Complex numbers are a fundamental concept in mathematics that extends the realm of real numbers. They are composed of a real part and an imaginary part, where the imaginary part is a multiple of the imaginary unit, denoted by “i”.

## What is a complex number?

Complex numbers are represented in the form $a + bi$ where “$a$” represents the real part and “$b$” represents the imaginary part. The study of complex numbers has significant applications in various branches of mathematics, physics, and engineering. This introductory article aims to provide a comprehensive overview of complex numbers, their properties, and their applications in different fields.

## Modulus of complex numbers

The modulus of a complex number is defined as the distance between the origin and the point representing the complex number in the complex plane. This modulus is denoted by $|z|$, where $z$ is the complex number.

The modulus of a complex number can be expressed in terms of its real and imaginary parts. Specifically, if $z = a + bi$, where a and b are real numbers and i is the imaginary unit, then $$|z| = \sqrt{a^2 + b^2}. $$This formula is derived from the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.

The modulus of a complex number has several important properties that make it a useful tool in mathematical analysis. For example, it is invariant under rotation, meaning that if a complex number is rotated by an angle $\theta$, its modulus remains unchanged. Additionally, the modulus of a product of complex numbers is equal to the product of their moduli, and the modulus of a quotient of complex numbers is equal to the quotient of their moduli.

## Argument of Complex Number

The argument of a complex number is a fundamental concept in mathematics that plays a crucial role in understanding the geometric interpretation of complex numbers. It is defined as the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane.

The argument of a complex number is denoted by the symbol $\arg(z)$, where $z$ represents the complex number. It is measured in radians and can take any value between $-\pi$ and $\pi$, inclusive. The argument of a complex number is unique up to an integer multiple of $2\pi$.

The argument of a complex number can be calculated using the Arctan function, which takes the imaginary part of the complex number divided by its real part. In fact if $z=a+ib$, then $$ \arg(z)=\arctan\left(\frac{b}{a}\right).$$ This yields the ratio of the lengths of the sides of a right triangle formed by the complex number and the positive real axis. The arctan function then returns the angle between the positive real axis and the line connecting the origin to the complex number.

The argument of a complex number has several important properties.

- Firstly, the argument of the product of two complex numbers is equal to the sum of their arguments.
- Secondly, the argument of the quotient of two complex numbers is equal to the difference of their arguments.

## Different Forms of Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, and they are represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. There are various forms of complex numbers that are used in different mathematical contexts.

### Rectangular form

One form of complex numbers is the rectangular form, which is the standard form of representing complex numbers. In this form, the real part and the imaginary part of a complex number are written as separate terms, with the real part being written first.

### Polar form

Another form of complex numbers is the polar form, which represents a complex number in terms of its magnitude and argument. The magnitude is the distance from the origin to the complex number in the complex plane, while the argument is the angle between the positive real axis and the line connecting the origin to the complex number.

### Exponential form

In addition to these forms, there are also other forms of complex numbers that are used in specific mathematical contexts. For example, the exponential form of a complex number is used in complex analysis and is written as $re^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument.

### Trigonometric form

The trigonometric form of a complex number is another form that is used in trigonometry and is written as $r(\cos\theta + i \sin\theta), where $r$ is the magnitude and $\theta$ is the argument.

## The Geometrical Representation of Complex Numbers

In the geometrical representation of complex numbers, the real part is represented on the horizontal axis, often referred to as the real axis, while the imaginary part is represented on the vertical axis, known as the imaginary axis. This two-dimensional coordinate system is commonly referred to as the complex plane.

The complex plane allows for the visualization of complex numbers as points in this plane. Each complex number corresponds to a unique point in the complex plane, where the real part determines the position along the horizontal axis and the imaginary part determines the position along the vertical axis.

Furthermore, the distance from the origin of the complex plane to a specific point represents the magnitude or modulus of the complex number. This magnitude can be calculated using the Pythagorean theorem, where the real and imaginary parts of the complex number form the two sides of a right triangle.

