What it is
The square of a real number is never negative: 3² = 9, (−3)² = 9 and 0² = 0. So the equation x² = −1 has no solution among the real numbers. The way out is to introduce a new number, the imaginary unit i, defined by
i² = −1
Calculate with i as with any letter, replacing i² by −1, and the rest follows:
- The powers of i repeat every fourth step: i¹ = i, i² = −1, i³ = i² · i = −i, i⁴ = (i²)² = 1, then i⁵ = i again. So i¹⁰ = i² = −1, since 10 = 4 · 2 + 2.
- Roots of negative numbers: √(−9) = 3i, because (3i)² = 9 · i² = −9. The equation x² + 1 = 0 has the solutions x = i and x = −i.
- Addition and multiplication: (3 + 2i) + (1 − i) = 4 + i, and (3 + 2i)(1 − i) = 3 − 3i + 2i − 2i² = 3 − i + 2 = 5 − i. A special case: (1 + i)² = 1 + 2i + i² = 2i.
Numbers of the form a + bi, with a and b real, are called complex numbers. In 3 + 2i the real part is 3 and the imaginary part is 2, not 2i. Every real number is complex: 3 = 3 + 0i.
Not the imaginary unit: −1 is not i (i² = −1, but (−1)² = 1), and no real number x has x² = −1.
Why it matters
With i every quadratic equation has solutions. The equation x² + 2x + 5 = 0 gives x = −1 ± √(1 − 5) = −1 ± 2i, and checking x = −1 + 2i works: x² = −3 − 4i, 2x = −2 + 4i, and with 5 the sum is 0. See quadratic equations with complex roots.
Multiplying by i turns a point a quarter turn counter-clockwise in the complex plane: 1 → i → −1 → −i → 1. That is the basis of polar form and of why complex numbers are used for oscillations, alternating current and signals.
- Multiplying complex numbers uses nothing but i² = −1.
- The conjugate flips the sign of the imaginary part, and the product with the conjugate is real.
Common misconceptions
- “Imaginary numbers do not exist.” The name is historical. i is as well defined as √2: a number with a stated property, and you calculate with it by the usual rules.
- “√a · √b = √(ab) always holds.” Not for negative numbers. √(−4) · √(−9) = 2i · 3i = 6i² = −6, while √36 = 6.
- “The imaginary part of 3 + 2i is 2i.” It is 2. The imaginary part is the real number in front of i.
- “i is an unknown.” i is a particular number, like π, not something to solve for.
- “i² = 1.” It is −1. It is i⁴ that equals 1.
Common questions
Is i the same as √−1?
i is a number whose square is −1, but −i has the same property: (−i)² = −1. So the notation √−1 is
ambiguous, and it is safer to rely on the definition i² = −1.
What are imaginary numbers used for?
Oscillations and waves in electronics and signal processing, rotations, and making every polynomial
equation have solutions.
How do I work out i to a large power?
Divide the exponent by 4 and look at the remainder. Remainder 0 gives 1, 1 gives i, 2 gives −1 and 3 gives
−i. So i²⁰²⁶ = i², since 2026 = 4 · 506 + 2, that is −1.