Romania Clasa a IX-a Mathematics (natural sciences)

Chapters: 7

1. Algebra: Real numbers

Algebraic structure of real numbers · Order structure of real numbers

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).

Propositions, predicates and reasoning methods

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'.

Polynomials

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].

Progressions and interest

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.

Lines and circles in coordinates

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).

Families of functions

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.

Unit circle and basic formulas · Trigonometry in triangles

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.
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.