2 sin(180°/n)), r = a / (2 tan(180°/n)), and area = ½ · perimeter · r. Special cases: triangle R = a/√3, r = a/(2√3); square R = a/√2, r = a/2; hexagon R = a, r = a√3/2.
• Relations and Functions (Class 12) – A relation R on a set A is any set of pairs (a, b) taken from A × A. R is reflexive if every element is related to itself, symmetric if (a, b) in R always brings (b, a), and transitive if (a, b) and (b, c) always bring (a, c). A relation with all three is an equivalence relation; it cuts A into separate equivalence classes. A function f: A → B sends every element of A to exactly one element of B. It is one-one (injective) if different inputs give different outputs, onto (surjective) if every element of B is hit, and bijective if it is both.
• Relations and Functions: Cartesian Product, Domain, Range, Graphs – An ordered pair (a, b) has a first and a second place, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, and n(A × B) = n(A) × n(B). A relation from A to B is any subset of A × B; its domain is the set of first elements and its range the set of second elements, while B is the co-domain. A function is a special relation where every element of A has exactly one image. We study constant, identity, polynomial, rational, modulus, signum, exponential, log and greatest integer functions with their graphs, and add, subtract, multiply and divide functions point by point.
• Remainder Theorem and Factor Theorem – When a polynomial p(x) is divided by (x − a), the remainder is p(a). So you can find the remainder without long division: just put x = a. If p(a) = 0, the remainder is 0 and (x − a) is a factor of p(x). This is the factor theorem.
• Right Triangles – A right triangle has one 90° angle. The side across it is the hypotenuse; the other two sides are legs. Its two acute angles add up to 90°. The median to the hypotenuse is half the hypotenuse. If an angle is 30°, the leg across from it is half the hypotenuse. Two right triangles are congruent by HL, LL, LA or HA.
• Roots of Polynomials and Polynomial Identities – If a polynomial's roots are known, its coefficients are fixed, and the other way round. For ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a (Vieta's formulas). These let you find expressions in the roots without solving, and build new equations whose roots are changed (transformed roots) by a substitution. A polynomial identity is an equation true for every value of the variable; we prove it by expanding or factorising one side until it equals the other.
• Sampling Distributions and the Central Limit Theorem – A statistic (like a sample mean x̄ or a sample proportion p̂) changes from sample to sample. If you took every possible sample and plotted the statistic, you would get its sampling distribution. Its centre is the true population value (μ or p), its spread is the standard error (σ/√n for means, √(p(1−p)/n) for proportions), and by the Central Limit Theorem its shape becomes close to normal when n is large enough.
• Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.
• Scatter Plots and Association – A scatter plot shows pairs of numbers as dots on a grid, one dot for each person or thing. The pattern of dots tells us the association: positive (both go up together), negative (one goes up, the other goes down) or none. We also look for clusters (clumps of dots), outliers (dots far from the rest) and whether the pattern is a straight line (linear) or a curve (non-linear).
• Scientific Notation (Standard Form) – Scientific notation writes any number as a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number (an integer). For big numbers the decimal point moves left and n is positive. For small numbers it moves right and n is negative. To multiply, multiply the a's and add the powers; to divide, divide the a's and subtract the powers; then fix a so it is between 1 and 10. Unit prefixes like kilo (10³), mega (10⁶), milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹) are powers of ten with names.
• Sections of Solids – A section (cross-section) is the flat shape you get when a plane cuts a solid. Its number of sides equals the number of faces the plane passes through. A cube can give a triangle, a quadrilateral (square, rectangle), a pentagon or a hexagon. A plane parallel to a pyramid’s base gives a smaller copy of the base. In technical drawing, a section shows only the cut face; a cut (sectional view) also shows what lies behind it.
• Sequences and Progressions – A sequence is a list of numbers in a fixed order. We can describe it with a recursive rule (how to get the next term from the last one) or an explicit rule (a formula for term n). In an arithmetic progression (AP) we add the same number every time. In a geometric progression (GP) we multiply by the same number every time. Fractals and the Tower of Hanoi are fun patterns that hide these rules.
• Sequences and Series: AP, GP, AM and GM – A sequence is a list of numbers in a definite order: a₁, a₂, a₃, … A series is what we get when we add the terms. In an arithmetic progression (AP) we add the same number d each time; aₙ = a + (n − 1)d and Sₙ = n/2[2a + (n − 1)d]. The arithmetic mean (AM) of a and b is (a + b)/2; n AMs between a and b use d = (b − a)/(n + 1). In a geometric progression (GP) we multiply by the same number r each time; aₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If |r| < 1 the terms shrink and the infinite sum is S∞ = a/(1 − r). The geometric mean (GM) of two positive numbers a and b is √(ab), and n GMs between them use r = (b/a)^(1/(n+1)). For positive a and b, AM ≥ GM, with equality only when a = b.
