Complex Numbers Detailed Discussion on Modulus for JEE

VSAT 2022

Complex Numbers: A Detailed Discussion on Modulus and Complex Number Properties

The Modulus of a complex number is nothing but its distance from the origin in the argand plane. In other words we can also define the modulus of a complex number as a point on the argand plane and its distance from the origin. Conjugate of a complex number on the other hand is the image of the complex number on the real axis of the argand plane. Both modulus and conjugate of a complex number have certain properties that make mathematical calculations much easier.  


Modulus of Complex Number

So, what is the modulus of a complex number and how is it represented? As mentioned above the modulus of a complex number is its distance from origin in the argand plane. Let us first understand what is an argand plane?


Simply put, an argand plane is used to represent a complex number. For example if a complex number such as z=a+ib is given, it is represented as  point (a,b) on the argand plane.


The diagram below represents the mod of complex number


(image will be uploaded soon)


Image: Pictorial Representation of a Complex number


How to Find Modulus of a Complex Number

The mod of a complex number is represented as |z|. It is calculated as follows.


Suppose a complex number z = x + iy is given. Then in the argand plane it will be represented as (x,y). To find its distance from the origin we take the square root of the complex number, i.e.


Check out some of the examples below to understand the modulus of a complex number in depth.


Examples of Modulus of Complex Numbers

1. Calculate the modulus of complex number z = -4 + 3i.

Solution: We know that the modulus of complex number z = -4 + 3i is given by,

$\begin{align} &Z=\sqrt{4^{2}+3^{2}} \\ &Z=\sqrt{16+9} \\ &Z=\sqrt{25} \\ &Z=5 \end{align}$

Therefore, |z| = 5


2. Determine the modulus of the complex number v = (1 + 3i)(-2 + 4i). Use the modulus property to find the modulus of a complex number.

Solution: We have v = (1 + 3i)(-2 + 4i). Assuming, a = (1 + 3i), b = (-2 + 4i).

We know the modulus property |ab| = |a||b|Therefore, we can determine the modulus of complex numbers a and b separately.

$\begin{align} &|a|=\sqrt{1^{2}+3^{2}}=\sqrt{1+9}=\sqrt{10} \\ &|b|=\sqrt{(-2)^{2}+4^{2}}=\sqrt{4+16}=\sqrt{20} \\ &\text { Thus, }|v|=|a b|=|a||b|=\sqrt{10} \sqrt{20}=\sqrt{200}=10 \sqrt{2} \end{align}$

Therefore, the modulus of complex number (1 + 3i)(-2 + 4i) is $10 \sqrt{2}$


Properties of Modulus of Complex Numbers

Listed below are some of the properties of mod of a complex number.

(i) |z1 z2| = |z1||z2|

Proof:  Let us assume z1= m + in and z2 = p + iq

Then, |z1 z2| = | (m + in)(p + iq)|

⇒ |mp+ imq + inp + i2nq|

⇒ |mp + imq + inp – nq|

⇒ |mp – nq + i (mq + np)|

⇒ (mp – nq) 2+ (mq+np) 2

⇒ (mp) 2+ (nq) 2 – 2mnpq + (mq) 2 + (np) 2 + 2mnpq

On solving the above equation we get,

⇒ (m2 + n2) (p2 + p2)

⇒ |z1||z2|.

Thus, |z1 z2| = |z1||z2|

(ii) $\left|\dfrac{z_1}{z_2}\right|=\dfrac{(|z_1|)}{(|z_2|)}$.

Proof: $\left|\dfrac{z_{1}}{ z_{2}}\right|=\left|z_{1} \cdot \dfrac{1}{z_{2}}\right|$

By using the multiplicative property of modulus we get,

$\Rightarrow \left|z_{1}\right|\left|\dfrac{1}{z_{2}}\right| \\ \Rightarrow\left|z_{1}\right|\dfrac{1} {\left(\left|z_{2}\right|\right)} \\ \Rightarrow \dfrac{\left(\left|z_{1}\right|\right)}{\left(\left|z_{2}\right|\right)}$ 

Thus, $\left|\dfrac{z_1}{z_2}\right|=\dfrac{(|z_1|)}{(|z_2|)}$.

Besides, the properties derived above there are also some other important results. These are known as the triangle inequalities and are as listed below:

|z1 + z2| ≤ |z1| + |z2|

|z1 + z2| ≥ |z1| – |z2|

|z1 – z2| ≥ |z1| – |z2|.


Conclusion

The modulus of a complex number is nothing but the distance of a point representing a complex number in the argand plane from the origin. This article summarises the modulus of a complex number and its properties. The article also discusses the representation of the modulus of a complex number in the argand or Gaussian plane. 

The article also discusses how to calculate the modulus of a complex number. Complex numbers is an important topic and a keen attention must be paid on the properties of the modulus of complex numbers when solving the questions related to the same. It must also be noted that the modulus of complex numbers is always positive.

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FAQs on Complex Numbers Detailed Discussion on Modulus for JEE

1. What are some of the properties of argument of a complex number?

The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the argand plane. It is denoted by “and is measured in “radians”, which is its SI unit. 

Now we know in polar form, a complex number is represented as $r(\cos{\theta}+i\sin{\theta})$, where $\theta$ is the argument. The argument function is denoted by arg(z), where z denotes the complex number, i.e. z = a+ ib. Argument can be calculated as: 

$\arg (z)=\tan ^{-1}\left(\dfrac{b}{ a}\right)$

Therefore, $\theta$ is represented as: 

$\theta=\tan ^{-1}\left(\dfrac{y}{ x}\right)$

Now let us discuss a few properties shared by the arguments of complex numbers. If z is a complex number (non-zero) and n is some random integer, then

arg (zn) = n arg (z)

Let us assume, z1 and z2 be the two complex numbers, then the following identities or properties can be used to find the argument.

$\begin{align} &\arg \left(\dfrac{z_{1}}{z_{2}}\right)=\arg \left(z_{1}\right)-\arg \left(z_{2}\right) \\ &\arg \left(z_{1} z_{2}\right)=\arg \left(z_{1}\right)+\arg \left(z_{2}\right) \end{align}$

2. What are some properties of conjugate of a complex number?

The conjugate of a complex number z = a+ ib is denoted as.  For example, the conjugate of the complex number z = 5 -6i is denoted as 5+ 6i.  As for modulus and argument there are some properties of a complex number for finding the conjugate as well. These properties help to make the calculations much easier. The complex conjugate of a complex number is a number with an equal real part and an imaginary part of equal magnitude but opposite sign in mathematics.


Below are some properties of a complex number:

$\begin{align} &\overline{z_{1} \pm z_{2}}=\overline{z_{1}} \pm \overline{z_{2}} \\ &\overline{z_{1} z_{2}}=\overline{z_{1} z_{2}} \\ &\left(\dfrac{\overline{z_{1}}} {\overline{{z_{2}}}}\right)=\dfrac{\overline{z_{1}}} {\overline{z_{2}}} \\ &\overline{\bar{\bar{z}}}=z \\ &\bar{z} \bar{z}=|z|^{2} \end{align}$


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