ACT Math Extension

ACT-specific concepts that extend beyond the core SAT curriculum, including complex numbers, vectors, matrices, logarithms, sequences and series, permutations and combinations, probability, advanced trigonometry, and conic sections. Use this deck to prepare for the broader mathematical scope of the ACT.

Powers of i

The powers of i repeat every four: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Reduce any exponent modulo 4.

Operations with complex numbers

Add or subtract real and imaginary parts separately. Multiply by distribution using i^2 = -1. To divide, multiply numerator and denominator by the denominator's conjugate.

Vectors

A vector from A(x1,y1) to B(x2,y2) is . Its magnitude is sqrt[(x2 - x1)^2 + (y2 - y1)^2]; add vectors component by component.

Matrices

An m x n matrix has m rows and n columns. Add matrices only when dimensions match. AB exists only when columns of A equal rows of B. A system can be written AX = B; if A is invertible, X = A^(-1)B.

Logarithms

log_b(x) = y means b^y = x, with b > 0, b not equal to 1, and x > 0. Logarithms undo exponential functions.

Logarithm properties

log_b(MN) = log_b(M) + log_b(N); log_b(M/N) = log_b(M) - log_b(N); log_b(M^p) = p log_b(M).

Arithmetic sequence

An arithmetic sequence has a constant difference d. The nth term is a_n = a_1 + (n - 1)d.

Arithmetic series

The sum of the first n terms is S_n = n/2(a_1 + a_n), equivalently S_n = n/2[2a_1 + (n - 1)d].

Geometric sequence

A geometric sequence has a constant nonzero ratio r. The nth term is a_n = a_1 r^(n - 1).

Factorial

For a positive integer n, n! = n(n - 1)(n - 2)…1, and 0! = 1.

Permutations and combinations

Use P(n,r) = n!/(n - r)! when order matters. Use C(n,r) = n!/[r!(n - r)!] when order does not matter.

Probability rules

Complement: P(not A) = 1 - P(A). Independent events: P(A and B) = P(A)P(B). Overlapping events: P(A or B) = P(A) + P(B) - P(A and B).

Law of Sines

For any triangle, a/sin(A) = b/sin(B) = c/sin(C). Use it when you know an angle-side opposite pair and one other side or angle.

Law of Cosines

For any triangle, c^2 = a^2 + b^2 - 2ab cos(C). Use it for SAS or SSS information; it becomes the Pythagorean theorem when C = 90°.

Conic sections

Recognize a circle, parabola, ellipse, or hyperbola from its equation. For axis-aligned forms, completed squares reveal the center or vertex and the denominators control the axes.