The document introduces complex numbers and their properties. It defines the imaginary unit i as the square root of -1. Complex numbers have both a real and imaginary part and can be added, subtracted, multiplied and divided. Powers of i rotate through the values of i, -1, -i, and 1, depending on whether the exponent is 1, 2, 3, or 4 modulo 4. Real and imaginary numbers are subsets of complex numbers.

1. Complex Numbers

3. Since there is not real number whose square is -1, the equation has no real solution. French mathematician Rene Descartes (1596 -1650) proposed that i be defined such that,

4. Complex Numbers Note that squaring both sides yields: therefore and so and And so on…

5. Finding Powers of i The successive powers of i rotate through the four values of i , -1, - i , and 1. i n = i if n = 1, 5, 9, … i n = -1 if n = 2, 6, 10, … i n = - i if n = 3, 7, 11, … i n = 1 if n = 4, 8, 12, …

6. *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:

10. Examples

16. CAUTION!

18. Dividing Examples: