Example 1.1.7 (d): Simple Complex Numbers
Which complex number (a,b) should correspond to our symbol i?
The symbol i has the property that i2 = -1. Translating that property into the language or our definition means that we are looking for a number z=(x,y) such that
(x,y)*(x,y) = (x*x-y*y,xy + yx) = (-1, 0)
which results in the system of equations
(1) x*x-y*y = -1
(2) 2xy = 0
The second equation means that either x = 0 or y = 0 or both. But together with the first equation we see that y can not be zero so that:
x = 0 and y = 1
Therefore the symbol i corresponds to the complex number (0,1).