EECS20N: Signals and Systems

Imaginary arithmetic


The sum of i and i is written 2i or i2. Sums and differences of imaginary numbers simplify like real numbers:

i3 + i2 = i5,

i3 - i4 = -i.

If iy1 and iy2 are two imaginary numbers, then

iy1 + iy2 = i(y1 + y2),

iy1 - iy2 = i(y1 - y2).


The product of a real number x and an imaginary number iy is

x ´ iy = iy ´ x = ixy.

To take the product of two imaginary numbers, we must remember that i 2 = -1, and so for any two imaginary numbers, iy1 and iy2, we have

iy1 ´ iy2 = -y1 ´ y2

The result is a real number. We can use this rule repeatedly to multiply as many imaginary numbers as we wish. For example,

i ´ i = -1,

i 3 = i ´ i 2 = -i,

i 4 = 1.


The ratio of two imaginary numbers iy1 and iy 2 is a real number

iy1 / iy2 = y1 / y2.