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MathidiaMathematics Academy
Complex numbers
Lesson 1 of 75
Introduction

The imaginary unit

Objective. Use i²=−1 to simplify powers and solve equations.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The imaginary unit i is defined by i² = −1, extending the real numbers so that negative quantities have square roots. Every power of i cycles through four values: i, −1, −i, 1, so i^n can be simplified by reducing n mod 4. This lets equations like x² + 9 = 0 be solved as x = ±3i rather than declared unsolvable. For example, i³ = i² × i = −i, and i^10 = (i²)^5 = (−1)^5 = −1. Mastering this cycle underpins all later complex-number arithmetic, from Cartesian form to polynomial roots with no real solutions.

Worked example

Simplify i^23 and hence solve x² + 49 = 0, giving roots in terms of the imaginary unit.

23 … 49
Illustrated card showing 23 … 49
  1. 123 = 4×5 + 3, so i^23 = (i^4)^5 × i^3 = 1 × i^3
  2. 2i^3 = i² × i = −i, so i^23 = −i
  3. 3For x² + 49 = 0, x² = −49
  4. 4x = ±√49 × i = ±7i

Answer: i^23 = −i; x = ±7i

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Common mistake. Students often write √(−9) = −3i or treat i² as +1. Avoid this by always writing √(−9) = √9 × i = 3i, and remembering i² = −1 is the defining rule, not i = −1.

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