### Complex Numbers

**Chapter 2 Arithmetic of complex numbers**

**Chapter 2.1 Definition**

A

**complex**number

**z**, as the name suggests, is a

**complex**(set) of

**2 numbers**. It consists of a

**real**and

**imaginary**part. It can be represented equivalently as a

**pair**or as a

**sum**. E.g.

**z=(3,2**) or

**z=3+2j**

**Fig. 2-1**

Equivalent notations for the complex number

**z**

While the

**real**numbers can be represented on one real

**x**axis,

**complex**numbers require a plane with the real axis

**Re**

**z**and the imaginary axis

**Im z**. So it’s a bit like a

**vector**, but only similar!

Let’s see what some typical complex numbers

**z1,… z9**look like.

**Fig. 2-2**

Several typical

**complex numbers**in the

**z-plane**as

**pairs**.

They can also be represented equivalently as

**sums**, e.g.

**z3=(1,1)**is the same as

**z3=1+j1**. Notice that the

**complex numbers**

**z6, z1,**and

**z2**have only a

**real part**. So these are ordinary

**real numbers -1,0,+1**.

**Chapter 2.2 The most “absurd” assumption for complex numbers**

Note the quotation marks in the title. Maybe absurd, but not that much.**Fig. 2-3**Equivalent definitions of the imaginary number

**j**

**1.**The imaginary number

**j**is the

**square**root of

**-1**

**2.**The square of the imaginary number

**j**is equal to

**-1**

**3.**The product of

**2**imaginary numbers

**j**is equal to

**-1**

For something like this in

**elementary**and even

**high school**, you can get your hands slapped! However, it turns out that making such an assumption makes calculations in the

**real**world much easier! Causes

**2 complex**numbers to be

**rotated**when

**multiplying**. In combination with

**addition**, this facilitates operations on

**sine waves**. And

**sine waves**are

**electrical**engineering!

In it,

**resistance, induction**and

**capacitance**:

– change the

**amplitude**of the current

**sine**wave

– they move the

**sine**wave of the

**current**

relative to the

**voltage sine**wave.

The

**amplitude**and

**shift**of the

**current**can also be calculated using simple

**trigonometry**, but the formulas are then

**very complicated**compared to formulas

**based**on

**complex numbers**.

**Note**:

Mathematicians use the symbol

**i,**electricians use the symbol

**j**.

**Chapter 2.3 Basic operations on complex numbers****Addition**

The **real** component of the **sum** is **equal** to the **sum** of its components. It’s the same with the** imaginary** component.**Fig. 2-4z3=z1+z2**

**z3=(-5+4j)+(7+5j)=-5+4j+7+5j=2+9j**

Notice that in the addition operation, complex numbers behave like vectors.

**Subtraction**

The

**real**component of the

**difference**is equal to the

**difference**of its

**components**. It’s the same with the imaginary component.

Subtraction can also be treated as

**adding**the

**opposit**e number, i.e.

**-z**.

**Fig. 2-5**

z3=z1-z2

z3=z1-z2

**z3=(-5+4j)-(7+5j)=-5+4j-7-5j=-12-1j**

**Multiplication**

Same as

**multiplying binomials**, e.g.

**(1+x)(2+y)=2+y+2x+2xy**. Just remember the “absurd” rule

**j*j=-1**.

**Example**

**z1=0.4+1.6j**

**z2=1-0.8j**

**z3=z1*z2=(0.4+1.6j)*(1-0.8j)=0.4+0.32j+1.6j-1.28j*j=1.68+1.28j**

It turns out that the

**multiplication of complex numbers**is their

**rotation**!

This is easier to see if we represent a

**complex number**as a

**vector**of length

**|z|**and slope

**α**. In other words, a

**complex number**with module

**|z|**and argument

**α**.

**Fig. 2-6**

A

**complex number**as a modulus

**|z|**and angle argument

**α**

Here

**z=4+6j**–>

**tgα=6/4**–>

**α=arctan(6/4)=56.31º**

Fig. 2-7

Fig. 2-7

We will not prove it, but from ordinary trigonometry it follows that the product

**z1*z2**is a number

**z3**whose

**modulus**and

**argument**are:

**|z3|=|z1|*|z2|**

**α3=α1+α2**

Both notations of the number

**z=a+jb**and

**|z|**and

**α**are

**equivalent**, i.e. they show the same number

**z**on the plane of

**complex numbers**.

Sometimes the

**first**save is more convenient, and sometimes the

**second**.

**Note**that:

**j*1=1, j*j=-1, j*(-1)=-j**and

**j*(-j)=1**

And this is nothing more than successive rotations of the number by

**90º**of the number

**1**when

**multiplying**it by

**j**.

**Fig. 2-8**

Successive rotations of the number

**1**

Here, for example, it is clear that

**-1**is another

**2**turns of the number

**+1**.

So

**-1=j*j*(+1)**!

**Dividing**

We will use a trick. Multiply the

**numerator**and

**denominator**by the so-called

**conjugate number**. The new

**denominator**will always be a

**real**number.

So the main job is to

**multiply**the

**denominator**.

Example

**Fig. 2-9**

How to divide

**z1**by

**z2**?

By the way, you learned what

**conjugate**

**numbers**are and what is their

**product**. Not only is it a

**real**number, it’s easy to count! Try to prove it.

When

**multiplying**, the

**product**changes its

**modulus**and

**rotates**according to the formula:

**|z3|=|z1|*|z2|**

**α3=α1+α2**

There is also

**rotation**when

**dividing**, but in the opposite direction and the analogous formula looks like this:

**|z3|=|z1|/|z2|<**–quotient

**α3=α1-α2**

i.e. we

**divide**the

**modules**and

**subtract**the

**angle**arguments.

**Note**

**α3**positive is a positive rotation, i.e.

**counter-clockwise**motion.