Definition 1.1.6: Complex Numbers

The set of complex numbers C is defined as the set of all pairs (x,y) , x, y R, where for z = (x,y) and w = (u, v) we define:
  • z + w = (x,y) + (u,v) = (x+u, y+v)
  • z * w = (x,y) * (u,v) = (x*u - y*v, x*v + y*u)
Two complex numbers z1 = (x1,y1) and z2 = (x2,y2) are called equal if both x1 = x2 and y1 = y2. We will frequently use the symbol 0 to mean (0,0).
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Note that two complex numbers being equal results in two equations that need to be true simultaneously. You should also observe that we have defined equality of two complex numbers, but not inequality. In other words, we can not decide if one complex number is less or greater than another!

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007