Imaginary and Complex Numbers

Definition: When the square root of a negative number is taken, the result is an imaginary number. The imaginary number  i  is defined as the square root of -1:

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Complex numbers are numbers that have a real part and an imaginary part and are written in the form

a + bi

where a is real and bi is imaginary.

Complex conjugates are complex numbers that have equal and opposite imaginary parts. For example 1 + 2i would have a complex conjugate of 1 – 2i.

Examples:

a)  Find i³.

b)  Add: (2 + 3i) + (1 – 2i)

c)  Multiply: (1 + 2i) x (3 – 4i)

d)  Divide: (1 + i) / (2 – i)

Solutions: 

a) To simplify imaginary powers, find pairs of (i * i), simplify them to -1, and multiply:

png.latex-%5Cdpi%7B150%7D%20%5Cbg_white%20%5Clarge%20i%5E%7B3%7D%5C%5C%3Di%20%5Ccdot%20i%5Ccdot%20i%5C%5C%3D%28-1%29%5Ccdot%20i%5C%5C%3D-i

b) To add complex numbers, add the real parts first, then add the imaginary parts separately:

png.latex-%5Cdpi%7B150%7D%20%5Cbg_white%20%5Clarge%20%282%20+%203i%29%20+%20%281%20-%202i%29%5C%5C%3D%5C%3B%5C%3B2%20+%201%20+%203i%20-%202i%5C%5C%3D%5C%3B%5C%3B3%20+%20i

c) To multiply complex numbers, use the FOIL method:

png.latex-%5Cdpi%7B150%7D%20%5Cbg_white%20%5Clarge%20%5C%21%281+2i%29%283-4i%29%5C%5C%3D%283-4i+6i-8i%5E2%29%5C%5C%3D%5C%3B3+2i+8%5C%5C%3D%5C%3B11+2i

d) To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator using the FOIL method. Simplify:

png.latex-%5Cdpi%7B150%7D%20%5Cbg_white%20%5Clarge%20%5Cfrac%7B1%20+%20i%7D%7B2%20-%20i%7D%5C%5C%5C%5C%3D%20%5Cfrac%7B1%20+%20i%7D%7B2%20-%20i%7D%5Ccdot%20%5Cfrac%7B2%20+%20i%7D%7B2%20+%20i%7D%5C%5C%5C%5C%3D

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