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.

1. Firstly, the argument of the product of two complex numbers is equal to the sum of their arguments.
2. Secondly, the argument of the quotient of two complex numbers is equal to the difference of their arguments.
These properties are analogous to the properties of exponents in real numbers.

## 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.

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