# Complex Arithmetic

**Complex Arithmetic:**

**The Real Numbers:**

We assume the reader to be familiar with the real number system R. We let R^{2} = {(x, y) : ∈ R , y ∈ R} . These are ordered pairs of realnumbers. As we shall see, the complex numbers are nothing other than R^{2} equipped with a special algebraic structure.

**The Complex Numbers:**

The complex numbers C consist of R^{2} 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.