Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Modulus and Conjugate of a Complex Number

ffImage
Last updated date: 14th Jul 2024
Total views: 88.5k
Views today: 0.88k
hightlight icon
highlight icon
highlight icon
share icon
copy icon

Know About Modules and Conjugate of a Complex Number

In mathematics, a complex number is said to be a number that can be expressed in a + bi form, where a and b are real numbers and i is the imaginary unit.


Here, a is named as the real part of the number, and b is referred to as the imaginary part of a number.


The following table provides a representation of the complex numbers.


Representation of the Complex Numbers

Complex Number

Standard Form 

(a + bi)        

Explanation  

5i + 7  

7 + 5i

Real part is 7 and the imaginary part is 5 

2i

0 + 2i

Here the real part is 0 and the imaginary part is 2

-3 – 5i 

-3 + (-5)i

Here the real part is -3 and the imaginary part is -5


Modulus and conjugate of a complex number are discussed in detail in chapter 5 of class 11 NCERT book of mathematics. It is a very complex concept and therefore students who want to make a strong foundation of The concept of modulus and conjugate of complex numbers should go through the notes provided by Vedantu, these are thoroughly researched notes and are up-to-date as the CBSE keeps on making minute changes to the syllabus every year. Students who are preparing for IIT, JEE examinations and want to get a glance over the basic concepts that were discussed in class 11 and 12 mathematics and science can also refer to these notes provided by Vedantu These notes will give you a comprehensive understanding of complex numbers and they act as a reference guide for students preparing during the examination period.


We have studied earlier that the square of a real number is not negative, this chapter mainly deals with finding the square root of negative numbers as the square numbers cannot really have a real square root. Iota or i for positive square root is a concept that was introduced by Euler. The chapter complex numbers and quadratic equations mainly deal with iota or i.


The modulus of a complex number helps to calculate the distance of the complex number from the origin in the argand plane, while, complex conjugate helps to find out the polynomial roots and the conjugate of a complex number gives the deflection of the real axis in the argand plane.


Key Concepts studied in Relation to Modulus and Conjugate of a Complex Number-

  • Imaginary numbers

  • Integral powers of i

  • Complex numbers

  • Algebra of complex numbers

  • Addition of complex numbers

  • Multiplication of complex numbers

  • Conjugate of a complex number

  • Modulus of a complex number

  • Properties of modulus of a complex number

  • Argand Plane

  • The polar form of a complex number

  • Solution of a quadratic equation


What is the Modulus of Complex Numbers?

Modulus of complex number defined as | z | where if z = a + bi is a complex number. The modulus of the complex number will be defined as follows:                

\[ | z | =a + bi  | z | =0 \] then it indicates a=b=0 

\[ | -z | = | z | \]

Imagine \[z_{1}~and~z_{2}\] are two complex numbers, then 

\[| z_{1}.z _{2} | = | z_{1} | | z _{2} |\]

\[| z_{1} + z _{2} | \leq  | z_{1} | + | z _{2} |\]

\[\left | \frac{z_{1}}{z _{2}} \right | = \frac{\left | z _{1} \right |}{\left | z _{2} \right |}\]


Modulus of a Complex Number

There seems to be a method to get a sense of how large these numbers are. We consider the conjugate complex and multiply it by the complex number specified in (1). Therefore we describe the product \[\overline{zz}\] as the square of a complex number's Absolute value or modulus. So let's write \[\overline{zz}\] =  \[\left | z \right |^{2}\].


As per the explanation, \[\overline{zz}\] provides a calculation of the absolute value or magnitude of the complex number. When you learn about the Argand Plane, the exact explanation for that concept will become apparent.


Therefore, \[\left | z \right |^{2}\] = \[a^{2} + b^{2}\]   Using(1)


Hence, \[\left | z \right | = \sqrt{a^{2} + b^{2}} ...(2)\]


The equation above is the modulus or absolute value of the complex number z.


Conjugate of a Complex Number

The complex conjugate of a complex number is the number with the same real part and the imaginary part equal in magnitude, but are opposite in terms of their signs. 