• Sets and Subsets: Elements, Belongs To and Subset – A set is a clear collection of different things. The things inside are its elements. We write 4 ∈ A when 4 is in A and 5 ∉ A when 5 is not in A. If every element of B is also in A, then B is a subset of A, written B ⊂ A. One element outside A is enough to break it. The empty set ∅ and the set A itself are always subsets of A.
• Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.
• Shortest Path by Reflection – To go from A to a straight road and then to B (both on the same side), flip B over the road to get B′. The shortest route is the straight line from A to B′; where it crosses the road is the best stop P. It works because PB = PB′ for every point P on the road, so AP + PB = AP + PB′, and a straight line is the shortest way from A to B′.
• Showing Data: Dot Plots, Bar Graphs, Histograms and Line Graphs – Graphs turn a list of numbers into a picture. A dot plot puts one dot per item above a number line. A bar graph compares separate categories with gaps between bars. A histogram shows grouped numerical data in class intervals with bars that touch. A line graph shows change over time. A graph can mislead if its axis does not start at 0, its scale is uneven or its pictures are the wrong size.
• Sigma Notation (∑) – The sign ∑ is a short way to write "add up these terms". ∑ from k = 1 to n of a_k means a_1 + a_2 + ... + a_n. Constants come out of the sum, sums can be split, and there are ready formulas for the sums of k, k² and k³. Telescoping lets most terms cancel.
• Similar Triangles – Two figures are similar when they have the same shape but maybe a different size. Two triangles are similar if their matching angles are equal and their matching sides are in the same ratio. Basic Proportionality Theorem (BPT): a line parallel to one side of a triangle cuts the other two sides in the same ratio; its converse is also true. You can prove similarity with AA, SSS or SAS.
• Simultaneous Equations – Simultaneous equations are two or more equations that must be true at the same time. Their solution is the set of values that fits every equation. On a graph, each solution is a point where the graphs meet. With one straight line and one curve, put the line into the curve (substitution) to get a quadratic; its discriminant tells you if there are 2, 1 or 0 meeting points. With three linear equations, eliminate one letter at a time.
• Slope Fields, Euler's Method and Logistic Models – A differential equation dy/dx = f(x, y) gives the slope at every point. Drawing tiny lines with those slopes makes a slope field, and solution curves follow the lines. Euler's method walks along the field in straight steps: yₙ₊₁ = yₙ + h·f(xₙ, yₙ). The logistic model dP/dt = kP(1 − P/K) grows fastest at P = K/2 and levels off at the carrying capacity K. A falling object with drag, m dv/dt = mg − bv, approaches the terminal velocity v_t = mg/b.
• Social-Emotional Skills and Mathematical Processes – Doing maths well is not only about numbers. Social-emotional skills help you notice your feelings, calm stress, keep going after mistakes, work well with others and believe you can grow. The mathematical processes are the habits good problem solvers use: understand the problem, make a plan, carry it out and look back; reason and prove; connect ideas; choose tools; represent ideas in words, tables, graphs and models; and communicate clearly.
• Solid Geometry: Points, Lines and Planes in Space – Solid geometry studies figures in three dimensions. Three points not on one line fix a plane. Two lines in space can be parallel, intersecting or skew (not in one plane). A line can lie in a plane, cut it, or be parallel to it; it is perpendicular to a plane if it is perpendicular to two intersecting lines of that plane. Angles in space are found by projecting onto a plane and using right triangles.
• Solids of Revolution – A solid of revolution is the 3D shape you get when a flat shape turns a full 360° around a straight line (the axis). A rectangle makes a cylinder, a right triangle makes a cone, a half-circle makes a sphere and a trapezium makes a frustum. Their volumes are V = πr²h, V = ⅓πr²h, V = ⁴⁄₃πr³ and V = ⅓πh(R² + Rr + r²). In calculus, any curve y = f(x) turned about the x-axis gives V = π∫y² dx.
