Definition
The powers of i return cyclic values:| (repeats the pattern from blue area) |
| (repeats the pattern from blue area) |
The imaginary number i is defined solely by the property that its square is −1:
With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.
Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers by treating i as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of i 2 with −1. Higher integral powers of i can also be replaced with −i, 1, i, or −1:
Similarly,
Read more about this topic: Imaginary Unit
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