Training Complex Numbers Complex Number Basics
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Complex Number Basics

22 min Complex Numbers
Complex numbers extend the real numbers by introducing i = √(−1), so every complex number has the form z = a + bi where a is the real part and b is the imaginary part. Addition and subtraction work component-wise: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. Multiplication uses the distributive property and i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The complex plane plots a on the horizontal axis and b on the vertical axis, turning algebraic operations into geometric ones—addition becomes vector addition, and multiplication involves rotation and scaling. Every polynomial of degree n has exactly n complex roots (counting multiplicity), guaranteeing that the complex numbers are algebraically closed.

Complex Number Basics

Complex Number

A complex number has the form $z = a + bi$ where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$ so that $i^2 = -1$.

Arithmetic Rules

Addition: $(a+bi)+(c+di) = (a+c)+(b+d)i$

Subtraction: $(a+bi)-(c+di) = (a-c)+(b-d)i$

Multiplication: $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$

Powers of $i$

$i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then the pattern repeats every 4.

Example 1

Compute $(3+2i)+(5-4i)$.

  1. $(3+5)+(2-4)i = 8 - 2i$
Example 2

Compute $(2+3i)(4-i)$.

  1. $8 - 2i + 12i - 3i^2 = 8 + 10i - 3(-1) = 11 + 10i$
Example 3

Simplify $i^{23}$.

  1. $23 = 4(5) + 3$
  2. so $i^{23} = i^3 = -i$.
Example 4

Write $\sqrt{-49}$ as a complex number.

  1. $\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i$

Practice Problems

1. Compute $(4+5i)+(3-2i)$.
2. Compute $(6+i)-(2+4i)$.
3. Compute $(1+i)(1-i)$.
4. Compute $(3i)^2$.
5. Simplify $i^{50}$.
6. Write $\sqrt{-12}$ in terms of $i$.
7. Compute $(2+i)(3+2i)$.
8. Find the real and imaginary parts of $z = -3 + 7i$.
9. Compute $(5-3i)+(2+3i)$.
10. Simplify $i^{100}$.
11. Compute $(1+2i)^2$.
12. Is $z = 4$ a complex number? Explain.
Show Answer Key

1. $7 + 3i$

2. $4 - 3i$

3. $1 - i^2 = 1 + 1 = 2$

4. $9i^2 = -9$

5. $50 = 4(12)+2$; $i^{50} = i^2 = -1$

6. $2i\sqrt{3}$

7. $6 + 4i + 3i + 2i^2 = 4 + 7i$

8. Real: $-3$, Imaginary: $7$

9. $7$

10. $i^{100} = (i^4)^{25} = 1$

11. $1 + 4i + 4i^2 = -3 + 4i$

12. Yes, $4 = 4 + 0i$

ℂ Complex Arithmetic Calculator
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