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($x_1$, $y_1$) to Q($x_2$, $y_2$), found by subtracting initial coordinates from terminal coordinates: $<x_2 - x_1, y_2 - y_1>$.

Vector addition

Combining vectors component-wise:

Magnitude of a vector

The length of a vector $<a, b>$, found with the Pythagorean theorem: $\sqrt{a^2 + b^2}$.

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.