Russia 8 класс Algebra (advanced)
Chapters: 4
1. Numbers and calculations
Square roots and irrational numbers · Number sets · Remainders · Standard form
- 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.
- 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.
2. Algebraic expressions
Algebraic fractions · Expressions with roots · Integer exponents
- 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.
- 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.
3. Equations and inequalities
Quadratic equations · Fractional-rational equations · Inequalities
- 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.
- 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.
- 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.
4. Functions
Functions and graphs
- 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).