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The pics. After all the hype, they're probably anticimactic :-(
The three cube roots of 1, listed above, can all be seen to have moduli of 1 and phases of 0 degrees, 120 degrees, and 240 degrees respectively. When raised to the third power, the phases are tripled, obtaining 0, 360, and 720 degrees. But they are all the same angle—zero. So the cubes of these numbers are all just one. Similarly, <math>i\,</math> and <math>-i\,</math> have phases of 90 degrees and 270 degrees. When those numbers are squared, the phases are 180 and 540, both of which are the same angle—180 degrees. So their squares are both -1.
The three cube roots of 1, listed above, can all be seen to have moduli of 1 and phases of 0 degrees, 120 degrees, and 240 degrees respectively. When raised to the third power, the phases are tripled, obtaining 0, 360, and 720 degrees. But they are all the same angle—zero. So the cubes of these numbers are all just one. Similarly, <math>i\,</math> and <math>-i\,</math> have phases of 90 degrees and 270 degrees. When those numbers are squared, the phases are 180 and 540, both of which are the same angle—180 degrees. So their squares are both -1.
One can apply this property to the n<sup>th</sup> roots of any complex numbers. They lie equally spaced on a circle. This is DeMoivre's theorem.
One can apply this property to the n<sup>th</sup> roots of any complex number. They lie equally spaced on a circle. This is DeMoivre's theorem.
[[File:Argandcbrt1.jpg|thumb|left|400px|The three cube roots of +1.]]
[[File:Argandsqrtm1.jpg|thumb|right|400px|The two square roots of -1.]]
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===Euler's formula===
===Euler's formula===