Complex Numbers
A complex number is an expression of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$. The real number $a$ is the real part and the real number $b$ is the imaginary part.
Quick Reference
| Concept | Notation | Example |
|---|---|---|
| Complex number | $z = a + bi$ | $z = 3 - 5i$ |
| Real part | $\operatorname{Re}(z) = a$ | $\operatorname{Re}(3 - 5i) = 3$ |
| Imaginary part | $\operatorname{Im}(z) = b$ | $\operatorname{Im}(3 - 5i) = -5$ |
| Purely imaginary | $a = 0$, $b \neq 0$ | $7i$ |
| Real number as complex | $a + 0i$ | $5 = 5 + 0i$ |
| Set of all complex numbers | $\mathbb{C}$ | $\mathbb{R} \subset \mathbb{C}$ |
Definition
Definition. A complex number is an expression of the form $a + bi$ (or equivalently $a + ib$), where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$. We write:
$\operatorname{Re}(a + bi) = a, \qquad \operatorname{Im}(a + bi) = b.$For example,
$ \operatorname{Re}(2 - 3i) = 2, \qquad \operatorname{Im}(2 - 3i) = -3. $Note that the imaginary part is the real number $b$, not $bi$. In the number $2 - 3i$, the imaginary part is $-3$, not $-3i$.
The set of all complex numbers is denoted $\mathbb{C}$:
$ \mathbb{C} = \{a + bi \mid a, b \in \mathbb{R} \text{ and } i^2 = -1\}. $
Equality of Complex Numbers
Two complex numbers $a + bi$ and $c + di$ are equal if and only if their real parts are equal and their imaginary parts are equal:
$ a + bi = c + di \quad \Longleftrightarrow \quad a = c \quad \text{and} \quad b = d. $This means that a single complex equation is really two real equations in disguise. This property is extremely useful when solving equations involving complex numbers.
Example. Find $x$ and $y$ such that $2x + (y - 3)i = 6 + 5i$.
Solution. Matching real and imaginary parts:
- Real parts: $2x = 6$, so $x = 3$.
- Imaginary parts: $y - 3 = 5$, so $y = 8$.
Special Cases
Real numbers as complex numbers. The complex number $a + 0i$ is identified with the real number $a$. This means the real numbers are a subset of the complex numbers:
$ \mathbb{R} \subset \mathbb{C}. $Every real number is a complex number whose imaginary part is zero.
Purely imaginary numbers. When the real part is zero and $b \neq 0$, the number $0 + bi = bi$ is called purely imaginary. For example, $5i$ and $-\sqrt{2}\,i$ are purely imaginary.
Zero. The complex number $0 = 0 + 0i$ is the only number that is simultaneously real and purely imaginary.
Shorthand notation. Complex numbers are often denoted by a single letter, typically $z$ or $w$:
$ z = a + bi. $
Powers of the Imaginary Unit
Since $i^2 = -1$, higher powers of $i$ follow a repeating cycle of period 4:
$ \begin{aligned} i^1 &= i \\ i^2 &= -1 \\ i^3 &= i^2 \cdot i = (-1)i = -i \\ i^4 &= i^2 \cdot i^2 = (-1)(-1) = 1 \\ i^5 &= i^4 \cdot i = i \\ &\vdots \end{aligned} $In general, $i^n$ takes only four possible values: $1, i, -1, -i$, corresponding to the remainder when $n$ is divided by 4.
Rule for powers of $i$. For any integer $n$, compute $n \bmod 4$ (the remainder when dividing by 4):
- Remainder $0$: $i^n = 1$
- Remainder $1$: $i^n = i$
- Remainder $2$: $i^n = -1$
- Remainder $3$: $i^n = -i$
Example. Simplify $i^{47}$.
Solution. Divide $47$ by $4$: $47 = 4 \times 11 + 3$, so the remainder is $3$. Therefore $i^{47} = -i$.
Frequently Asked Questions
What is the imaginary part of $3 - 5i$?
The imaginary part is $-5$, not $-5i$. The imaginary part is always a real number. Formally, $\operatorname{Im}(3 - 5i) = -5$.
