Mathematics Basic cheat sheet

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

Absolute value |x| is the distance from x to 0 on the number line, so |x| = x when x ≥ 0 and |x| = −x when x < 0.

Rules of exponents: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n.

Fraction multiplication multiplies numerators and denominators; division by a fraction equals multiplication by its reciprocal: (a/b) ÷ (c/d) = (a/b)(d/c).

Percent change = (new − old) / old × 100; a percent increase of p% multiplies by (1 + p/100) and a decrease by (1 − p/100).

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying shifts the exponent by addition, dividing by subtraction.

A number is rational if it can be written as a ratio of integers; its decimal either terminates or repeats. Irrational numbers such as √2 and π have non-terminating, non-repeating decimals.

To simplify a radical, factor out perfect squares: √(a²b) = a√b for a ≥ 0, and rationalize denominators by multiplying by the conjugate or the radical itself.

Unit rates convert quantities by multiplying by conversion factors written as fractions equal to 1, arranging units so unwanted units cancel.

Slope of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have slope 0 and vertical lines have undefined slope.

Slope-intercept form y = mx + b shows slope m and y-intercept b; point-slope form is y − y₁ = m(x − x₁).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1, i.e. negative reciprocals.

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

The discriminant b² − 4ac determines the number of real solutions: positive gives two, zero gives one, negative gives none.

Factoring: find two numbers whose product is ac and sum is b, then split the middle term and group; check by expanding.

Solving a linear inequality: adding or subtracting preserves the direction, but multiplying or dividing by a negative number reverses the inequality sign.

Systems of two linear equations can be solved by substitution or elimination; parallel distinct lines give no solution, identical lines give infinitely many.

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x, and log_b b = 1.

A function assigns exactly one output to each input; the vertical line test confirms a graph is a function.

Function notation: f(a) means the output when the input is a; to evaluate, substitute a for every x in the rule.

Domain is the set of valid inputs, range is the set of outputs; denominators cannot be zero and even roots require non-negative radicands.

A linear function f(x) = mx + b has constant rate of change m; an exponential function f(x) = a·b^x changes by a constant ratio b.

Exponential growth occurs when b > 1 and decay when 0 < b < 1; the value a is the initial amount at x = 0.

Transformations: f(x) + k shifts the graph up k units, f(x − h) shifts right h units, and −f(x) reflects across the x-axis.

Composition (f ∘ g)(x) = f(g(x)) applies g first, then f; the inverse f⁻¹ reverses the mapping and satisfies f(f⁻¹(x)) = x.

A function is even if f(−x) = f(x) (symmetric about the y-axis) and odd if f(−x) = −f(x) (symmetric about the origin).

Average rate of change of f over [a, b] is (f(b) − f(a)) / (b − a), the slope of the secant line between the two points.

Area formulas: rectangle A = lw, triangle A = ½bh, trapezoid A = ½(b₁ + b₂)h, circle A = πr².

Circumference of a circle is C = 2πr; arc length of a sector with central angle θ in degrees is (θ/360)(2πr).

Pythagorean theorem: in a right triangle, a² + b² = c², where c is the hypotenuse; converse tests for right angles.

Volume: rectangular prism V = lwh, cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³.

Sum of interior angles of an n-gon is (n − 2) × 180°; each interior angle of a regular n-gon measures (n − 2) × 180° / n.

Similar figures have equal corresponding angles and proportional sides; the ratio of areas equals the square of the scale factor.

Distance between points is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Complementary angles sum to 90°, supplementary angles sum to 180°; vertical angles are equal and angles on a straight line sum to 180°.

Mean = sum of values / count; median is the middle value of ordered data; mode is the most frequent value.

Range = maximum − minimum; standard deviation measures typical distance from the mean, so larger values mean more spread.

For a data set, adding a constant shifts mean and median by that constant but leaves standard deviation unchanged.

Probability of an event = favorable outcomes / total equally likely outcomes, always between 0 and 1 inclusive.

Complement rule: P(not A) = 1 − P(A); addition rule: P(A or B) = P(A) + P(B) − P(A and B).

Independent events satisfy P(A and B) = P(A) × P(B); mutually exclusive events have P(A and B) = 0.

Conditional probability: P(A | B) = P(A and B) / P(B) for P(B) > 0.

Expected value = Σ (value × probability); the mean of a symmetric distribution equals its median.

In a normal distribution, roughly 68% of values lie within one standard deviation of the mean and about 95% within two.