Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev

Algebra for IIT JAM Mathematics

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Mathematics : Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev

The document Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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Conjugate of A Complex Number
Complex numbers are represented in a binomial form as (a + ib). It almost invites you to play with that ‘+’ sign. What happens if we change it to a negative sign? Let z = a + ib be a complex number. We define another complex numberModulus and Conjugate of a Complex Number Mathematics Notes | EduRevsuch thatModulus and Conjugate of a Complex Number Mathematics Notes | EduRev= a – ib. We call Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev or the complex number obtained by changing the sign of the imaginary part (positive to negative or vice versa), as the conjugate of z. Let us now find the product zModulus and Conjugate of a Complex Number Mathematics Notes | EduRev= (a + ib)×(a – ib)
Hence, zModulus and Conjugate of a Complex Number Mathematics Notes | EduRev = {a2 -i(ab) + i(ab) + b2} = (a2 + b2)…(1)
If a and b are large numbers, the sum in (1) will be greater. So one can use this equation to measure the value of a complex number.
Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev

Modulus of A Complex Number
There is a way to get a feel for how big the numbers we are dealing with are. We take the complex conjugate and multiply it by the complex number as done in (1). Hence, we define the product zModulus and Conjugate of a Complex Number Mathematics Notes | EduRev as the square of the Absolute value or modulus of a complex number. Therefore, we can write zModulus and Conjugate of a Complex Number Mathematics Notes | EduRev=|z|2
This is done because as we just discussed, zModulus and Conjugate of a Complex Number Mathematics Notes | EduRevgives a way to measure the Absolute value or magnitude of our complex number. The actual reason for this definition will get clear once you learn about the Argand Plane.
Therefore, |z2 = (a2 + b2 ) [Using (1)]
Hence, |z| =Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev…(2)
The above equation represents the modulus or the absolute value of our complex number z.

Important Identities
• If |z|2 = 1 = |z|, i.e. z is a complex number of unit modulus, then from (1), we have
Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev
• Also If z1 and z2 is any complex number, we have the following identities:
|z1 z2| = |z1| |z2|
•Modulus (z1 / z2)= Modulus of z1 /Modulus of z2
Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev
Modulus and Conjugate of a Complex Number Mathematics Notes | EduRevwith the accord that z2 ≠ 0

Solved Examples For You
Example 1: If z = 2 – i, findModulus and Conjugate of a Complex Number Mathematics Notes | EduRev?
Solution: All we need to do is to change the sign of the imaginary term. Therefore, we denote it as follows:z = 2 – i or (Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev)= (Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev)Now ¯ or ‘bar’ or ‘dagger’ becomes an operation that transforms the imaginary part of a complex number in sign. In other words, we can write: Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev = 2 + i. Note that there is no change in the sign of 2.

Example 2: If z = a+ib, show that |z2 = zModulus and Conjugate of a Complex Number Mathematics Notes | EduRev.
Solution: We know that Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev
Therefore, we haveModulus and Conjugate of a Complex Number Mathematics Notes | EduRev [using the definition of conjugate]
Hence, Modulus and Conjugate of a Complex Number Mathematics Notes | EduRev

Example 3: The maximum value of ∣z∣ when z satisfies the condition ∣z+ 2/z ∣ = 2 is:
A) 1
B) √3+1
C) √3–1
D) -1
Solution: (B) We have the condition that modulus (z+2/z) = 2. Therefore, can we not treat this as an equation in one variable? We can but the equation would be second order in z and we won’t be able to get a relation for |z| alone. So we do the following:
|z| can be written as =|z+[2/z]–[2/z]|≤|z+[2/z]|≤|z+[2/z]|+ 2/|z|
Therefore,|z|≤2+2/|z| Or |z|≤(2|z|+2)/|z|
Hence,|z|2 ≤ 2|z|+2 Or |z|2 – 2|z| ≤ 2
To get the value of |z|, let’s convert the LHS of the above equation into a perfect square. After adding 1 on both the side, we have |z|2–2|z|(1)+(1)2 ≤  2 +(1)2
(|z| – 1)2 ≤ 3
Therefore,–√3 ≤ (|z| – 1) ≤ √3
Hence, the maximum value of |z| will be √3 + 1

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