EECS20N: Signals and Systems

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• Imaginary numbers
• Imaginary arithmetic
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Imaginary arithmetic

Sums

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).

Products

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.

Ratios

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

iy₁ / iy₂ = y₁ / y₂.