Complex Numbers

Definition:  \/(-1) = i is called an imaginary number

\/(-16) = \/16 \/(-1) = 4i

Powers of i:

 i1 = i   i5 = i(i4) = i(1) = i   i9 = i(i8) = i(1) = i 
 i2 = -1   i6 = i2(i4) = -1(1) = -1   i10 = i2(i8) = -1(1) = -1
 i3 = i2(i) = -1(i) = -i   i7 = i3(i4) = -i(1) = -i  i11 = i3(i8) = -i(1) = -i
i4 = i2(i2) = (-1)(-1) = 1  i8 = i4(i4) = 1(1) = 1 i12 = i4(i8) = 1(1) = 1

Notice the pattern that repeats itself.

To find i33, simply divide the exponent by 4 and use the remainder as the power on i.

   33/4 = 8 R 1

so i33 = i1 = i

Definition:  Complex Numbers are numbers in the form a + bi where a and b are real numbers and i is an imaginary number.

Add:

(2 + 3i) + (3 + 5i) = 5 + 8i   (Just add like terms)

Subtraction works the same way except remember to change the signs on the second number.

Multiply:

(5 + 2i)(3 - 4i) = 15 - 20i + 6i - 8i2 = 15 - 14i - 8(-1) = 15 - 14i + 8 = 23 - 14i
                   (Use FOIL)

Divide:

Definition:  The conjugate of a + bi is a - bi

 4 - 2i  =  4 - 2i (2 - 3i)  =  8 - 12i - 4i + 6i2  =  8 - 16i - 6  =  2 - 16i
2 + 3i      2 + 3i (2 - 3i)             4 - 9i2                  4 + 9              13

(Multiply the numerator and the denominator by the conjugate of the denominator.  Essentially,     you are rationalizing the denominator since i is a square root.)

To graph a complex number, create a complex coordinate system.  The horizontal axis is the real number axis, and the vertical axis is the imaginary number axis.


Problems

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