Additionally, the angle formed between the positive real axis and the line connecting the origin and the point representing the complex number is known as the argument or phase of the complex number. This argument can be determined using trigonometric functions such as sine and cosine.

The geometrical representation of complex numbers provides a powerful tool for understanding and analyzing these mathematical entities. It allows for the visualization of complex operations such as addition, subtraction, multiplication, and division, as well as the interpretation of complex numbers in terms of magnitude and phase. This representation is widely used in various branches of mathematics, physics, engineering, and other scientific disciplines

## How to determine the Square root of a complex number

In order to determine the square root of a complex number, let us consider the example of finding the square root of the complex number $\lambda=4+3i$. To do this, we need to find a complex number $z$ such that $z^2=\lambda$. We can represent $z$ as $z=a+ib$, where $a$ and $b$ are real numbers.

Expanding $z^2$, we have $z^2=a^2+2iab+(ib)^2$. Since $i^2=-1$, we can simplify this expression to $z^2=a^2-b^2+i(2ab)$. Now, let us consider the equation $a^2-b^2+i(2ab)=4+i3$. By comparing the real and imaginary parts of both sides of the equation, we obtain the following system of equations: $a^2-b^2=4$ and $2ab=3$.

To solve this system, we introduce a third equation involving $a^2$ and $b^2$. Taking the modulus of $z$, we have $|z|^2=a^2+b^2=|4+i3|=\sqrt{16+9}=5$. This equation allows us to eliminate one of the squares, either $a^2$ or $b^2$. By adding the two equations containing $a^2$ and $b^2$, we find that $2a^2=9$, which implies $a=\pm \frac{3}{2}\sqrt{2}$. Furthermore, we have $2b^2=2a^2-4=9-4=5$, leading to $b=\pm \sqrt{\frac{5}{2}}$. Therefore, the complex number $\lambda$ has two square roots, given by $\frac{3}{2}\sqrt{2} +i \sqrt{\frac{5}{2}}$ and $ā \frac{3}{2}\sqrt{2} -i \sqrt{\frac{5}{2}}$.

## How to solve a system of complex numbers

In this section, we will discuss the process of solving a system of complex equations. While systems of equations with real numbers are commonly solved using determinants or elimination techniques, we will focus on solving systems with complex numbers. To illustrate this, we will consider a specific system of equations: \begin{align*}z_1z_2=i,\qquad z_1-z_2=1+i,\end{align*} where $z_1$ and $z_2$ are complex numbers. Our goal is to determine the expressions for $z_1$ and $z_2$.

It is important to approach this problem with caution and avoid complex calculations. Instead of solving for $z_1$ in terms of $z_2$ and substituting the expression into the second equation, we will introduce a concise method to solve the complex system. We can rewrite the system as: \begin{align*} z_1 (-z_2)=-i,\quad z_1+(-z_2)=1+i.\end{align*} From this, we can deduce that $z_1$ and $(-z_2)$ are solutions of the equation: \begin{align*}\tag{E}t^2-(1+i)t-i=0.\end{align*}

The discriminant associated with this equation is: \begin{align*} \Delta&= 2i+4i=6i\cr &= \left(\sqrt{6}\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\right)\right)^2\end{align*} Hence, $\Delta$ has square roots: \begin{align*}\Delta_1=\sqrt{3}+\sqrt{3}i\quad \text{and}\quad \Delta_2=-\sqrt{3}-\sqrt{3}i.\end{align*} The roots of equation $(E)$ are: \begin{align*}zā&=\frac{1+i+\Delta_1}{2}\cr &= \frac{1+\sqrt{3}}{2}(1+i)\end{align*} and \begin{align*}zā&=\frac{1+i+\Delta_2}{2}\cr &= \frac{1-\sqrt{3}}{2}(1+i).\end{align*} Therefore, we have $z_1=zā$ and