ACT ADVANCED MATH

Imaginary Numbers

i^2 = -1, used to solve quadratic equations with negative roots. For example, √-4 = 2i.

Rational vs. Irrational Numbers

Rational numbers can be expressed as a fraction (e.g., \frac{1}{2}, 0.75), while irrational numbers are non-repeating, non-terminating decimals (e.g., \pi, \sqrt{2}).

Arithmetic Series

Arithmetic Series: Sum = \frac{n}{2} \times (first + last). Geometric Series: Each term is multiplied by a constant factor.

Matrix Operations

Master addition, subtraction, and multiplication of 2 \times 2 matrices.

Log Rules

\log(ab) = \log a + \log b, \log(\frac{a}{b}) = \log a - \log b, \log(a^n) = n \cdot \log a

Asymptotes

Vertical Asymptotes: Occur where the denominator of a rational function equals zero. Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator.

Asymptotes — Vertical: denominator = 0; Horizontal: degree of numerator vs. denominator

$$\sin^2{\theta} + \cos^2{\theta} = 1