Complex conjugates are responsible for finding polynomial roots. According to the complex conjugate root theorem, if a complex number in one variable with real coefficients is a root to a polynomial, so is its conjugate.


How to Find Conjugate of a Complex Number

If you are wondering how to find the conjugate of a complex number, then go through this. Each complex number has a relationship with another complex number known as its complex conjugate. You can find the conjugate complex by merely changing the symbol of the imaginary part of the Complex numbers.


For Example

We alter the sign of the imaginary part to find the complex conjugate of 4 + 7i. So the complex conjugate is 4 − 7i.


Example: 

We alter the sign of the imaginary part to find the complex conjugate of 1−3i. So the complex conjugate is 1 + 3i.


Example:

We alter the sign of the imaginary component to find the complex conjugate of −4 − 3i. So the complex conjugate is −4 + 3i.


It is to be noted that the conjugate complex has a very peculiar property.


If we multiply a complex number by its complex conjugate, think about what will happen.


Let’s take this example:

Multiply \[(4 + 7i) by (4 - 7i): (4 + 7i)(4 - 7i) = 16 - 28i + 28i - 16 + 49i^{2} = 65\]


We find the answer to this is a strictly real number; there is no imaginary part. It often occurs when a complex number is multiplied by its conjugate, the consequence is a real number.


Modulus of the Sum of Two Complex Numbers

To add two complex numbers of the x plus iy form, we have to add the real parts and the imaginary parts individually.


Let z = a + ib reflect a complex number. Module of z , referred to as z, is defined as the real number \[(a^{2} + b^{2})^{\frac{1}{2}} z = (a^{2} + b^{2})^{\frac{1}{2}}\]


Conjugate of Complex Number Class 11

Numerical: Evaluate the modulus of (3-4i)

\[z = (a^{2} + b^{2})^\frac{1}{2} = (3^{2} + 4^{2})^\frac{1}{2} = 5\]

Let z = a + ib reflect a complex number. Z conjugate is the complex number

 a - ib, i.e., = a - ib.

Z * = Z

Or Z–1 = / Z (Useful to find a complex number in reverse)


Properties of Complex Numbers

Properties of complex numbers are mentioned below:

1. In case of a and b are real numbers and a + ib = 0 then a = 0, b = 0

Proof:

According to the property, 

a + ib = 0 = 0 + i ∙ 0,

Therefore, we conclude that x = 0 and y = 0.


2. For any three the set complex numbers z1, z2 and z3 satisfies the commutative, associative and distributive laws.

  1. \[z_{1} + z_{2} = z_{2} + z_{1}\] (Commutative law for addition)

  2. \[z_{1} . z_{2} = z_{2} . z_{1}\] (Commutative law for multiplication)

  3. \[(z_{1} + z_{2}) + z_{3} = z_{1} (z_{2} + z_{3})\] (Associative law for addition)

  4. \[(z_{1}z_{2}) + z_{3} = z_{1} (z_{2}z_{3})\] (Associative law for multiplication)

  5. \[ ( z_{1}(z_{1} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}\] (Distributive law)


3. The sum of two complex conjugate numbers is real.

Proof:

Let, z = a + ib (a, b are real numbers) be a complex number. Then, a conjugate of z is\[\overline{z}\] = a - ib.

Now, z + \[\overline{z}\]  = a + ib + a - ib = 2a, which is real.


4. The product of two complex conjugate numbers is real.

Proof:

Let, z = a + ib (a, b are real numbers) be a complex number. Then, a conjugate of z is 

\[\overline{z}\] = a - ib.


\[z * \overline{z}\] = \[(a + ib)(a - ib) = a^{2} - i^{2} b^{2} = a^{2} + b^{2}, (Since i^{2} =  -1)\], which is real.


Thus, the modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number.


5. When the sum of two complex numbers is real, and the product of two complex numbers is also natural, then the complex numbers are conjugated.

Proof:

According to the property,

\[z_{1} + z_{2}\] = a+ ib + c + id = (a + c) + i(b + d) is real.

Therefore, b + d = 0

\[\Rightarrow  d = -b \]

And,

\[z_{1}, z_{2}\] = (a + ib)(c + id) = (a + ib)(c +id) = (ac– bd) + i(ad + bc) is real.

