1. The Number System
1.1. Natural Numbers
1.1.1. N = 1,2,3,…
1.2. Whole Numbers
1.2.1. N° = 0,1,2,3,…
1.3. Integers
1.3.1. Z = …,-3,-2,-1,0,1,2,3,…
1.4. Rational Numbers
1.4.1. Integers
1.4.2. Fractions
1.4.3. Terminating Decimals
1.4.4. Recurring Decimals
1.4.4.1. Let 𝑥 = 0,363636… ∴ 𝑥 = 0,363636… (1) ∴ 100𝑥 = 36,363636… (2) ∴ 100𝑥 - 𝑥 = 36,3636… - 0,363636 ∴ 99𝑥 = 36 ∴ 𝑥 = 36/99
1.5. Irrational Numbers
1.5.1. Non-Terminating Decimals
1.5.2. Non-Recurring Decimals
1.5.3. Surds
1.5.3.1. Locating Surds
1.5.3.1.1. √11 = √3 < √11 < √4
1.5.3.1.2. -√11 = -√4 < -√11 < -√ 3
1.5.4. Transcendental Numbers
1.6. Non-Real Numbers
1.6.1. Even Roots of (-) Numbers
1.7. Undefined Numbers
1.7.1. Division by 0
2. Representations
2.1. Set Builder Notaion
2.1.1. { 𝑥 : 0 ≤ 𝑥 < 3 ; 𝑥 ∈ R }
2.2. Interval Notation
2.2.1. 𝑥 ∈ ( -1 ; 3 ]
2.2.1.1. Included - [ ]
2.2.1.2. Excluded ( )
2.3. Number Line
2.3.1. Real Numbers
2.3.1.1. Line Connecting The Dots
2.3.2. Integers
2.3.2.1. Dots on Each Number
2.3.3. Excluded ○
2.3.4. Included 🌑
3. Simplification
3.1. Distributive Law
3.1.1. a(𝑥+𝑦) = a𝑥+a𝑦
3.2. Two Binomials
3.2.1. (a+b)(𝑥+𝑦) = a𝑥 + a𝑦 + b𝑥 +b𝑦
3.2.2. First Outers Inners Lasts
3.3. Three Binomials
3.3.1. (a+b)(𝑥+𝑦+ 𝑧)
3.4. Squaring Binomials
3.4.1. (𝑥+𝑦)² = 𝑥² + 2𝑥𝑦 + 𝑦²
3.4.2. (𝑥-𝑦)² = 𝑥² - 2𝑥𝑦 + 𝑦²
3.5. Difference of Two Squares
3.5.1. (𝑥+𝑦)(𝑥-𝑦) = 𝑥² - 𝑦²
3.6. Expand and Simplify
3.6.1. (a+2b)² (a-2b)² = [(a+2b)(a-2b)]²
3.7. Sum or Difference of Two Cubes
3.7.1. (𝑥 + 𝑦)(𝑥² - 𝑥𝑦 + 𝑦²) = 𝑥³ + 𝑦³
3.7.2. (𝑥 - 𝑦)(𝑥² + 𝑥𝑦 + 𝑦²) = 𝑥³ - 𝑦³
4. Exponents
4.1. Laws of Exponents
4.1.1. a² x a³ = a²⁺³
4.1.2. a³ — = a³⁻² a²
4.1.3. (a³)² = a³∘²
4.1.4. (ab)² = a²b²
4.1.5. a⁰ = 1
4.1.6. a⁻² 1 a ⁻² b ² — = — or — = — 1 a² b a
5. Algebraic Fractions
5.1. Simplification
5.1.1. ap p — = — aq q
5.2. Multiplication
5.2.1. a b ab — x — = — x y xy
5.3. Division
5.3.1. a x a y ay — ÷ — = — x — = — b y b x bx
5.4. Addiction and Subtraction
5.4.1. a b a ± b — ± — = ——— x x x
6. Factorisation
6.1. Common Factors
6.1.1. a𝑥 + a𝑦 = a(𝑥 + 𝑦)
6.2. Difference of Two Squares
6.2.1. 𝑥² - 𝑦² = (𝑥+𝑦)(𝑥-𝑦)
6.3. Sum or Difference of Two Cubes
6.3.1. 𝑥³ + 𝑦³ = (𝑥 + 𝑦)(𝑥² - 𝑥𝑦 + 𝑦²)
6.3.2. 𝑥³ - 𝑦³ = (𝑥 - 𝑦)(𝑥² + 𝑥𝑦 + 𝑦²)
6.4. Quadratic Trinomials
6.4.1. 𝑥² - 9𝑥 + 8= (𝑥 - 1)(𝑥 -8)
6.4.2. 𝑥2 + 3𝑥 - 4 = (𝑥 - 1)(𝑥 + 4)
6.4.3. 𝑥² - 𝑥 - 12 = (𝑥 + 3)(𝑥 - 4)
6.4.4. 𝑥² + 5𝑥 + 6= (𝑥 + 2)(𝑥 + 3)
6.5. Grouping
6.5.1. 9 + 8ab - 4a² - 4b² = 9 - 4a² + 8ab - 4b² = 9 - 4 (a² - 2ab + b²) = 9 - 4 (a - b)² = [3 - 2 (a - b)] [3 + 2 (a - b)] = (3 - 2a + 2b)(3 + 2a - 2b) = (3 - 2a + 2b)(3 + 2a - 2b)