Complex Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Complex Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Numbers of the form a + bi where a, b are real and i = \sqrt{-1}; they extend the real numbers to solve x^2 = -1.

Extending numbers into a second dimension to solve equations like x^2 = -1.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Adding an imaginary axis creates a number plane instead of just a line.

Common stuck point: Getting past the name 'imaginary' - they're as real as real numbers.

Sense of Study hint: Write out the powers of i in order: i, i squared = -1, i cubed = -i, i to the fourth = 1, then the cycle repeats.

Worked Examples

Example 1

easy
Simplify i^2, i^3, and i^4.

Solution

  1. 1
    i^2 = -1 by definition of the imaginary unit.
  2. 2
    i^3 = i^2 \cdot i = (-1) \cdot i = -i.
  3. 3
    i^4 = i^3 \cdot i = (-i) \cdot i = -i^2 = -(-1) = 1.

Answer

i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
The powers of i cycle with period 4: i, -1, -i, 1, i, -1, -i, 1, \ldots Knowing this cycle allows rapid simplification of any power of i by finding the remainder when the exponent is divided by 4.

Example 2

medium
Multiply (3 + 2i)(1 - i) and write the result in standard form a + bi.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Add (4 + 3i) + (2 - 5i).

Example 2

medium
Simplify i^{23}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

real numbersquadratic formula