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Aug 18

## Complex numbers: time to turn on the beacon

Math is motivated by analogy and the desire to solve new problems, among other things. We'll see that this is the pattern that leads from real numbers to complex numbers and further to matrix algebra.

### Complex numbers are not that complex

Problem. For any real number $x,$ the square $x^2$ is non-negative. Therefore the equation $x^2=-1$ doesn't have solutions in the set of real numbers.

Idea. Expand the set of real numbers in such a way that a) the square root $\sqrt{-1}$ exists in the new set of numbers and b) the properties of real numbers involving addition/subtraction and multiplication/division are preserved.

Solution to the problem. Step 1. Formally introduce the imaginary unit $i=\sqrt{-1}.$ This implies, in particular, that $i^{2}=-1$ and that everywhere $i^{2}$ is encountered it should be replaced by $-1.$

Step 2. Formally introduce complex numbers as linear combinations $c=a+ib$ of the real unit $1$ and imaginary unit $i$ with real $a,b.$ Manipulate them using the properties of real numbers involving addition/subtraction and multiplication/division. For example,

(1) $(a-ib)(a+ib)=a^2-abi+abi-b^2i^2=a^2+b^2.$

Step 3. A complex number $c=a+ib$ can be interpreted as a vector $(a,b)$ on the plane because summation of complex numbers

$c_1+c_2=a_1+ib_1+a_2+ib_2=(a_1+a_2)+i(b_1+b_2)$

corresponds to summation of vectors. In particular, use the norm of the vector $c$ as the absolute value of $c:$

(2) $|c|\overset{def}{=}(a^2+b^2)^{1/2}=\|c\|.$

These formalities plus the geometric interpretation is all one needs to know about the set of complex numbers $C.$

Definition 1. The number $\bar{c}=a-ib$ is called a conjugate of the number $c=a+ib.$

With this definition, we have from (1) and (2)

(3) $\bar{c}c=a^2+b^2,$ $|c|=(\bar{c}c)^{1/2}.$

Besides, it's easy to see that $c=\bar{c}$ if and only if $b=0.$ This is the way to identify real numbers in the set of complex numbers:

(4) a number $c$ is real if and only if $c=\bar{c}.$

Exercise 1. Express the ratio $u/v$ of two complex numbers in the form $a+ib.$

The set of complex numbers $C$ corresponds to the whole plane and the set of real numbers $R$ corresponds to the $x$ axis. Similarly to $R^n$ we can define $C^n$ as the set of all vectors $x=(x_1,...,x_n)$ with components $x_i\in C.$

Definition 2. The scalar product in $C^n$ is defined by $x\cdot y=\sum{x_i}\bar{y}_i.$

Exercise 2. Check that it has all properties of the scalar product in $R^n$ except that instead of $x\cdot (ay+bz)=ax\cdot y+bx\cdot z$ one has $x\cdot(ay+bz)=\bar{a}x\cdot y+\bar{b}x\cdot z$ where $a,b\in C.$

### Polar representation

Figure 1. Polar form

Let us write
$c=a+ib=|c|\left(\frac{a}{|c|}+i\frac{b}{|c|}\right).$ Here the numbers $a_1=\frac{a}{|c|},$ $b_1=\frac{b}{|c|}$ satisfy $a_1^2+b_1^2=1.$ Therefore there exists an angle $\theta$ such that $a_1=\cos\theta,$ $b_1=\sin\theta$ (see Figure 1, sorry about the notation discrepancy). This implies

(5) $c=|c|\left(\cos\theta+i\sin\theta\right).$

Euler established a wonderful formula

(6) $e^{i\theta}=\cos\theta+i\sin\theta.$

For example, with $\theta=\pi$ we have an interesting relationship between three most important numbers in mathematics: $e^{i\pi }=-1.$ Combining (5) and (6) we get the polar representation:

(7) $c=|c|e^{i\theta}.$

The additive form $c=a+ib$ is better for addition/subtraction and the polar (multiplicative) form is better for multiplication/division. In particular, by (7) $c^n=|c|^ne^{in\theta}.$