• Solving Equations and Inequalities Graphically – The graph of an equation is the set of all its solutions. To solve f(x) = g(x), draw y = f(x) and y = g(x): the x-coordinates where they cross are the solutions. f(x) > g(x) is true for the x-values where the graph of f is above the graph of g. A linear inequality in two variables, like y > 2x + 1, is solved by a half-plane: draw the boundary line (dashed for < or >, solid for ≤ or ≥) and shade the side that a test point says is true. A system of inequalities is solved by the overlap of the shaded half-planes.
• Solving Polynomial and Rational Equations – To solve a polynomial equation p(x) = 0, find where the graph of y = p(x) meets the x-axis. For a cubic or quartic: guess one root from the divisors of the constant term, divide it out, and solve what is left. Biquadratics like x⁴ − 5x² + 4 = 0 become quadratics with t = x². Rational equations P(x)/Q(x) = 0 need P(x) = 0 and Q(x) ≠ 0. Inequalities use a sign chart.
• Speed, Distance and Time (with Time and Work) – Speed = distance ÷ time. To change km/h into m/s multiply by 5/18. Average speed = total distance ÷ total time. Two bodies moving towards each other close the gap at the sum of their speeds; in the same direction, at the difference. A train must cover its own length (plus the platform or other train). Downstream speed = boat + stream; upstream = boat − stream. Work and pipes use the same idea: add rates (per hour), subtract leaks.
• Square Roots: Build a Square, Find Its Side – The square root of a number n is the number that, multiplied by itself, gives n. √25 = 5 because 5 × 5 = 25. Picture n tiles arranged in a square: the root is the side. Perfect squares (1, 4, 9, 16, …) have whole-number roots. Other roots are irrational and lie between two whole numbers; we estimate them or simplify them, like √50 = 5√2.
• Statistical Inference – Statistical inference means using a sample to say something about a whole population. A number that describes the population (like the true proportion p or the mean μ) is a parameter. A number worked out from a sample (like p̂ or x̄) is a statistic, and we use it as a point estimate. Different random samples give different answers: this is sampling variability. If we took many samples, their statistics would form the sampling distribution, centred on the true value, with spread called the standard error: SE = √(p(1−p)/n) for a proportion and σ/√n for a mean. Bigger samples give smaller spread. A 95% confidence interval is estimate ± 1.96 × SE; about 95 of every 100 such intervals catch the true value. Simulation helps us check whether a claimed model fits the data. Good inference needs random sampling, and an association in data does not prove cause.
• Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.
• Statistics: Mean, Median and Mode of Grouped Data – When data is put into classes (like marks 20–30), we cannot see each value. We use the middle of each class to find the mean, the running total to find the median, and the tallest bar to find the mode. All three tell us the 'centre' of the data in different ways.
• Straight Lines (Class 11): Slope, Forms of the Equation and Distance – The slope of a line is rise ÷ run = (y₂ − y₁)/(x₂ − x₁) = tan θ. Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1. The angle between two lines is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. A line can be written as y = b or x = a (parallel to an axis), y − y₁ = m(x − x₁) (point-slope), y = mx + c (slope-intercept), the two-point form, or x/a + y/b = 1 (intercept form). The distance of (x₁, y₁) from Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).
• Student Fitness Survey: A Data Project from Start to Finish – A survey project has two parts. First collect data fairly: one question, one test, same rules for everyone, results written with units. Then analyse it: sort, find the mean, median and quartiles, draw a box plot, and write a short report with a clear finding and one honest limit.
• Surface Area and Similarity of Solids: Homothety in Space – A homothety (scaling) with centre O and ratio k moves every point to k times its distance from O. Every length is multiplied by k, every surface area by k², and every volume by k³, while angles and shape stay the same. Surface area formulas: cylinder 2πr(h + r), cone πr(l + r), sphere 4πr². Cutting a cone or pyramid by a plane parallel to its base gives a smaller similar solid.
• Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.
• Surface Areas and Volumes: Combined Solids – Many real objects are two simple solids stuck together, like a cone on a cylinder. The surface area is only the outside skin you can touch, so the hidden joint is left out. The volume is the space inside, so you simply add the volumes of the parts.
• Surveying in Civil Engineering – Surveying is measuring the ground so that roads, bridges and buildings can be planned and set out. Distances are measured with tapes, ranging poles and electronic instruments, and must be corrected to horizontal (H = L cos θ). Angles are measured with a theodolite or a compass. Together, distance and angle fix any point on a plan.
