The complex number is defined as the number in the form a+ib, where a is the real part while ib is the imaginary part of the complex number where i is known as iota and b is a real number. The value of i is √(-1). That is, a complex number is a combination of real and imaginary numbers. For example, 5+11i, 10+20i, etc.

**Real number:**A real number is a number present in the number system that can be positive, negative, integer, rational, irrational, etc. For example, 23, -3, 3/6.**imaginary number:**Imaginary numbers are those numbers that are not real numbers. For example, √3, √11, etc.**Complex number zero:**A complex number zero is a number whose real and imaginary parts are equal to zero. For example, 0+0i.

**Graphical representation of the complex number:**

The graphical representation of the complex number is shown in the following image:

Here, the real part of the complex number is plotted on the horizontal axis, while the imaginary part of the complex number is plotted on the vertical axis.

### Absolute value

The absolute value (Modulo) of a number is the distance of the number from zero. The absolute value is always represented in module (|z|) and its value is always positive. Thus, the absolute value of the complex number Z = a + ib is

|z| = √ (uno

^{2}+b^{2})

Thus, the absolute value of the complex number is the positive square root of the sum of the square of the real part and the square of the imaginary part, that is,

**Rehearsal:**

Let us assume that the mode of the complex number z extends from 0 to z and the mode of a, b real numbers extends from a to 0 and b to 0. Thus, these values create a right triangle in which 0 is the vertex of the acute angle

So, using the Pythagorean theorem, we get,

|z|^{2}= |and|^{2}+ |b|^{2}

|z| = √ a^{2}+b^{2}

Now, in sets of complex numbers z_{1}> z_{2}o z_{1}<z_{2}has no meaning, but |z_{1}| > |z_{2}| o |z_{1}|_{ }< |z_{2}| has meaning because |z_{1}| e |z_{2}| is a real number.

**Properties of the module of a complex number:**

- |of| = 0 <=> z = 0i, oseja, Re(z) = 0 and Im(z) = 0
- |z| = || = |-z|
- -|z| ≤ Re(z) ≤ |z|, -|z| ≤ Im(z) ≤ |z|
- z.= |z
^{2}| - |z
_{1}z_{2}| = |z_{1}||z_{2}| - |z
_{1}/z_{2}| = |z_{1}|/|z_{2}| - |z
_{1}+z_{2}|^{2}= |z_{1}| + |z_{2}| + 2Re(z_{1}) - |z
_{1}– z_{2}|^{2}= |z_{1}| + |z_{2}| – 2Re(z_{1}) - |z
_{1}+z_{2}|^{2}≤ |z_{1}| + |z_{2}| - |z
_{1}– z_{2}|^{2}≥ |z_{1}| – |z_{2}| - that is
_{1}- beauty_{2}|^{2}+ |bz_{1}+ es_{2}|^{2}= (one^{2}+b^{2})(|z_{1}|^{2}+ |z_{2}|^{2}) o |z_{1}– z_{2}|^{2}+ |z_{1}+z_{2}|^{2}= 2(|z_{1}|^{2}+ |z_{2}|^{2}) - |z
^{norte}| = |z|^{norte} - 1/z = a – ib/a
^{2}+b^{2}=/|z|^{2}

**Example:**

(yo) z = 3 + 4i|z| = √(3

^{2}+4^{2})= √(9 + 16)

= √25

= 5

(ii) z = 5 + 6i|z| = √(5

^{2}+6^{2})= √(25

^{ }+36)= √61

### Angle of complex numbers

The angle of a complex number or complex number argument is the tilt angle of the real axis in the direction of the complex number it represents in the complex plane or argand plane.

θ = tons

^{-1}(LICENSED IN LETTERS)o

arg(Z) = bronceado

^{-1}(LICENSED IN LETTERS)

Here, Z = a + one

**Properties of the angle or argument of a complex number:**

- argument (Z
^{norte}) = n argument (Z) - argument (Z
_{1}/Z_{2}) = argument (Z_{1}) – argument (Z_{2}) - argument (Z
_{1}Z_{2}) = argument (Z_{1}) + argument (Z_{2})

**Examples:**

(yo) z = 2 + 2i(Video) What is the absolute of a complex numberθ = tons

^{-1}(2/2)= tan

^{-1}(1)= 45°

(ii) z = -4 + 4iθ = tons

^{-1}(4/-4)= tan

^{-1}(-1)= -45°

It is important to note here that the angle θ = -45° is at 4

^{º}clock face,while we always measure the angle with the positive x-axis.

So we'll have to add 180° to the answer to get the actual opposite angle.

So θ = 180° + (-45°)

= 135°

So the complex number above will make an angle of 135° with the positive x-axis.

### Polar form of the complex number

The polar form of the complex number is also a way to represent a complex number. Usually, we represent a complex number as Z = a + ib, but in polar form, the complex number is represented in the combination of modulo and argument. Here, the modulus of the complex number is known as the absolute value of the complex number and the argument is known as the angle of the complex number.

