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The Strange World of Imaginary Numbers

The Strange World of Imaginary Numbers

By Math Tools ·

The Strange World of Imaginary Numbers

Here's a calculation that looks like it should produce something wildly imaginary:

i^i

An imaginary number raised to an imaginary power. The result? About 0.2079, an ordinary real number. More precisely, its principal value is e^(−π/2).

Imaginary numbers are full of surprises like this. But once you see what i really does, rather than what its name suggests, the strangeness turns into something elegant.

"Imaginary" Numbers Are No Less Real Than Negative Numbers

The word "imaginary" was a put-down. René Descartes used it in 1637 to dismiss square roots of negative numbers as fictions. The label stuck for almost 400 years.

But negative numbers were once called "absurd" too. You can't hold −3 apples, yet −3 is perfectly useful for debt and temperature. Imaginary numbers are the same kind of thing: a tool that describes a structure the real world genuinely has, in this case rotation.

Meet i

The imaginary unit i is defined by one rule:

i² = −1

Everything else follows. Look at what happens with powers of i:

Power Value
i⁰ 1
i¹ i
i² −1
i³ −i
i⁴ 1
i⁵ i

The pattern repeats every 4 steps. To find any power, divide the exponent by 4 and use the remainder: i¹⁰³ = i³ = −i, because 103 = 25 × 4 + 3. That's modular arithmetic at work; see the number theory formulas.

i Is a Quarter Turn

Why a cycle of 4? Picture the complex plane: real numbers on the horizontal axis, imaginary numbers on the vertical axis.

  • Start at 1 (pointing right)
  • Multiply by i → i (pointing up), a 90° turn
  • Multiply by i → −1 (pointing left), another 90°
  • Multiply by i → −i (pointing down)
  • Multiply by i → back to 1

Multiplying by i rotates any number 90° counterclockwise around zero. Four quarter-turns make a full circle. And i² = −1 is just the statement that two quarter-turns make a half-turn, which flips a number to its negative.

An Insider Reference: Gauss Wanted a Better Name

Carl Friedrich Gauss helped make complex numbers respectable. In 1831, he wrote that the confusion around these numbers came largely from their names. If +1, −1 and √−1 had been called direct, inverse and lateral units instead of positive, negative and imaginary, he argued, people would hardly have found them obscure.

"Lateral" is a perfect description: i points sideways, off the number line, into a second dimension.

Complex Numbers: Two Numbers in One

A complex number combines a real part and an imaginary part:

z = a + bi

It's a point (a, b) on the plane. You can also describe it by its distance from zero, r, and its angle, θ:

r = √(a² + b²)
θ = arctan(b / a)   (adjusted for the correct quadrant)

That's polar form. Convert between the two views with the polar to rectangular converter.

In polar form, multiplication becomes beautifully simple: multiply the distances, add the angles. That's why multiplying by i (distance 1, angle 90°) is a pure rotation.

Euler's Formula: Where It All Connects

In 1748, Leonhard Euler published a formula connecting exponentials and trigonometry:

e^(iθ) = cos θ + i sin θ

As θ increases, e^(iθ) travels around a circle of radius 1. At θ = π (180°), cos π = −1 and sin π = 0, which gives Euler's identity:

e^(iπ) + 1 = 0

It links five fundamental constants (e, i, π, 1 and 0) in a single short equation. Check the cosine and sine values with the cosine calculator and sine calculator.

Solving the i^i Mystery

Now we can explain the opening puzzle. At θ = π/2 (90°), Euler's formula gives:

e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i = i

So i = e^(iπ/2). Raise both sides to the power i:

i^i = (e^(iπ/2))^i = e^(i × i × π/2) = e^(−π/2) ≈ 0.2079

The two i's multiply to −1, and the result is real. (Because angles repeat every 2π, i^i actually has infinitely many real values, e^(−π/2 − 2πk) for integer k. The one above is the principal value.)

Even Stranger Places

  • Quaternions: In 1843, William Rowan Hamilton extended complex numbers to four dimensions with three imaginary units i, j, k, where i² = j² = k² = ijk = −1. They now handle 3D rotations in games and spacecraft.
  • Complex logarithms: ln(−1) = iπ. Logarithms of negative numbers exist once you allow complex answers.
  • The Mandelbrot set: one of the most famous images in mathematics comes from repeatedly squaring complex numbers: z → z² + c.

Two Concepts Worth Knowing

Complex Conjugate

The conjugate of a + bi is a − bi, its mirror image across the real axis. Multiplying a number by its conjugate always gives a real number: (a + bi)(a − bi) = a² + b².

Modulus

The modulus |a + bi| = √(a² + b²) is the distance from zero, the complex version of absolute value. It's exactly the Pythagorean theorem. Try it with the right triangle calculator.

Quick Answer: What Is an Imaginary Number?

An imaginary number is a multiple of i, where i is defined so that i² = −1. Geometrically, multiplying by i rotates a number 90° on the complex plane. Combined with real numbers, imaginary numbers form complex numbers a + bi, which are essential in physics, engineering and mathematics.

Try Them Yourself

Plot 1, i, −1 and −i on graph paper, then multiply 2 + i by i and plot the result. You'll see the quarter-turn with your own eyes.

Next → Why You Can't Take the Square Root of a Negative Number — Or Can You?