### Theorem 1.1.8: Complex Numbers are a Field

 The set of complex numbers C with addition and multiplication as defined above is a field with additive and multiplicative identities (0,0) and (1,0). It extends the real numbers R via the isomorphism (x,0) = x. We define the complex number i = (0,1). With that definition we can write every complex number interchangebly as z = (x,y) = x + i*y = x + i y Back

We need to prove the field axioms for our definition of addition and multiplication:

• z + w = (x,y) + (u,v) = (x+u, y+v)
• z * w = (x,y) * (u,v) = (x*u - y*v, x*v + y*u)

1. Both + and * are associative, which is obvious for addition. For multiplication we nned to show that a*(b*c)=(a*b)*c. Let a=(x1,y1), b=(x2,y2), and c=(x3,y3). Then:

2. Both + and * are commutative, i.e. a+b=b+a and a*b=b*a

Exercise

3. The distributive law holds, i.e. a*(b+c)=(a*b)+(a*c)

Exercise

4. The additive identity is (0,0), and the multiplicative identity is (1,0), which you can easily confirm.

5. The additive inverse to (x,y) is (-x,-y). The multiplicative inverse to (x,y) is (x/(x2+y2), -y/(x2+y2).

That shows that C is a field.

We now identify the real number x with the complex number (x,0). Thus, the (real) numbers 0 and 1 are the same as the complex numbers (0,0) and (1,0), respectively. Technically, the map

f: R {(x,y) C: y = 0}
defined via f(x)=(x,0) is an isomorphism (a bijection such that it and its inverse are homomorphisms) that identifies the real numbers with a subset of the complex numbers.

The complex number (0,1) has the property that (0,1)*(0,1)=(-1,0), which is the same property as our symbol i.

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth