**Definition: **The imaginary number i is defined as the square root of -1. (So i^2 = -1) Complex numbers are numbers that have a real part and an imaginary part. Complex numbers are of the form a + bi. Complex conjugates are complex numbers that have equal and opposite imaginary parts. For example 1 + 2i would have a complex conjugate of 1 – 2i.

**Example: **Find i^3.

(2 + 3i) + (1 – 2i) =

(1 + 2i) x (3 – 4i) =

(1 + i) / (2 – i) =

**Solution: **

To simplify imaginary powers, find pairs of i * i and simplify those to -1, and multiply. i^3 = i * i * i = (-1) * i = – i

To add/subtract complex numbers, add the real parts first, then add the imaginary parts separately. (2 + 3i) + (1 – 2i) = 3 + i

To multiply complex numbers, FOIL the parts. (1 + 2i) x (3 – 4i) = (3 – 4i + 6i – 8i^2) = 3 + 2i + 8 = 11 + 2i

To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. (1 + i) / (2 – i) = [(1 + i)*(2 + i)] / [(2 – i) * (2 + i)] = (2 + i + 2i + i^2) / (4 + 2i – 2i – i^2) = (2 + 3i – 1) / (4 + 1) = (1 + 3i) / 5 = 1/5 + 3i/5

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