ACT Math Strategic Review: Mastering Number & Quantity

Natural Numbers

Counting numbers starting from 1: {1, 2, 3, …}.

Whole Numbers

Natural numbers including zero: {0, 1, 2, …}.

Integers

Whole numbers and their negatives: {…, -2, -1, 0, 1, 2, …}.

Rational Numbers

Numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.

Irrational Numbers

Numbers that cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-repeating.

Imaginary Unit (i)

Defined as i = √−1 and satisfies i² = −1.

Complex Number

A number in the form a + bi, where a is the real part and b is the imaginary part.

Magnitude of a Vector

The length of a vector, calculated using the Pythagorean theorem: ||v|| = √(a² + b²).

Vector Addition

Adding two vectors by combining their corresponding components.

Scalar Multiplication

Multiplying a vector by a real number, which scales its length.

Matrix

A rectangular array of numbers arranged in rows and columns.

Matrix Dimensions

Written as Rows x Columns, indicating the size of the matrix.

Determinant of a Matrix

For a 2x2 matrix A = [[a, b], [c, d]], the determinant is calculated as det(A) = ad - bc.

Product Rule of Exponents

States that x^a ⋅ x^b = x^{a+b}, applicable to the same base.

Quotient Rule of Exponents

States that x^a / x^b = x^{a-b}, applicable to the same base.

Power Rule of Exponents

States that (x^a)^b = x^{a⋅b}.

Negative Exponent Rule

States that x^{-a} = 1/x^a, valid only when x ≠ 0.

Zero Exponent Rule

States that x^0 = 1, valid only when x ≠ 0.

Rational Exponent

An exponent that represents a root, with the formula x^{m/n} = √[n]{x^m}.

Complex Conjugate

For a complex number a + bi, the conjugate is a - bi.

Simplifying i^n

To find i^n, divide n by 4 and examine the remainder (R) to determine the result.

FOIL Method

A method for multiplying two binomials: First, Outer, Inner, Last.

Matrix Addition

Can only add matrices with identical dimensions by adding corresponding elements.

Matrix Multiplication Condition

The number of columns in the first matrix must equal the number of rows in the second matrix.

Common Mistake: Distributing Exponents

Do not distribute exponents over addition: (a + b)^2 ≠ a^2 + b^2.

Common Mistake: Order of Matrix Multiplication

Matrix multiplication is not commutative; A × B usually ≠ B × A.