CA Foundation Exam June 2023 » CA Foundation Study Material » Business Mathematics » Modulus and Argument of a Complex Number

# Modulus and Argument of a Complex Number

How to find an argument of a complex number? What are the Modulus and principal arguments of a complex number? What are their properties?

## Introduction

Complex numbers are imaginary numbers, and the complex plane represents these numbers. Among the two forms of these numbers, one form is z = a + bi, where i is the square root of -1 (i =√−1), and a and b are real numbers. In another form, z=reiθ, r = modulus and θ = argument. Both these Modulus and argument are of z.

Before understanding how to find an argument of a complex number, it is essential to understand the meaning of Modulus and argument.

## Modulus of a Complex Number

The distance of a complex number on an Argand plane or complex plane from its origin is the Modulus of a complex number. The Modulus is the non-negative value and the absolute value of a complex number. In other words, when we add and sum the squares of real and imaginary numbers and take out its square root, the resultant number is the Modulus of a complex number.

Modulus of a complex number is z =a+ib or Mod z = √(a2 + b2).

## Properties of Modulus of a Complex Number

We know that Modulus is the non-negative value, which means the value of -2 is 2, and 2 is also 2. We take out the absolute value of a number after considering its distance from zero on the number line of real numbers.

We represent modulus of a complex number as |z|or mod z or |x + iy|, and it has following properties:

• |z| = 0 => z = 0 + i0
• |z1 – z2 |⩾|z1|-|z2|
• |z1 + z2 |≤|z1|+|z2|
• –|z| ≤ Re(z) ≤ |z|
• –|z| ≤ Im(z) ≤ |z|
• |z| ≤ |Re(z)| + |Im(z)| ≤ |z|
• |z| = √z
• |z1z2| = |z1| |z2|

## The Argument of a Complex Number

We know that the complex plane has a horizontal and perpendicular axis. The former represents real numbers, while the latter is for imaginary numbers.

The inclined angle towards the complex number on the horizontal axis is the argument. We denote the argument by “θ” or “φ,” and measure radians, the standard unit.

## Properties of Argument of a Complex Number

Before understanding the properties of an argument, it is essential to know how to find argument of a complex number. We represent the complex number on a complex plane, and the angle which falls on the real axis and is inclined towards the complex number is the argument of a complex number. The standard unit of the argument is ‘radians.’

There are three properties of argument of a complex number and these are same as the properties of a logarithm:

• arg(z1z2) = Θ1 + Θ2 = arg(z1) + arg(z2)
• arg (z1/z2) = Θ1 – Θ2 = arg(z1) – arg(z2)
• arg (zn) = n arg(z)

Hence, we now know how to find argument of a complex number.

## How To Find Out the Principal Argument of a Complex Number

How to find out the argument of a complex number? The argument is a multi-valued angle of polar representation. The principal argument of the complex number is the unique value of the multi-valued anglel. Thus, the principal argument of a complex number is individual data, and the argument of a complex number is multiple data.

Before understanding the principle argument of a complex number example, we have to know how to find argument of a complex number. Let us study this principle argument of a complex number example which lies on the imaginary axis:

Z=i= P (0,1)

Here, the complex number arg(z)= π/2+2nπ and π/2 is the principal argument of a complex number.

## How To Find Out Modulus and Argument of a Complex

We can represent a complex number either geometrically or algebraically. Though, in both conditions, the value of the argument is multiple. Therefore, how to find out Modulus and argument of a complex number in this condition? We can find the Modulus and argument through the following steps:

• Firstly, how to find argument of a complex number? For this, take a graph paper and mark complex numbers on it. It will help you determine where that complex number falls on a complex plane and ultimately determine its argument
• Secondly, we proceed to find the Modulus of a complex number. We can do so with the formula-|z|=√a^2+b^2
• Thirdly, use the formula arg(z)=tan-1(b/a) to determine the argument of a complex number and then adjust the result of the argument

### Conclusion

There are many uses of Modulus and arguments of a complex number in real life. Modulus equals the remainder of an integer division. It is instrumental in scientific calculations as it helps to calculate the remaining seconds from the total seconds. With the help of Modulus, we get minutes, whole seconds, and the remainder seconds, as well. Similarly, an argument of a complex number is helpful in quantum mechanics, circuit analysis, and so on. It solves complex geometrical problems with ease.