Why invent the number i

No real number has a negative square: x² is always positive or zero. Yet by deciding that such a number exists, we get a consistent theory, simpler than the previous one, and one that has become indispensable in physics and engineering.

The number i

We postulate the existence of a number, written i, satisfying:

i² = -1

This number is not real: it sits nowhere on the real line. So a plane will be needed to house it, which is the subject of the next chapter.

Algebraic form

A complex number is written:

z = a + i b        with a and b real

a = Re(z)   the REAL PART
b = Im(z)   the IMAGINARY PART   (a REAL number, despite the name)

This form is unique: two complex numbers are equal if and only if their real parts and their imaginary parts agree.

a + ib = a' + ib'      <=>      a = a'   AND   b = b'

So a single complex equation is equivalent to two real equations — a mechanism used constantly in calculations.

z = 3 - 2i     ->   Re(z) = 3        Im(z) = -2   (not -2i)
z = 5          ->   a real number: Im(z) = 0
z = 4i         ->   purely imaginary: Re(z) = 0

The set of complex numbers is written C, and it contains R: every real is a complex number with zero imaginary part.

   N  ⊂  Z  ⊂  Q  ⊂  R  ⊂  C

Where they come from: cubic equations

Contrary to the legend, complex numbers were not born from a wish to solve x² = -1 — that equation was simply declared impossible.

They appeared in the 16th century among Italian algebraists (Cardano, Bombelli) while solving cubic equations. The solution formula went through square roots of negative numbers even when the three final solutions were perfectly ordinary real numbers. There was no avoiding them: you had to pass through the imaginary to come back to the real.

Long regarded as suspect — Descartes named them "imaginary" out of distrust — these numbers were only fully legitimised in the 19th century, with their geometric representation.

Powers of i

They cycle with period 4:

i⁰ = 1        i¹ = i        i² = -1       i³ = -i
i⁴ = 1        i⁵ = i        i⁶ = -1       ...

    1  ->  i  ->  -1  ->  -i  ->  1  ->  ...

To compute i^n, divide n by 4 and look at the remainder:

remainder 0 -> 1     remainder 1 -> i     remainder 2 -> -1    remainder 3 -> -i

i^2026 :  2026 = 4 × 506 + 2      ->  i^2026 = i² = -1

Summary

• We posit i² = -1: i is not a real number.
• Every complex is written z = a + ib uniquely: a = Re(z), b = Im(z).
• One complex equation is equivalent to two real equations.
• C contains R; the reals are the complex numbers with zero imaginary part.
• Powers of i cycle with period 4: work with the remainder of n modulo 4.
• Historically, complex numbers came from cubic equations.