Unit 1 – The Complex Number System Review

Vocabulary and key concepts covering radicals, rational exponents, complex numbers, imaginary units, real number sub-sets, and exponent rules from Unit 1.

Complex Number System Diagram

A visual classification hierarchy showing that Complex Numbers contain Real Numbers and Imaginary Numbers, where Real Numbers consist of Rational Numbers and Irrational Numbers, and Rational Numbers encompass Integers, Whole Numbers, and Natural Numbers.

Complex Numbers

The set of numbers of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Imaginary Unit ($i$)

The unit defined such that $i = \frac{1}{-1}$ is not valid, but $i = \text{sqrt}(-1)$ or i = \root 2 \text{of} -1, satisfying $i^2 = -1$.

Real Numbers

The set of all numbers that can be represented on a continuous number line, comprising both rational and irrational numbers.

Rational Numbers

Numbers that can be expressed as a ratio or fraction $\frac{a}{b}$ of two integers, where $b \neq 0$.

Irrational Numbers

Real numbers that cannot be expressed as a quotient of two integers; their decimal expansions are non-repeating and non-terminating.

Integers

The set of numbers containing positive whole numbers, zero, and negative whole numbers: $\text{\{..., -3, -2, -1, 0, 1, 2, 3, ...\}}$.

Whole Numbers

The set of non-negative integers including zero: $\text{\{0, 1, 2, 3, ...\}}$.

Natural Numbers

The set of positive counting numbers starting from one: $\text{\{1, 2, 3, ...\}}$.

Simplest Radical Form

An expression with radicals where the radicand has no perfect power factors corresponding to the index, no fractions exist inside a radical, and no radicals appear in the denominator.

Rational Exponent Rule

The equivalence relationship x^{\frac{m}{n}} = \root n \text{of} x^m = (\root n \text{of} x)^m between fractional exponents and radical expressions.

Powers of $i$ Cycle

The repeating four-step pattern for integer powers of the imaginary unit $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$.

Product Rule for Exponents

The law stating that when multiplying exponential expressions with the same base, you add the exponents: $x^a \times x^b = x^{a+b}$.

Quotient Rule for Exponents

The law stating that when dividing exponential expressions with the same base, you subtract the denominator exponent from the numerator exponent: $\frac{x^a}{x^b} = x^{a-b}$.

Power of a Power Rule

The law stating that raising an exponential term to another power requires multiplying the exponents together: $(x^a)^b = x^{a \times b}$.

Negative Exponent Rule

The law stating that a term raised to a negative exponent is equal to its reciprocal with a positive exponent: $x^{-a} = \frac{1}{x^a}$.

Radical Conjugate

A binomial formed by changing the sign between two terms involving radicals, used to rationalize denominators (e.g., the conjugate of a + b\root 2 \text{of} c is a - b\root 2 \text{of} c).