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The Complex Field
Adrian Down
August 30, 2005
1 Definitions
Definition. The set C of all complex numbers is the set of all possible pairs of real numbers (x, y)
Want to make an algebra, so must define addition and multiplication.
Definition.
(x, y) + (u, v) = (x + u, y + v) (x, y)(u, v) = (xu − yv, xv + yu)
Theorem. C with the above definition of +, · is a field
This is too much to prove, so we only prove one of the field properties.
Proof. Multiplicative inverses exist. Given z = (x, y) 6 = (0, 0). Claim the inverse is
w =
x x^2 + y^2
y x^2 + y^2
Want to get that z · w = 1. Multiply out and do algebra.
This is how you prove a simple existence statement: find one and show that it has the desired properties.
Definition. ı is the complex number (0, 1). We can write
−1 instead of ı. Identify R with a subset of C via the one-to-one correspondence x → (x, 0). This correspondence preserves sums and products.
Definition. x + ıy is the complex number (x, y), where x, y ∈ R.
Now we are back to the usual notation. To multiply, expand and use the identity ı^2 = −1.
2 Absolute value
Definition.
|z| =
x^2 + y^2
Definition. Complex conjugate
z¯ = x − ıy if z = x + ıy
Facts:
z · z¯ = |z|^2
z ·
¯z |z|^2
z
z¯ |z|^2
, ∀z 6 = 0
|z · w| = |z| · |w| |z + w| ≤ |z| + |w|
The last of these is called the triangle inequality. Note that there is no reasonable ordering (≤ and ≥) for the complex numbers, as there is for the reals.
Proof. Proof of the triangle inequality to practice with inequalities and com- plex numbers. Note that |z + w|, |z| + |w| are positive real numbers ⇒
|z + w| ≤ |z| + |w| ⇔ |z + w|^2 ≤ (|z| + |w|)^2 |z + w|^2 = |(x + u) + ı(y + v)|^2 = (x + u)^2 + (y + v)^2 (|z| + |w|)^2 = |z| + 2|z| · |w| + |w|^2 = (x^2 + y^2 ) + (u^2 + v^2 ) + 2|z| · |w|
So our proof is finished if we can prove
x^2 + 2xu + u^2 + y^2 + 2yv + v^2 ≤ x^2 + y^2 + u^2 + v^2 + 2|z| · |w| 2(xu + yv) ≤ 2 |z| · |w|
The magnitudes still contain a square root, so square again to get rid of it. Note that this time the left hand side could be negative. Want to get
xu + yv ≤ |z| · |w|
