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Modulus and Conjugate of a Complex Number | Algebra - Mathematics PDF Download

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 | Algebra - Mathematicssuch thatModulus and Conjugate of a Complex Number | Algebra - Mathematics= a – ib. We call Modulus and Conjugate of a Complex Number | Algebra - Mathematics 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 | Algebra - Mathematics= (a + ib)×(a – ib)
Hence, zModulus and Conjugate of a Complex Number | Algebra - Mathematics = {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 | Algebra - Mathematics

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 | Algebra - Mathematics as the square of the Absolute value or modulus of a complex number. Therefore, we can write zModulus and Conjugate of a Complex Number | Algebra - Mathematics=|z|2
This is done because as we just discussed, zModulus and Conjugate of a Complex Number | Algebra - Mathematicsgives 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 | Algebra - Mathematics…(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 | Algebra - Mathematics
• 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 | Algebra - Mathematics
Modulus and Conjugate of a Complex Number | Algebra - Mathematicswith the accord that z2 ≠ 0

Solved Examples For You
Example 1: If z = 2 – i, findModulus and Conjugate of a Complex Number | Algebra - Mathematics?
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 | Algebra - Mathematics)= (Modulus and Conjugate of a Complex Number | Algebra - Mathematics)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 | Algebra - Mathematics = 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 | Algebra - Mathematics.
Solution: We know that Modulus and Conjugate of a Complex Number | Algebra - Mathematics
Therefore, we haveModulus and Conjugate of a Complex Number | Algebra - Mathematics [using the definition of conjugate]
Hence, Modulus and Conjugate of a Complex Number | Algebra - Mathematics

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

The document Modulus and Conjugate of a Complex Number | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Modulus and Conjugate of a Complex Number - Algebra - Mathematics

1. What is the modulus of a complex number?
The modulus of a complex number is the distance between the origin and the point representing the complex number in the complex plane. It can be calculated as the square root of the sum of the squares of the real and imaginary parts of the complex number.
2. How do I calculate the modulus of a complex number?
To calculate the modulus of a complex number, you can use the formula: modulus = √(real^2 + imaginary^2). Simply square the real part, square the imaginary part, add them together, and take the square root of the sum.
3. What is the conjugate of a complex number?
The conjugate of a complex number is obtained by changing the sign of its imaginary part. For example, if a complex number is written as a + bi, its conjugate would be a - bi.
4. How do I find the conjugate of a complex number?
To find the conjugate of a complex number, simply change the sign of its imaginary part. If the complex number is written as a + bi, its conjugate would be a - bi.
5. What is the relationship between the modulus and conjugate of a complex number?
The modulus of a complex number and its conjugate are related in the following way: if z is a complex number, then the product of z and its conjugate is equal to the square of its modulus. In other words, |z|^2 = z * z̅.
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