• Symmetry of Regular Polyhedra – A symmetry of a solid is a move that puts it back exactly on its own outline. The regular polyhedra have mirror planes, rotation axes and (all except the tetrahedron) a centre of symmetry. Tetrahedron: 6 planes, axes 4×3-fold and 3×2-fold, no centre. Cube and octahedron: 9 planes, axes 3×4-fold, 4×3-fold, 6×2-fold, with centre. Dodecahedron and icosahedron: 15 planes, axes 6×5-fold, 10×3-fold, 15×2-fold, with centre.
• Symmetry: Mirror Lines, Centres and Bisectors – A figure is symmetric when one half is a perfect copy of the other. In line (axial) symmetry the copy is a mirror image across a line. In point (central) symmetry the copy is the figure turned half a circle (180°) about a centre. The perpendicular bisector of a segment and the bisector of an angle are lines of symmetry, so every point on them is the same distance from two things.
• Systems of Equations and Equivalent Systems – A system is a group of equations that must all be true together. Two systems are equivalent when they have exactly the same solutions. Swapping equations, multiplying one by a number that is not zero, and adding a multiple of one equation to another all keep a system equivalent, so we can make it simpler without losing the answer.
• Tables in Maths: Reading, Building and Spotting Proportion – A table lists pairs of values of two variables. You can read it, compare rows, and build one from a graph, a formula or a short story. If y ÷ x is always the same, the table shows direct proportion. If x × y is always the same, it shows inverse proportion.
• Tallying and Analysing Information – Businesses use statistics to turn many answers into a few clear facts. Tally marks count answers; a frequency table lists the counts; a bar graph shows them as bars. Mean (average), median (middle) and mode (most common) summarise the data, and percentages show shares and change. Comparing with the average helps you spot a problem, ask why, try a fix and check again.
• Tangent to a Circle – A tangent is a line that touches a circle at exactly one point. At that point it makes a 90° angle with the radius, and two tangents drawn from one outside point are always equal in length.
• The Axiomatic Method and Modelling – In the axiomatic method we start from a few statements called axioms, which we accept without proof, and from clear rules of logic. Everything else, the theorems, must be proved from them step by step. A good set of axioms is consistent (no contradiction) and each axiom is independent (not provable from the others). A model is a real or mathematical thing in which all axioms are true. The same method is used for arithmetic (Peano), geometry (Euclid, Hilbert) and probability (Kolmogorov).
• The Cartesian Plane: Coordinates, Distance and Midpoint – Two number lines that cross at right angles turn a flat surface into a map where every point has an address (x, y). The axes meet at the origin O(0, 0) and cut the plane into four quadrants. Distance between two points comes from Pythagoras: square the x-gap and the y-gap, add, take the square root. The midpoint is the average of the x values and the average of the y values. Three points are collinear when the two short distances add up to the long one, and they make a right angle when the squares of the two short sides add up to the square of the long side.
• The Circle: Chords, Tangents and Triangle Circles – A circle is all points at one fixed distance (the radius) from a centre. A chord joins two points on it; the longest chord is the diameter. A line can cut a circle at two points (secant), touch it at one (tangent) or miss it. A tangent is perpendicular to the radius. Every triangle has an incircle inside and a circumcircle around it.
• The Golden Ratio – Cut a length into a big part a and a small part b so that whole ÷ a = a ÷ b. That shared ratio is the golden ratio φ (phi) = (1 + √5) ÷ 2 ≈ 1.618. A rectangle with sides in this ratio is a golden rectangle: cut off a square and the piece left is golden again. Ratios of neighbouring Fibonacci numbers get closer and closer to φ.
• The Indian System of Numeration – In the Indian system, the first comma comes after 3 digits from the right and then after every 2 digits: ones, thousands, lakhs, crores. 1 lakh = 1,00,000 = 100 thousand; 1 crore = 1,00,00,000 = 10 million. The international system puts a comma after every 3 digits: thousands, millions, billions. India also gave the world zero and the decimal place-value idea, with old names for every power of ten.
• The Law of Large Numbers – If you repeat the same random experiment many times, the relative frequency of an event gets closer and closer to its probability p. More precisely, for any small distance ε, the chance that the frequency (or the sample mean) is farther than ε from the true value goes to 0 as the number of trials n grows. Chebyshev's inequality gives a bound: P(|X̄ − μ| ≥ ε) ≤ σ² ÷ (nε²).
• The Maths of Voting – A vote count is a rule, and the rule can change the winner. With the same ranked ballots, plurality (most first choices), runoff (drop the last and recount) and Borda count (points for each rank) can pick three different winners. A Condorcet winner beats every other candidate one-to-one, but it may not exist. No method is perfect for three or more candidates.
