ACT Math — Number & Quantity: Deep Understanding Notes

Real numbers

All numbers that can be placed on a number line, including both rational and irrational numbers.

Natural numbers

Counting numbers such as 1, 2, 3; some definitions include 0, but ACT problems usually avoid that ambiguity.

Whole numbers

The natural numbers together with 0: 0, 1, 2, 3, and so on.

Integers

Positive and negative whole numbers, including 0; they are closed under addition, subtraction, and multiplication, but not always under division.

Rational numbers

Numbers that can be written as a ratio of integers a/b with b not equal to 0; their decimals terminate or repeat.

Irrational numbers

Numbers that cannot be written as a ratio of integers; their decimals do not terminate or repeat.

Absolute value

The nonnegative distance of a number from 0 on the number line; for example, |x| = a with a > 0 has solutions x = a and x = -a.

Complex number

A number in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Imaginary unit

The number i defined by i^2 = -1, which allows square roots of negative numbers to be written in complex form.

Complex conjugate

For a complex number a + bi, the conjugate is a - bi; multiplying conjugates removes the imaginary part and gives a^2 + b^2.

Zero exponent

For any nonzero base a, a^0 = 1.

Negative exponent

For any nonzero base a, a^(-n) means 1/a^n; it indicates a reciprocal, not a negative value.

Product of powers

When multiplying like bases, add the exponents: a^m times a^n equals a^(m+n).

Power of a power

When raising a power to another power, multiply the exponents: (a^m)^n = a^(mn).

Rational exponent

A fractional exponent that represents a root: a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a^m.

Scientific notation

A way to write numbers as a times 10^n, where 1 is less than or equal to a and a is less than 10, and n is an integer.

Vector

A quantity with both magnitude and direction, often written in 2D as

Displacement vector

The vector from P(x1, y1) to Q(x2, y2), found by subtracting initial coordinates from terminal coordinates: .

Vector addition

Combining vectors component-wise:

Magnitude of a vector

The length of a vector

Unit vector

A vector with magnitude 1; for a nonzero vector v, a unit vector in the same direction is v divided by its magnitude.

Matrix

A rectangular array of numbers arranged in rows and columns.

Matrix dimensions

The size of a matrix written as rows by columns; for example, a 2 x 3 matrix has 2 rows and 3 columns.

Matrix multiplication

A row-by-column operation defined when the number of columns in the first matrix equals the number of rows in the second; in general, AB does not equal BA.

Identity matrix

A square matrix that acts like 1 in multiplication; for example, the 2 x 2 identity matrix leaves a compatible matrix unchanged when multiplied.