

# Representation of complex numbers

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part $a$ is the coefficient of the real axis, and the imaginary part $b$ is the coefficient of the imaginary axis.

Some examples of complex numbers are:

- $3 + 2i$
- $5 - 4i$
- $8i$
- $1 + 0i$

## Exercise
Find the complex number that represents the sum of the following two complex numbers:

- $2 + 3i$
- $1 - 4i$

### Solution
$3 + -1i$

# Addition and subtraction of complex numbers

To add or subtract complex numbers, we follow the rules of addition and subtraction for real numbers and use the distributive law for multiplication. For addition, we add the real parts and the imaginary parts separately. For subtraction, we subtract the real parts and the imaginary parts separately.

- $2 + 3i + 1 - 4i = 3 - i$
- $5 + 2i - 3 + 4i = 2 + 6i$

## Exercise
Instructions:

Find the sum of the following two complex numbers:

- $1 + 2i$
- $3 - 4i$

### Solution
$4 - 2i$

# Multiplication of complex numbers

To multiply complex numbers, we multiply the real parts and the imaginary parts separately. Then, we use the distributive law for multiplication.

- $(2 + 3i)(1 + 2i) = (2 \cdot 1 + 2 \cdot 2i)(1 + 2i) = 2 + 6i - 4 - 4i^2 = 2 + 6i - 4 + 4 = 6 + 2i$
- $(3 - 2i)(4 + 5i) = (3 \cdot 4 - 2 \cdot 5i)(4 + 5i) = 12 + 20i - 10 - 25i^2 = 12 + 20i - 25 = 12 + 20i - 25 + 0i = 12 + 20i - 25$

## Exercise
Instructions:

Find the product of the following two complex numbers:

- $2 + 3i$
- $4 - i$

### Solution
$8 - 7i$

# Division of complex numbers

To divide complex numbers, we use the formula:

$$\frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$$

- $\frac{2 + 3i}{1 + 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{2 + 3i

# The modulus and argument of complex numbers

The modulus of a complex number is the distance of the number from the origin on the complex plane. It is denoted by $|z|$, where $z$ is the complex number. The modulus of a complex number $a + bi$ is calculated as:

$$|a + bi| = \sqrt{a^2 + b^2}$$

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. It is denoted by $\arg(z)$, where $z$ is the complex number. The argument of a complex number $a + bi$ is calculated as:

$$\arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right)$$

- Let $z = 3 + 4i$. Find the modulus and argument of $z$.

To find the modulus of $z$, we use the formula:

$$|3 + 4i| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

To find the argument of $z$, we use the formula:

$$\arg(3 + 4i) = \tan^{-1}\left(\frac{4}{3}\right)$$

## Exercise
- Find the modulus and argument of the complex number $2 + 3i$.

# Applications of complex numbers in engineering and mathematics

Complex numbers have numerous applications in various fields of engineering and mathematics. Some of these applications include:

- In electrical engineering, complex numbers are used to analyze AC circuits. The impedance of a circuit is represented as a complex number, which takes into account both the resistance and reactance of the circuit.

- In signal processing, complex numbers are used to represent the amplitude and phase of sinusoidal signals. This allows for the analysis of signals in the time and frequency domains.

- In fluid dynamics, complex numbers are used to analyze the behavior of waves in fluid systems. The wave number and group velocity are represented as complex numbers, which provide information about the propagation of waves in the system.

- In quantum mechanics, complex numbers are used to describe the wave functions of quantum systems. The wave function is represented as a complex number, which encodes the probability amplitude of finding a particle at a specific position in space.

- In number theory, complex numbers are used to study the properties of complex numbers themselves. For example, the study of the Gaussian integers, which are complex numbers of the form $a + bi$, where $a$ and $b$ are integers, has numerous applications in number theory.

## Exercise
- Explain the application of complex numbers in the field of fluid dynamics.

- In fluid dynamics, complex numbers are used to analyze the behavior of waves in fluid systems. For example, consider a fluid system with a wave number $k = 2 + 3i$. The wave number represents the number of wavelengths in the fluid system. The real part of the wave number, $2$, represents the number of wavelengths in the positive real direction. The imaginary part of the wave number, $3$, represents the number of wavelengths in the positive imaginary direction.

- The group velocity of the wave is represented as a complex number $v_g = 1 + 2i$. The real part of the group velocity, $1$, represents the velocity of the wave in the positive real direction. The imaginary part of the group velocity, $2$, represents the velocity of the wave in the positive imaginary direction.

- The wave function of a quantum system can be represented as a complex number $ψ = 2 + 3i$. The real part of the wave function, $2$, represents the probability amplitude of finding the particle in the positive real direction. The imaginary part of the wave function, $3$, represents the probability amplitude of finding the particle in the positive imaginary direction.