Therefore, ad + bc = 0

\[\Rightarrow  -ab + bc = 0 (since, d = -b)  \]

\[\Rightarrow  b (c - a) = 0 \]

\[\Rightarrow  c = a (since, b  \ne 0)  \]

Hence, \[z_{2}\] = c + id = a + i(-b) = a - ib =  \[\overline{z1}\]

Therefore, we conclude that z1 and z2 are conjugate to each other.

\[|z_{1} + z_{2}| \leq |z_{1}| + |z_{2}|\], for two complex numbers \[z_{1} and z_{2}\]

FAQs on Modulus and Conjugate of a Complex Number

1. What is the modulus of a complex number?

Modulus of complex number defined as | z | where if z = a + bi is a complex number. The modulus of the complex number will be defined as follows:                     

| Z | =a + bi 

| z | =0 then it indicates a=b=0 

| -z | = | z |

Imagine \[z_{1} and z_{2}\] are two complex numbers, then 

\[|z_{1} . z_{2}| = |z_{1}|  |z_{2}|\]

\[|z_{1} + z_{2}| \leq |z_{1}| + |z_{2}|\]

\[|z_{1} / z_{2}| = |z_{1}| / |z_{2}|\]

2. What is the conjugate of a complex number?

Complex conjugate helps to find out the polynomial roots and the conjugate of a complex number gives the deflection of the real axis in the argand plane. Conjugation of a complex number can be further studied in detail in the study notes provided by Vedantu.

3. Where can I find the detailed notes on modulus and conjugate of a complex number?

The detailed notes on modulus and conjugate of a complex number are available on Vedantu’s website, the study material can be accessed by downloading the link for free in a pdf format. The notes are of high quality and are up to date as per the guidelines set by the CBSE.

4. How to find the conjugate of a complex number?

Each complex number has a relationship with another complex number known as its complex conjugate. You can find the conjugate complex by merely changing the symbol of the imaginary part of the Complex numbers.


For example, We alter the sign of the imaginary part to find the complex conjugate of 4 + 7i. So the complex conjugate is 4 − 7i.


Example: We alter the sign of the imaginary part to find the complex conjugate of 1−3i.

So the complex conjugate is 1 + 3i.

5.  What do you mean by the algebra of complex numbers?

There are certain equations that have no real solution such as the equation \[x^{2} = -1\]. These kinds of equations arise naturally and therefore it is important to use its roots. This compels us to create a different symbol for the roots which we refer to as a complex number. In such a case the solution to the equation  \[x^{2} = -1\]  is the symbol +,- i. These new numbers are considered to be complex numbers which can be written as follows- root -1 = +,-i.


It is important to understand the fundamental theorem of algebra in order to use complex numbers. If we use complex roots it helps us to analyze that there are expected numbers of roots of every polynomial.


The fundamental theorem of algebra mainly asserts that there are exactly n complex numbers in a polynomial of degree n, it is a situation where the repeated roots have to be counted with multiplicity.


Complex numbers are extremely useful mathematical obstructions, students perceive this topic as extremely difficult as complex numbers appear more confusing and mysterious than real numbers which are written in an extremely simplified manner such as 2 or 341.07. But we must understand that all complex numbers are equally meaningful.

6. What is a Complex Conjugate Number?

Complex numbers are called a complex conjugate of each other when in two complex numbers, the sign of the imaginary part is differing. Complex numbers in the binomial form are depicted as (a + ib). It invites you to work with the "+" symbol. What about if we turn it to a minus sign? Let z = a + ib represent a complex number. We describe another complex number \[\overline{z}\] such that \[\overline{z} = a – ib\]. We are calling \[\overline{z}\] or even the complex number acquired by altering the symbol of the imaginary part (positive to negative or vice versa), as the conjugate of z. Let's just find the product \[\overline{zz} = (a + ib) * (a - ib)\].


Thus, \[\overline{zz} = {a^{2} - i(ab) + i(ab) + b^{2}} = (a^{2} + b^{2}) ....(1)\]


If a and b are big numbers, then the sum in (1) becomes more significant. And one can use this equation to determine a complex number's value.