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:


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.


a)  Find i³.

b)  Add: (2 + 3i) + (1 – 2i)

c)  Multiply: (1 + 2i) x (3 – 4i)

d)  Divide: (1 + i) / (2 – i)


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


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


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


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


Still need help with imaginary and complex numbers? Download Yup and get help from an expert math tutor 24/7.