ORT Mathematics cheat sheet

Order of operations: evaluate grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Exponent rules: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^0 = 1 for a ≠ 0.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; to multiply, multiply the coefficients and add the exponents.

Percent change = (new − old) / old × 100%; successive percent changes multiply as factors (1 ± r), they never simply add.

Divisibility tests: by 3 or 9 when the digit sum is divisible by 3 or 9; by 4 when the last two digits form a multiple of 4; by 6 when divisible by both 2 and 3.

A number is rational exactly when it can be written p/q with integers p and q, q ≠ 0; √n is irrational unless n is a perfect square.

For a ≥ 0, |x| = a means x = a or x = −a; |x| is the distance from x to 0 on the number line, so |x| ≥ 0 always.

For positive integers a and b, gcd(a, b) × lcm(a, b) = a × b.

Complex numbers use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, and |a + bi| = √(a² + b²).

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Line equations: slope-intercept y = mx + b and point-slope y − y₁ = m(x − x₁); a vertical line has undefined slope and equation x = k.

Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0 with a ≠ 0.

Discriminant b² − 4ac: positive gives two distinct real roots, zero gives one repeated real root, negative gives no real roots.

A parabola y = ax² + bx + c has its vertex at x = −b/(2a); it opens upward when a > 0 and downward when a < 0.

Factoring patterns: a² − b² = (a − b)(a + b); a³ − b³ = (a − b)(a² + ab + b²); a³ + b³ = (a + b)(a² − ab + b²).

Logarithm laws: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^k) = k log a, and change of base log_b x = ln x / ln b.

Exponential models: y = A·b^t with b > 1 for growth and 0 < b < 1 for decay; A is the value at t = 0.

Composition (f ∘ g)(x) = f(g(x)); an inverse f⁻¹ satisfies f(f⁻¹(x)) = x and reverses domain and range.

A linear system has one solution when slopes differ, none when slopes match but intercepts differ, and infinitely many when both match.

Area: triangle ½bh, parallelogram bh, trapezoid ½(b₁ + b₂)h, circle πr².

Circle measures: circumference = 2πr; arc length = (θ/360°) × 2πr; sector area = (θ/360°) × πr² for a central angle θ in degrees.

Volume: prism and cylinder V = Bh (B = base area); pyramid and cone V = (1/3)Bh; sphere V = (4/3)πr³ with surface area 4πr².

Pythagorean theorem: a² + b² = c² in a right triangle with hypotenuse c; common triples are 3-4-5, 5-12-13, 8-15-17.

Special right triangles: 45°-45°-90° sides are in ratio 1 : 1 : √2; 30°-60°-90° sides are in ratio 1 : √3 : 2.

Similar figures with side ratio k have area ratio k² and volume ratio k³; corresponding angles remain equal.

Coordinate geometry: distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

The interior angles of an n-sided polygon sum to (n − 2) × 180°; in a regular n-gon each interior angle is (n − 2)180°/n.

Unit conversion multiplies by ratios equal to 1; converting area squares the linear factor and converting volume cubes it.

Mean = sum of values / count; median = middle value of ordered data (average of the two middle values for even counts); mode = most frequent value.

Range = maximum − minimum; standard deviation measures typical distance from the mean, so it is never negative and is 0 only for constant data.

Adding a constant c to every data value adds c to the mean but leaves the standard deviation unchanged; multiplying every value by k multiplies the standard deviation by |k|.

For equally likely outcomes, P(E) = favourable outcomes / total outcomes, and 0 ≤ P(E) ≤ 1 with P(not E) = 1 − P(E).

General addition rule: P(A or B) = P(A) + P(B) − P(A and B); the last term is zero for mutually exclusive events.

Independent events satisfy P(A and B) = P(A) × P(B); conditional probability is P(A | B) = P(A and B) / P(B) when P(B) > 0.

Counting: permutations nPr = n!/(n − r)! for ordered arrangements; combinations nCr = n!/(r!(n − r)!) for unordered selections.

Normal distribution: roughly 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3; z = (x − μ)/σ.

Correlation coefficient r lies in [−1, 1]; |r| near 1 indicates a strong linear association, and correlation alone does not establish causation.

Box plots: IQR = Q3 − Q1; values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR are flagged as outliers.

Modelling cycle: formulate a model from the real situation, solve it mathematically, interpret the output in context, validate against data, then revise assumptions if validation fails.

A linear model has a constant absolute rate of change; an exponential model has a constant percentage rate of change per equal time step.

Dimensional analysis: units are carried through each operation, and both sides of a valid equation must reduce to identical units.

Direct proportion means y = kx, so y/x is constant; inverse proportion means y = k/x, so the product xy is constant.

A rate equation relates quantities as distance = rate × time; combined work rates add as reciprocals, 1/t_total = 1/t₁ + 1/t₂.

Inductive reasoning generalises from observed patterns and yields conjectures, while deductive reasoning derives conclusions from stated premises and yields proofs.

A single counterexample disproves a universal statement, but confirming examples never prove one.

Extrapolation predicts outside the observed data range and carries far more risk than interpolation between known points.

Rounding error accumulates through intermediate steps, so precision is retained until the final answer and only then rounded to the required place.