GCE 'O' and 'N' EM Notes

By u/One_Wishbone_4439 Page 1 of 53 GCE 'O' and 'N' EM Notes Number and algebra Positive numbers: greater than 0. eg. 1, 2, 3, 4, 5 Negative numbers: less than 0. eg. −1, −2, −3, −4, −5 Prime numbers: have exactly two factors: 1 and itself. eg. 2, 3, 5, 7, 11 Composite numbers: have more than two factors. Eg. 4, 6, 8, 9, 12 ➢ 1 and 0 are NOT prime numbers. ➢ 2 is the ONLY even prime number. Rational numbers: Can be represented in the form of P Q where Q ≠ 0. Irrational numbers: Cannot be represented in the form of P Q . HCF – Highest Common Factor ➢ Look at the lowest power. LCM – Lowest Common Multiple ➢ Look at the highest power. Prime factorisation ➢ Only can divide by prime numbers. Significant figures (5 rules) ➢ All non-zero digits are significant. (eg. 211.8 has 4sf) ➢ All zeros that are found between nonzero digits are significant. (eg. 20007 has 5sf) ➢ Leading zeros (to the left of the first nonzero digit) are not significant. (eg. 0.0085 has 2sf) ➢ Trailing zeros for a whole number that ends with a decimal point are significant. (eg. 320 can be 2sf or 3sf) ➢ Trailing zeros to the right of the decimal place are significant. (eg. 12.000 has 5sf)

By u/One_Wishbone_4439 Page 2 of 53 Standard form: expressed as A × 10n where A must be 1 ≤ A < 10 and n is an integer. Common prefixes Power of 10 English word SI prefix Symbol 1012 trillion tera T 109 billion giga G 106 million mega M 103 thousand kilo K 10-3 thousandth milli m 10-6 millionth micro µ 10-9 billionth nano n 10-12 trillionth pico P Law of indices ➢ Zero indices: a0 = 1, a ≠ 0 and a < 0 ➢ Negative indices: a-n = 1 an, a ≠ 0 and a < 0 ➢ Rational indices: a m n = √amn = (√a n ) m , a ≤ 0 ➢ Law 1: am × an = am+n, if a > 0 ➢ Law 2: am ÷ an = am-n, if a > 0 ➢ Law 3: (am)n = amn, if a > 0 ➢ Law 4: an × bn = (a × b)n, if a, b > 0 ➢ Law 5: an ÷ bn = (a b) n , if a, b > 0 Inequality Signs Definitions How is it represented on a number line? < more than > less than ≤ more than or equal to ≥ less than or equal to Basic Four Operations of Algebraic Fractions ➢ Addition: a b + c d = ad + bc bd ➢ Subtraction: a b − c d = ad − bc bd ➢ Multiplication: a b × c d = ac bd ➢ Division: a b ÷ c d = a b × d c = ad bc

By u/One_Wishbone_4439 Page 3 of 53 ➢ Conditions* 1. Addition: If A < B, then A + c < B + c 2. Subtraction: If A < B, then A − c < B − c 3. Multiplication: If A < B, then cA < cB If A < B, then − cA > − cB 4. Division: If A < B, then A c < B c If A < B, then A − c > B − c *Note: For multiplication and division of negative numbers, the inequality sign must flip (shown in red). Algebraic expression ➢ The square of sum: (a + b)2 = (a2 + 2ab + b2) ➢ The square of difference: (a - b)2 = (a2 - 2ab + b2) ➢ The difference of two squares: a2 - b2 = (a + b)(a - b) Factorisation methods: ➢ Divide by HCF ➢ Algebraic expression ➢ Grouping ➢ Completing the square in the for