The symbol* i *is a number whose square is -1, ie.

**i² = -1**

The num* be*r i is cal

*led an imaginary uni*t. Numbers of the for

*bi, where a*

**m a +***b ∊*

**,***d i-imaginary unit, are calle*

**R, a****n***d complex numbers*.

The set of complex numbers is denoted by C **= {****a ****+ bi | a, b****∈ R****}***.*

The nota* tion* a + bi is

*called the*algeb

*raic or standard form of a complex number.*

The numb

*r a is call*

**e***ed the real*part,

*r, ie.*

**and b – the imaginary part of the complex numbe***a = Re (Z*),

*b = Im (Z),*i.

**z = a + b**For the complex numbe

**r z = a +**bi we adopt:– if b

*=*0, then a

*+ bi = a*is a real number;

– if

*b ≠*0, then a

*+ bi*is called an imaginary (unreal) number;

– if b

*≠ 0*and

*a =*0, then a

*+ bi*is called a purely imaginary number.

For the complex numbers* a + b*i a*nd c +* di we say that they are equal if their real parts are equal and their imaginary parts are equal, ie.

*a + bi* = *c + di* <=>

*a = *c ⋀ *b = d*

## 1. COMPLEX NUMBER OPERATIONS

#### 1.1. ZBIR

(*a + bi*) + (*c + di*) = (*a + c*) + (*b + d*)* i*

Example: *(2 + 4*i) + *(2 – 3*i) =* 2 + 4i + 2 – 3i* =* 4 + i*

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#### 1.2. PRODUCT

(*a + bi*) ·* (c + d*i) = *(ac – bd*) + (*ad + bc*)* i*

Example:

*(2 – 3i) · (5 + 2i) = 10 – 15i + 4i – 6i² =
= 10 – 15i + 4i -6 · (-1) = 10 – 15i + 4i +6 = 16 -11i*

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#### 1.3. DIFFERENCE

*(a + bi) – (c + di) = (a – c) + (b – d) i*

The complex numbe*r -z = -a – b*i is the opposite of the numb*er z = a +* bi, becaus*e (a + bi) + (-a – bi) = *0.

Example:

*(2 + 4i) – (6 – i) = 2 + 4i – 6 + i = -4 + 5j*

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#### 1.4. QUICK

The complex number ** z = a – bi **is a conjugate complex number of the number

**i. The following properties apply to co**

*z = a + b***juga**

*n**e complex numbers z and :z:*

**t**For the quotient of the complex numb* ers a* +

*c + di we use the rule:*

**bi and**The quotient of two complex numbers can be obtained through the following procedure:

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#### 1.5. ACCELERATION OF COMPLEX NUMBER

The definition of a complex number z is defined as the gradation of a real number, ie.

*z¹ = z*

*zⁿ⁺¹ = zⁿ ∙ z*

*z⁰ = 1*

*z⁻ⁿ = (z⁻¹) ⁿ*

For the degrees of the imaginary unit i * a*pplies:

**ⁿ ∈**

*i**}, means*

**{1, -1, i, -i***i¹ = i*

*i² = -1*

*i³ = -i*

*i⁴ = 1*

Example:

*i³⁹⁷ = i⁴͘͘͘͘͘ ⁹⁷⁺³ = i³ = -i*

## 2. Complex number module

The modulus or absolute value of a complex number ** z = a + bi i**s called the real negative number and is denoted by |

**Z |**

If the complex number * a + bi *is a real number, ie. if b

*=*0, then:

Since a²* + b² = (a + bi) · (a – bi)*, the equation is true:

**͞Z ∙ z = | z | ²**

The following applies to the comple*x n*umbe*rs *z₁ and z₂:

3. COMPLEX NUMBERS AS SUBJECTED DOUBLE NUMBERS

The ordered pair of real numbers z =* (a, b) = a + bi* is a complex number, ie.

* C = *{(

**a, b) | a, b***}*

**∊ R**The operations of addition, subtraction, multiplication and division are performed according to the following rules:

For complex numbers (a, b) a*nd (c, d) th*e* following *applies:

*(a, b) = (c, d) <=> a = c *⋀* b = d*

A complex number (0, 0) is called a complex zero.

For the complex number of imaginary units, ie. *i* = (0, 1). 3that any complex numb*er z = a* + bi can be written as follows:

*z = a + bi = (a, b) = (a, 0) + (b, 0) · (0, 1)*