i · Complex Numbers
The 90° turn that widened the line into a plane — i.
A number that squares to −1 — it exists nowhere on the number line. Why couldn't mathematicians throw it away?
Explore the connection between squaring and rotation.
The number once mocked as merely 'imaginary'.
Open the immersive version →A reading path containing this lesson: Beyond familiar numbers · 3/3
Why does this question matter?
The history of numbers is a history of blocked equations. x+3=1 could not be solved with naturals, so negatives were born; 3x=2 could not be solved with integers, so fractions were born. New numbers have always been created in front of a 'this cannot be solved' wall.
In 16th-century Italy, mathematicians solving cubic equations ran into numbers that squared to a negative. If they accepted such a number for a moment and pushed the calculation through, a perfectly ordinary real answer came out. It was a number they could neither discard nor believe.
The answer arrived 300 years later, in the age of Gauss: i is a number off the line, on a plane. The moment multiplying by i was read as a 90° rotation, every calculation matched the picture, and numbers grew from a line into a plane.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
x + 3 = 1
Known numbers: naturals 1, 2, 3, …Predict first — can you solve it with only the numbers you know so far?
Read the idea from another angle
New numbers are not discovered — blocked problems give birth to them.
Negatives and fractions were both treated as 'nonsense numbers' when they first appeared. i, born in front of x²=−1, simply walked the same road. Only 300 years later was its identity revealed: rotation.
The identity of i is not a size but a direction.
If multiplying by −1 is a 180° flip on the number line, multiplying by i is half of that: a 90° turn. Do the half-flip twice and you get the full flip — the equation i² = −1 explained by a single picture.
Now read the definitions with their conditions
Write the rotation you experienced in the lab in the language of mathematics, and it looks like this.
Formula anatomy — matched with the buttons you pressed in stage 2
=
Definition of the imaginary unit
A declaration accepting a new number i that squares to −1. It is exactly where you landed after pressing ×i twice in stage 2.
A complex number
A point (a, b) on the plane with real part a and imaginary part b. A number of the plane, not of the line — the very point you moved with the sliders.
Multiplying by i is a 90° rotation
The point (a, b) moves to (−b, a). Every point turns exactly 90° counterclockwise — the rule you verified in stage 2.
Period 4
Four turns bring you home, so the powers of i take only four values. However large the exponent, only the remainder after dividing by 4 survives.
Make one more distinction
An 'imaginary' number is, as the name says, imaginary — it doesn't exist in reality.
The name is just Descartes' half-mocking nickname that stuck. AC electricity, sound analysis, and in-game rotation are all computed with complex numbers. The power in your wall outlet is designed with this number every day.
Complex numbers are numbers, so they can be ranked by size.
Points on a plane cannot be lined up in a single row. You cannot ask whether i or 1 is bigger. The only thing you can compare is the distance from the origin — the absolute value.
Where else can this idea help?
AC electricity
The power in an outlet is a wave that flips direction dozens of times per second. The offset (phase) between voltage and current is computed with a single complex number — the standard language of electrical engineering.
Sound and signal processing
Speech recognition, noise canceling, and MP3 all stand on the Fourier transform, which splits sound into frequencies. The heart of that transform is complex rotation.
Rotation in games and graphics
Turning a character in a 2D game is literally complex multiplication. In 3D, its extension — quaternions — rotates cameras and robot joints.
The Mandelbrot fractal
Just repeat z → z²+c on the complex plane and an infinitely deep fractal appears — the most famous picture a single complex multiplication has ever drawn.
Explain it in your own words
Check the idea with practice problems (4)
Questions completed 0 / 4
1. What is i²?
2. What is i²⁰²⁶?
3. Compute (2 + i) + (1 − 3i).
4. Which of the following is a real number?
What question comes next?
A reading path containing this lesson: Beyond familiar numbers · 3/3
Previous concept
Square Roots
When x²=2 got stuck, √ was born — the previous chapter of the same story.
← Square Roots labLeads to next
Vector
Complex multiplication is a rotating arrow — see numbers on the plane through a vector's eyes.
Go to the Vector lab →Related lab
Quadratic Function
The roots that vanished when the discriminant went negative were on the plane all along.
Go to the Quadratic Function lab →Related lab
Trigonometric Ratios
The language of angles and rotation — the heart of complex multiplication lives here.
Go to the Trigonometric Ratios lab →