SKEDSOFT
Complex Arithmetic:
The Real Numbers:
We assume the reader to be familiar with the real number system R. We let R2 = {(x, y) : ∈ R , y ∈ R} . These are ordered pairs of realnumbers. As we shall see, the complex numbers are nothing other than R2 equipped with a special algebraic structure.
The Complex Numbers:
The complex numbers C consist of R2 equipped with some binary algebraic operations. One defines,
These operations of and · are commutative and associative. We denote (1, 0) by 1 and (0,1) by i. If a 
R, then we identify α with the complex number (α , 0). Using this notation, we see that
As a result, if (x, y) is any complex number, then
Thus every complex number (x, y) can be written in one and only one fashion in the form x·1 y·i with x, y R. As indicated, we usually write the number even more succinctly as x i y. The laws of addition and multiplication becomes
The symbols z,w,ζ are frequently used to denote complex numbers. We usually take z = x iy , w = u iv , ζ = ζ iη. The real number x is called the real part of z and is written x = Re z. The real number y is called the imaginary part of z and is written y = Im z. The complex number x − iy is by definition the complex conjugate of the complex number x iy. If z = x iy, then we denote the conjugate of z with the symbol z; thus z = x − iy. The complex conjugate is initially of interest because if p is a quadratic polynomial with real coefficients and if z is a root of p then so is z.
