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What happens when you square a complex number?
Complex numbers are often introduced by way of the mysterious imaginary number i (the square root of -1). But beginning with i is not the best way to give beginners an intuitive sense of what complex numbers really are, and why they are so awesome.

Complex numbers are two-dimensional numbers. Multiplying complex numbers creates a kind of rotation in the complex plane. And squaring a complex number (multiplying it by itself) has interesting behaviors.


Squaring the Fabric of the Complex Plane
The interactive view at left shows what happens when a random field of complex numbers are squared. When you select the button, "Square them!", an animation causes the numbers to travel to thier new squared locations in the plane. This animation helps to visualize a sort of gravitational field in the plane that gives a sense of what squaring does to the whole plane.

Experiments
If you select the "Real" button, a set of numbers are initialized randomly on the real axis. Then if you select "Square them!" repeatedly, you will notice that these numbers do just what you'd expect for real numbers: any number that is greater than 1 gets larger and eventually approaches infinity. Any number that is less than 1 and greater than 0 approaches 0. And any number that is less than 0 is "flipped" to the positive side of the real axis.

Rotating about i
Complex numbers that lie on the imaginary (vertical) axis have a different behavior than those that lie on the real axis. Instead of moving along the axis as a result of multiplication or squaring, the numbers rotate away from the axis. Drag the red dot to experience complex multiplication.

Four Important Results From Squaring
1. Numbers on the positive imaginary axis rotate to the negative real axis.
2. Numbers on the negative imaginary axis rotate to the negative real axis.
3. Numbers on the negative real axis flip to the positive side.
4. Numbers on the positive real axis converge to 0 or infinity.



Created by Jeffrey Ventrella - www.ventrella.com