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 need to prove the field axioms for our definition of addition and multiplication:
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: Rdefined 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.{(x,y)
C: y = 0}
The complex number (0,1) has the property that (0,1)*(0,1)=(-1,0), which is the same property as our symbol i.
