# Imaginary arithmetic

### Sums

The sum of*i*and

*i*is written 2

*i*or

*i*2. Sums and differences of imaginary numbers simplify like real numbers:

*i*3 +

*i*2 =

*i*5,

*i*3 - *i*4 = -*i*.

If *iy*_{1} and *iy*_{2} are two imaginary numbers,
then

*iy*

_{1}+

*iy*

_{2}=

*i*(

*y*

_{1}+

*y*

_{2}),

*iy*_{1} - *iy*_{2} =
*i*(*y*_{1} - *y*_{2}).

### 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, *iy*_{1}
and *iy*_{2}, we have

*iy*

_{1}´

*iy*

_{2}= -

*y*

_{1}´

*y*

_{2}

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*iy*

_{1}and

*iy*

_{2}is a real number

*iy*

_{1}/

*iy*

_{2}=

*y*

_{1}/

*y*

_{2}.