Complex Analysis/Complex numbers

The field of the complex numbers
Historically, it was observed that the equation $$x^2 = -1$$ has no solution for a real $$x$$ (since $$x^2 \ge 0$$ for $$x \in \mathbb R$$). Since mathematicians wanted to solve this equation, they just defined a number $$i$$, called the imaginary unit, such that $$i^2 = -1$$. Of course, there exists no such number. But if we write a two-tuple $$(a, b)$$ with $$a, b \in \mathbb R$$ as $$a + ib$$ and calculate with these two-tuples using the calculation rule $$i^2 = -1$$, that is,
 * $$(a + ib) + (c + id) = (a + c) + i(b + d)$$ and $$(a + ib)(c + id) = (ac - bd) + i(ad + bc)$$

(where we already wrote a two-tuple $$(x, y)$$ as $$x + iy$$, which we will continue to do throughout this book), then the set of all two-tuples with this addition and multiplication forms a field. Indeed, the required axioms for a commutative ring are easy to check, and an inverse of $$x + iy$$, $$x$$ and $$y$$ not both zero, is given by
 * $$(x + iy)^{-1} = \frac{x - iy}{x^2 + y^2}$$,

as can be checked by a direct computation.

Absolute value, conjugation
To each complex number, we can assign an absolute value as follows: A complex number $$z = x + iy$$ ($$x, y \in \mathbb R$$) is actually a two-tuple $$(x, y)$$, which is as such an element of $$\mathbb R^2$$. Now in $$\mathbb R^2$$, we have the Euclidean absolute value, namely
 * $$\|(x, y)\|_2 = \sqrt{x^2 + y^2}$$,

and thus we just define:

Note that the absolute value of the absolute value of a complex number is always a real number, since the root function maps everything in $$\mathbb R_{\ge 0}$$ to $$\mathbb R$$ (in fact to $$\mathbb R_{\ge 0}$$).

To each complex number $$z = x + iy$$ ($$x, y \in \mathbb R$$), we also assign a different quantity, which is obtained by reflecting $$z$$ along the first axis:

That is, the second component changed sign; if, in precise terms, $$z = (x, y)$$, then $$\overline z = (x, -y)$$.

We observe:

Proof:


 * $$\overline{zw} = \overline{xa - yb + i(xb + ya)} = xa - yb - i(xb + ya)$$

and
 * $$\overline z \overline w = (x - iy)(a - ib) = ax - yb - i(xb + ya)$$.

With this notation, we can write the absolute value of a complex $$z = x + iy$$ only in terms of $$z$$ without referring to $$x$$ or $$y$$:

Here juxtaposition denotes multiplication, as usual (albeit complex multiplication in this case).

Proof:


 * $$\sqrt{z \overline z} = \sqrt{(x + iy)(x - iy)} = \sqrt{x^2 - (iy)^2} = \sqrt{x^2 - i^2 y^2} = \sqrt{x^2 + y^2} = |z|$$.

From this follows that the absolute value has the following crucial property:

Proof:


 * $$|zw| = \sqrt{zw \overline{zw}} = \sqrt{zw \overline z \overline w} = \sqrt{z \overline z} \sqrt{w \overline w} = |z| |w|$$

by theorems 1.4 and 1.5. Note that the argument of the square roots was always real.

The complex plane
Since each complex number is in fact a two-tuple $$(x, y)$$, $$x, y \in \mathbb R$$, the set of all complex numbers $$x + iy$$ can be visualized as the plane, where $$x$$ is the first coordinate and $$y$$ the second coordinate. The situation is indicated in the following picture:



The horizontal axis (or $$x$$-axis) indicates the real part and the vertical (or $$y$$-) axis indicates the imaginary part.

Exercises

 * 1) Compute the absolute value of the following complex numbers: $$3 + 4i$$, $$3 + 2i$$, $$1 + \frac{1}{2}i$$.
 * 2) Assume that $$m$$ and $$n$$ are natural numbers which can be written as the sum of two squares of natural numbers: $$m = a^2 + b^2$$ and $$n = c^2 + d^2$$ for some $$a, b, c, d \in \mathbb N$$. Prove that the product $$m\cdot n$$ can also be written as the sum of two squares. Hint: Plug in that $$a^2 + b^2 = |a + ib|^2$$ (and similarly for $$c, d$$) and use the rules of computation for complex numbers.
 * 3) Prove the following relation connecting complex multiplication and the standard scalar product of $$\mathbb R^2$$: $$\langle (a, b), (x, y) \rangle = \operatorname{Re} \left[(a - ib)(x + iy)\right]$$.
 * 4) This exercise introduces central concepts in algebra. Make yourself familiar with the concept of a field in algebra. If $$\mathbb F$$ is a field, a subfield $$\mathbb E \subseteq \mathbb F$$ is defined to be a subset of $$\mathbb F$$ which is closed under the addition, multiplication, subtraction and division inherited from $$\mathbb F$$ and contains the elements $$0$$ and $$1$$ (ie. the neutral elements of addition and multiplication) of $$\mathbb F$$. Prove:
 * 5) Let $$(\mathbb E_\alpha)_{\alpha \in A}$$ be a family of subfields of a field $$\mathbb F$$. Prove that the intersection $$\bigcap_{\alpha \in A} \mathbb E_\alpha$$ is also a subfield of $$\mathbb F$$.
 * 6) Make yourself familiar with the concept of a partially ordered set, and prove that the set of subfields of a given field $$\mathbb F$$ is partially ordered by inclusion (ie. $$\mathbb E \le \mathbb E' :\Leftrightarrow \mathbb E \subseteq \mathbb E'$$). Prove that with regard to that order, any family of subfields $$(\mathbb E_\alpha)_{\alpha \in A}$$ has a greatest lower bound.
 * 7) Prove that a field $$\mathbb F$$ has a smallest subfield, called the prime field, and identify the prime field of $$\mathbb C$$.