Example 1.1.7 (c): Simple Complex Numbers
Find the additive and multiplicative inverses for the numbers z = (2,3), w = (-2,0), v = (0,2).
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.