• The Simplex Algorithm: Solving Linear Programs with Tableaux – The graphical method works for two variables, but real problems have many. The simplex algorithm solves them with a table (tableau). Add a slack variable to each ≤ constraint, start at the origin, and pivot: choose the most negative number in the objective row, use the ratio test to choose the row, and clear the column. Each pivot moves to a better corner of the feasible region. When the objective row has no negative numbers, the tableau is optimal and you read the answer. To minimise C, maximise −C.
• The Sine Wave y = A sin(ωx + φ) – The graph of y = A sin(ωx + φ) is a smooth wave. A is the amplitude (height), the period is T = 2π/ω, and the wave is shifted sideways by φ/ω (to the left if φ is positive). It models anything that repeats: a Ferris wheel, tides, sound and AC current.
• The Statistical Inquiry Cycle – A statistical inquiry answers a question with data in five steps: Problem (ask a clear question), Plan (decide who to ask, which variable, how), Data (collect and record), Analysis (tables, graphs, averages) and Conclusion (answer the question, state limits, ask new questions). Then the cycle can start again.
• The Unit Circle: Symmetry and Periodicity – The unit circle has radius 1 and centre at the origin. A point at angle θ has coordinates (cos θ, sin θ). Mirror images of this point give the rules for π − θ, π + θ and −θ, and going once round (2π) brings you back, so sine and cosine repeat every 2π and tangent every π.
• Theorem of the Three Perpendiculars – If PO is perpendicular to a plane, OH is perpendicular to a line l in that plane, and H is on l, then PH is also perpendicular to l. This lets us find distances in space. The distance from a point to a plane is the length of the perpendicular PO. The distance from a point to a line is PH, found from the right triangle POH: PH² = PO² + OH². The distance between two parallel planes is the length of any common perpendicular, and it is the same everywhere.
• Theorem of Total Probability and Bayes' Theorem – Split all possibilities into non-overlapping cases (a partition). The theorem of total probability adds up, case by case, the chance of an event A: P(A) = Σ P(Ei)P(A|Ei). Bayes' theorem runs this backwards: once A has happened, it tells us how likely each case is, P(Ei|A) = P(Ei)P(A|Ei) ÷ P(A).
• Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.
• Time Series: Moving Averages, Trend and Seasonal Variation – A time series is a set of data recorded at equal time intervals, such as sales every quarter or rainfall every month. We plot it as a line graph with time on the horizontal axis. Real series go up and down with the seasons (seasonal variation), in longer waves (cyclic variation) and with random noise. To see the long-term direction (the trend) we smooth the data with moving averages: the mean of each group of consecutive values, for example 4 values for quarterly data. A line of best fit through the moving averages is the trend line. Seasonal variation = actual value − trend value; the mean seasonal variation for each season, added to the trend, gives a forecast.
• Transformations of Functions – A transformation changes the graph of a parent function y = f(x) without changing its basic shape. Adding outside the function moves it vertically: f(x) + k moves up k. Changes inside the brackets act horizontally and the opposite way: f(x − h) moves right h. Multiplying outside, a·f(x), stretches vertically by a (and flips in the x-axis if a < 0). Multiplying inside, f(bx), squeezes horizontally by factor 1/b (and flips in the y-axis if b < 0). All together: y = a·f(b(x − h)) + k, and the point (0, 0) of the parent moves to (h, k).
• Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.
• Trapezium: Types and the Midline – A trapezium is a four-sided shape with one pair of parallel sides, called the bases. The other two sides are the legs. In an isosceles trapezium the legs are equal and the angles on each base are equal. In a right trapezium one leg is perpendicular to the bases. The midline joins the midpoints of the legs, is parallel to the bases, and its length is half the sum of the bases: m = (a + b) ÷ 2.
• Triangles: Types, Sides, Medians and Altitudes – A triangle is a closed shape with 3 sides, 3 corners (vertices) and 3 angles. The three angles always add up to 180°. By sides, a triangle is equilateral (3 equal sides), isosceles (2 equal) or scalene (none equal). By angles it is acute (all under 90°), right (one 90°) or obtuse (one over 90°). Any two sides together must be longer than the third side: this is the triangle inequality. A median goes from a corner to the middle of the opposite side, an angle bisector splits the angle in two equal parts, and an altitude drops straight down at 90°. A triangle cannot change shape when pushed, so it is the strongest shape for bridges, roofs and towers.
• Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
• Trigonometric Identities – An identity is an equation that is true for every allowed angle. From Pythagoras on a right triangle with hypotenuse 1: sin²A + cos²A = 1. Dividing by cos²A gives 1 + tan²A = sec²A (A ≠ 90°). Dividing by sin²A gives 1 + cot²A = cosec²A (A ≠ 0°). Use them to find one ratio from another and to prove other statements.
• Trigonometric Ratios (sin, cos, tan) – In a right triangle, pick one sharp (acute) angle θ. The side facing θ is the opposite, the side touching θ (not the longest) is the adjacent, and the longest side is the hypotenuse. sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. cosec, sec and cot are their flips. The ratios depend only on the angle, not on the size of the triangle. Learn the table for 0°, 30°, 45°, 60°, 90°.
• Trigonometric Ratios of Obtuse Angles (0° to 180°) – On a circle of radius 1, a point at angle θ has x = cos θ and y = sin θ. This works for every angle from 0° to 180°. Past 90° the point is on the left, so cos is negative. sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ. tan θ = sin θ ÷ cos θ, and it is not defined at 90°.
• Two Circles: Position and Common Tangents – Two circles with radii r1 and r2 and centre distance d have five positions. Apart: d > r1 + r2, 4 common tangents. Touching outside: d = r1 + r2, 3 tangents. Crossing: |r1 - r2| < d < r1 + r2, 2 tangents and 2 common points. Touching inside: d = |r1 - r2|, 1 tangent. One inside the other: d < |r1 - r2|, no tangent. The touching point lies on the line of centres.
• Two-Way Tables – A two-way table counts data for two categorical variables at once: one in the rows, one in the columns. Row and column totals are marginal totals. Dividing a cell by the grand total gives a joint relative frequency; dividing by its row (or column) total gives a conditional relative frequency. If conditional percentages differ a lot between groups, the variables are associated.
• Valuation of Bonds – A bond is a loan to a government or company. It pays a fixed coupon (coupon rate × face value) every period and returns the face value at maturity. Its fair price today is the present value of all these payments, discounted at the market yield r: P = C × [1 − (1 + r)⁻ⁿ] ÷ r + F ÷ (1 + r)ⁿ. When the market yield is below the coupon rate the bond sells at a premium; equal, at par; above, at a discount. Price and yield always move in opposite directions.
• Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
• Vector-Valued Functions – A vector has a length and a direction and is written in components, like ⟨3, 2⟩. A vector-valued function p(t) = ⟨x(t), y(t)⟩ gives a position vector for every value of t, so its tip traces a path. The velocity vector ⟨x′(t), y′(t)⟩ points along the path, and its length is the speed.
• Vectors: Column Vectors, Adding, Scaling and Proofs – A vector has a size and a direction. On a grid we write it as a column vector (across over up). Add vectors tip to tail by adding their parts; multiply by a number to stretch or reverse them. Its length is found with Pythagoras. In geometry, we write any path as a sum of known vectors to prove lines are parallel or points are midpoints.
• Volume of Solids: Prism, Pyramid and Frustum – Volume is the space a solid fills, counted in unit cubes. A prism or parallelepiped has V = base area × height. A pyramid has V = ⅓ × base area × height. A frustum (a pyramid with its top cut off) has V = h/3 × (B₁ + B₂ + √(B₁B₂)). All of these come from one idea: add up thin slices.
• Volumes of Prism, Cylinder, Pyramid, Cone, Sphere and Spherical Cap – Volume is the space inside a solid, counted in cubic units. Prism and cylinder: base area × height. Pyramid and cone: one third of that. Sphere: (4/3)πr³, which is 2/3 of the cylinder around it. A spherical cap of height k has volume πk²(3r − k)/3. Cavalieri's idea explains all of them: equal slice areas at every height give equal volumes.
• Word Problems: Solve Them with Algebra – A word problem is a story with a number hidden in it. The algebraic method has four steps: name the unknown with a letter, turn the story into an equation, solve the equation, and check the answer against the story. The same four steps work for numbers, ages, speed, work, shapes, and also for stories that need two unknowns or a quadratic equation.
• Yang Hui's (Pascal's) Triangle – Yang Hui's triangle starts with 1 at the top. Every number below is the sum of the two numbers above it, and the edges are all 1. Row n lists the coefficients of (a + b)ⁿ, so row 4 (1 4 6 4 1) gives (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴. Each row is symmetric and adds up to 2ⁿ. The same triangle is called Pascal's triangle in Europe and Meru Prastara in India.