Mathematics Knowledge
Number Systems and Properties
Types of numbers: - Natural numbers: 1, 2, 3, ... (counting numbers) - Whole numbers: 0, 1, 2, 3, ... - Integers: ..., −2, −1, 0, 1, 2, ... (positive and negative whole numbers) - Rational numbers: expressible as p/q (includes integers, fractions, terminating and repeating decimals) - Irrational numbers: cannot be expressed as a fraction (π, √2, √3) - Real numbers: all rational and irrational numbers combined
Properties: - Commutative: a + b = b + a; a × b = b × a - Associative: (a + b) + c = a + (b + c) - Distributive: a(b + c) = ab + ac
Order of operations (PEMDAS): Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right)
Absolute value |x|: Distance from zero. Always ≥ 0. |−7| = 7.
Factors, Multiples, and Primes
Prime number: Exactly two factors — 1 and itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23... (2 is the only even prime.)
Composite number: More than two factors.
Prime factorization: Express as a product of primes. 60 = 2² × 3 × 5.
GCF (Greatest Common Factor): Largest factor shared by two numbers. Use prime factorization: multiply all shared primes at their lowest powers. GCF(12, 18): 12 = 2²×3, 18 = 2×3² → GCF = 2×3 = 6.
LCM (Least Common Multiple): Smallest number divisible by both. Use prime factorization: take each prime at its highest power. LCM(4, 6): 4 = 2², 6 = 2×3 → LCM = 2²×3 = 12.
Divisibility rules: - By 2: last digit even · By 3: digit sum divisible by 3 · By 4: last two digits divisible by 4 · By 5: ends in 0 or 5 · By 9: digit sum divisible by 9 · By 10: ends in 0
Exponents and Roots
Rules: | Rule | Formula | |------|---------| | Product | xᵃ × xᵇ = xᵃ⁺ᵇ | | Quotient | xᵃ / xᵇ = xᵃ⁻ᵇ | | Power | (xᵃ)ᵇ = xᵃᵇ | | Zero | x⁰ = 1 (x ≠ 0) | | Negative | x⁻ⁿ = 1/xⁿ | | Fraction | x^(1/n) = ⁿ√x |
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512
Simplifying radicals: √72 = √(36 × 2) = 6√2. Factor out the largest perfect square.
Rationalizing the denominator: Eliminate radicals from denominators by multiplying numerator and denominator by the radical: 1/√3 × √3/√3 = √3/3.
Polynomials
- Monomial: one term (5x², −3y)
- Binomial: two terms (2x + 3)
- Trinomial: three terms (ax² + bx + c)
Adding/subtracting: Combine like terms (same variable and exponent).
FOIL for two binomials: (a + b)(c + d) = ac + ad + bc + bd
Special products — memorize these: - (a + b)² = a² + 2ab + b² - (a − b)² = a² − 2ab + b² - (a + b)(a − b) = a² − b² ← difference of squares - (a + b)(a² − ab + b²) = a³ + b³ ← sum of cubes - (a − b)(a² + ab + b²) = a³ − b³ ← difference of cubes
Factoring
Always check for GCF first. 6x² + 9x = 3x(2x + 3).
Difference of squares: a² − b² = (a + b)(a − b). Example: 4x² − 25 = (2x + 5)(2x − 5).
Factoring trinomials (leading coefficient = 1): x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b. Example: x² − 5x + 6 — need two numbers that multiply to 6 and add to −5: (−2)(−3) → (x − 2)(x − 3).
Factoring trinomials (leading coefficient ≠ 1): 2x² + 7x + 3 — multiply a × c = 6; find factors of 6 that add to 7: (6 + 1). Rewrite: 2x² + 6x + x + 3. Group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²). Example: x³ + 8 = (x + 2)(x² − 2x + 4).
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²).
Simplifying rational expressions: Factor numerator and denominator fully, cancel common factors. (x² − x − 6)/(x² − 4) = (x − 3)(x + 2) / (x − 2)(x + 2) = (x − 3)/(x − 2), where x ≠ ±2.
Solving Equations and Inequalities
Linear equations: Isolate the variable using inverse operations. Whatever you do to one side, do to the other.
Inequalities: Same rules as equations, except multiplying or dividing both sides by a negative number flips the inequality sign. −2x < 6 → x > −3.
Systems of equations: - Substitution: Solve one equation for one variable, substitute into the other. - Elimination: Add/subtract equations to cancel one variable.
Quadratic equations (ax² + bx + c = 0): - Factor and set each factor = 0. - Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a) - Discriminant (b² − 4ac): > 0 → two real solutions; = 0 → one; < 0 → no real solutions.
Geometry — Angles and Lines
Angle types: Acute (< 90°) · Right (= 90°) · Obtuse (90°–180°) · Straight (= 180°)
Complementary: sum to 90°. Supplementary: sum to 180°.
Vertical angles (formed by two intersecting lines) are always equal.
Angles in any triangle: sum to 180°. Angles in any quadrilateral: sum to 360°.
Parallel lines cut by a transversal: - Corresponding angles: equal - Alternate interior angles: equal - Co-interior (same-side) angles: supplementary (sum to 180°)
Geometry — Triangles
By sides: Equilateral (all equal, all 60°) · Isosceles (two equal sides, base angles equal) · Scalene (all different)
By angles: Acute (all < 90°) · Right (one = 90°) · Obtuse (one > 90°)
Pythagorean Theorem (right triangles only): a² + b² = c² where c is the hypotenuse.
Common right triangle ratios: 3-4-5 · 5-12-13 · 8-15-17 (and their multiples: 6-8-10, etc.)
Area: A = ½ × base × height (height must be perpendicular to the base)
Triangle inequality: Sum of any two sides must be greater than the third side.
Similar triangles: Same shape; sides are proportional. All three angles match.
Geometry — Quadrilaterals and Circles
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × w | 2(l + w) |
| Square | s² | 4s |
| Parallelogram | base × height | 2(l + w) |
| Trapezoid | ½(b₁ + b₂) × h | sum of sides |
Circle: - Circumference: C = 2πr = πd - Area: A = πr² - π ≈ 3.14159 (often use 3.14 or 22/7) - Diameter = 2 × radius
Geometry — 3D Shapes
| Shape | Volume | Surface Area |
|---|---|---|
| Rectangular prism | l × w × h | 2(lw + lh + wh) |
| Cube | s³ | 6s² |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | ⅓πr²h | πr² + πrl (l = slant height) |
| Sphere | (4/3)πr³ | 4πr² |
| Pyramid | ⅓ × base area × h | base + lateral faces |
Coordinate Geometry
Cartesian plane: x-axis (horizontal), y-axis (vertical), origin (0, 0).
Slope: m = (y₂ − y₁) / (x₂ − x₁). Positive = rises left to right. Negative = falls. Zero = horizontal. Undefined = vertical.
Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
Parallel lines: same slope. Perpendicular lines: slopes are negative reciprocals (m₁ × m₂ = −1).
Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Statistics and Probability
Mean: Sum ÷ count. Median: Middle value (ordered list). Mode: Most frequent. Range: Max − min.
Finding a missing value: Total = mean × count. Subtract known values.
Probability: P(event) = favorable outcomes / total possible outcomes. Range: 0 to 1.
Factorial: n! = n × (n−1) × ... × 1. Example: 5! = 120.