## Example 1.1.7 (c): Simple Complex Numbers |

Find the additive and multiplicative inverses for the numbers |

The additive inverses are easy and left as an exercise. But let's try
the multiplicative inverse of *z = (2,3)*. In other words, we need
to find a complex number *(x,y)* with the property that
*(x,y)*(2,3) = (1,0)*, since we
just before found *(1,0)*
to be the multiplicative identity. That's equivalent to:

(x,y)*(2,3) = (2x-3y, 3x+2y) = (1,0)

That results in a linear system of two equations:

(1) 2x-3y = 1

(2) 3x+2y = 0

Multiply the first equation by 2, the second by 3, and add both equations:

4x - 6y = 2 9x + 6y = 0 13x = 2

Thus *x = 2/13*. Using that in equation (1) gives
*y = -3/13*. You can confirm that indeed
*(2,3)*(2/13,-3/13)=(1,0)*

The remaining questions can be solved similarly and are left to the esteemed reader.