Z = r(cos θ + tú θ)

**Rehearsal:**

Consider that we have a complex number Z = a+ib. So we draw an Argand plane, it is a plane where we can represent complex numbers, it is also known as a complex plane. In the background, the horizontal line represents the real axis and the vertical line represents the imaginary axis. Now we graph a on the real axis and b on the imaginary axis. We now represent the vector Z as a position vector starting at 0 and pointing at coordinate (a, b). Assume that θ is the angle made by Z with respect to the real axis and that the distance between 0 and Z is r (also known as the magnitude/absolute value of vector Z). Here, the pair (r, θ) is known as the polar coordinates of Z.

According to the diagram, we have a right triangle.

So, using the Pythagorean theorem, we get,

r = |z|^{2}= |and|^{2}+ |b|^{2}

r = |z| = √ a^{2}+b^{2}

this is the module

Now we find the value of θ

tan θ = (a/b)

θ = tons^{-1}(a/b)

this is the argument

Now we find the polar form of the complex number:

So, using the trigonometric formula, we get

cos θ = a/r

Now we multiply both sides by r, we get

r cos θ = a

sen θ = b/r

Now we multiply both sides by r, we get

rs sen θ = segundo

So we have

Z = rcos θ + irsen θ

Z = r(cos θ + tú θ)

Here, r is the absolute value of the complex number and θ is the argument of the complex number.

**To exponential form of complex numbers:**

The exponential form of complex numbers uses the trigonometric ratios of sine and cosine to define the complex exponential as a rotating plane in exponential form. The exponential form of a complex number is usually given by**euler's Identity,**It is named after the famous mathematician Leonhard Euler. It is given as follows:

Z = re

^{yo}

**Example:**

(UE)Since r = 5 and θ = 45°. Find the polar form of the complex numberz = rcosA + (rsinA)i

z = 5cos45° + (5sen45°)i

= 5(1/√2) + (5 (1/√2))i

= 5/√2 + (5/√2)i

(ii)Since r = 6 and θ = 30°. Find the polar form of the complex numberz = rcosA + (rsinA)i

z = 6cos30° + (6sen30°)i

= 6(√3/2) + (6(1/2))i

= 3√3 + 3i

**Convert rectangular form to polar form:**

Let us analyze this concept with the help of an example:

Suppose we have a complex number i.e. Z = 3 + 4i

It's in rectangular form, so now we need to convert it to polar form.

Paso 1:So let's first calculate the modulus of the complex numbers and then the angle.|Z|= √(3

^{2}+4^{2})= √(9 + 16)

= √25

= 5

Paso 2:Now we find the angle of the complex number,θ = tons

^{-1}(s/x)= tan

^{-1}(4/3)= 53,1°

Stage 3:As we know that the formula of the polar form is:Z = r(cos θ + tú θ)

(Video) Python tutorial - Complex numbersNow plug the value of r and θ into this equation, we get

Z = 5(cos 53.1 + isen 53.1)

So the polar form of 3 + 4i is 5(cos 53.1 + isen 53.1)

### examples of problems

**Question 1. Find the absolute value of z = 4 + 8i**

**Solution:**

The given complex number is z = 4 + 8i

Since we know that the formula for absolute value is

|z| = √ (uno

^{2}+b^{2})So, a = 4 and b = 8, we get

|z| = √(4

^{2}+8^{2})|z| = √80

**Question 2. Find the absolute value of z = 2 + 4i**

**Solution:**

The given complex number is z = 2 + 4i

Since we know that the formula for absolute value is

|z| = √ (uno

^{2}+b^{2})So, a = 2 and b = 4, we have

|z| = √(2

^{2}+4^{2})|z| = √20

**Question 3. Find the angle of the complex number: z = √3 + i**

**Solution:**

The given complex number is z = √3 + i

As we know, this

θ = tons

^{-1}(LICENSED IN LETTERS)So, a = √3 and b = 1, we get

θ = tons

^{-1}(1/ √3 )θ = 30°

**Question 4. Find the angle of the complex number: z = 6 + 6i**

**Solution:**

The given complex number is z = 6 + 6i

As we know, this

θ = tons

^{-1}(LICENSED IN LETTERS)So a = 6 and b = 6, we get

θ = tons

^{-1}(6/6)(Video) Complex Numbers: Part 5/5 "Sage"θ = 45°

**Question 5. Convert z = 5 + 5i to polar form**

**Solution:**

The given complex number is z = 5 + 5i

How do we know that

Z = r(cos θ + tú θ) ...(1)

Now we find the value of r

r = √(5

^{2}+5^{2})r = √(25 + 25)

r = √50

Now we find the value of θ

θ = tons

^{-1}(5/5)θ = tons

^{-1}(1)θ = 45°

Now plug all these values into eq(1), we get

Z = √50(cos 45° + isen 45°)

**Question 6. Convert z = 2 + √3i to polar form**

**Solution:**

The given complex number is z = 2 + 2√3i

How do we know that

Z = r(cos θ + tú θ) ...(1)

Now we find the value of r

r = √(2

^{2}+ (2√3)^{2})r = √(4 + 12)

r = √16

r = 4

Now we find the value of θ

θ = tons

^{-1}(2√3/2)θ = tons

^{-1}(√3)θ = 60°

Now plug all these values into eq(1), we get

Z = 4(cos 60° + isem 60°)

my personal notes*arrow_fall_up*