Complex Numbers

Overview

The set $\mathbb{C}$ of complex numbers is defined by $\mathbb{C}=\{a+bi\mid a,b\in \mathbb{R}\}$, where $i$ is the imaginary number defined as $i^2=-1$.

For any complex number of the form $z=a+bi$, $a,b\in \mathbb{R}$, the number $a$ is called the real part of $z$ and the number $b$ is called the imaginary part of $z$. This is denoted as $\text{Re}(z)=a$ and $\text{Im}(z)=b$ respectively.

The modulus of $z$, denoted $|z|$, is defined as $|z|=\sqrt{a^2+b^2}$. The argument of $z$, denoted $\text{Arg}(z)$, is the angle $-\pi <\theta \le \pi$ between the complex number and the positive real axis.

Conjugates

Let $z=a+bi$ be a complex number. Then its complex conjugate, denoted as $\overline{z}$, is defined as $$\bar{z} = \overline{a + bi} = a - bi.$$

Arithmetic

Addition, subtraction, and multiplication of complex numbers are done in the normal way. Division works with the aid of conjugates.

Addition

For complex numbers $a+bi$ and $c+di$, we have that $$(a + bi) + (c + di) = (a + c) + (b + d)i.$$

Subtraction

For complex numbers $z_1$ and $z_2$, $$z_1 - z_2 = z_1 + (-z_2).$$

Multiplication

For complex numbers $a+bi$ and $c+di$, we have that $$(a + bi) \cdot (c + di) = (ac -bd) + (ad + bc)i.$$

Division

For complex numbers $a+bi$ and $c+di$, we have that $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(a + bi)(c - di)}{c^2 + d^2}.$$

Bibliography

• “Complex Plane,” in Wikipedia, December 14, 2024, https://en.wikipedia.org/w/index.php?title=Complex_plane.
• John B. Fraleigh, A First Course in Abstract Algebra, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
• Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• Wikipedia. “Argument (complex analysis).” August 3, 2025. https://en.wikipedia.org/w/index.php?title=Argument_(complex_analysis).