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 $x\in ℝ$). 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 $a,b\in ℝ$ 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 $z=x+iy$ ($x,y\in ℝ$) is actually a two-tuple $(x,y)$, which is as such an element of ${ℝ}^2$. Now in ${ℝ}^2$, we have the Euclidean absolute value, namely

$\Vert (x,y){\Vert }_2=\sqrt{x^2+y^2}$,

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 ${ℝ}_{\ge 0}$ to $ℝ$ (in fact to ${ℝ}_{\ge 0}$).

To each complex number $z=x+iy$ ($x,y\in ℝ$), 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 $\bar{z}=(x,-y)$.

We observe:

Proof:

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

and

$\bar{z}\bar{w}=(x-iy)(a-ib)=ax-yb-i(xb+ya)$.$◻$

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:

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

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

Proof:

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

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 $(x,y)$, $x,y\in ℝ$, 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 ${ℝ}^2$: $⟨(a,b),(x,y)⟩=\mathrm{Re}[(a-ib)(x+iy)]$.
4. This exercise introduces central concepts in algebra. Make yourself familiar with the concept of a field in algebra. If $𝔽$ is a field, a subfield $𝔼\subseteq 𝔽$ is defined to be a subset of $𝔽$ which is closed under the addition, multiplication, subtraction and division inherited from $𝔽$ and contains the elements $0$ and $1$ (ie. the neutral elements of addition and multiplication) of $𝔽$. Prove:
   1. Let $({𝔼}_{\alpha }{)}_{\alpha \in A}$ be a family of subfields of a field $𝔽$. Prove that the intersection $\bigcap _{\alpha \in A}{𝔼}_{\alpha }$ is also a subfield of $𝔽$.
   2. Make yourself familiar with the concept of a partially ordered set, and prove that the set of subfields of a given field $𝔽$ is partially ordered by inclusion (ie. $𝔼\le {𝔼}^{\prime }:\iff 𝔼\subseteq {𝔼}^{\prime }$). Prove that with regard to that order, any family of subfields $({𝔼}_{\alpha }{)}_{\alpha \in A}$ has a greatest lower bound.
   3. Prove that a field $𝔽$ has a smallest subfield, called the prime field, and identify the prime field of $ℂ$.