How to find the Absolute Value of a Complex Number?
Last Updated :
30 Dec, 2023
The distance between the origin and the given point on a complex plane is termed the absolute value of a complex number. The absolute value of a real number is the number itself and is represented by modulus, i.e. |x|.
Therefore the modulus of any value gives a positive value, such that;
|6| = 6
|-6| = 6
Now, finding the modulus has a different method in the case of complex numbers,
Suppose, z = a+ib is a complex number. Then, the modulus of z will be:
|z| = √(a2+b2), when we apply the Pythagorean theorem in a complex plane then this expression is obtained.
Hence, mod of complex number, z is extended from 0 to z and mod of real numbers x and y is extended from 0 to x and 0 to y respectively. Now they form a right-angled triangle, where the vertex of the acute angle is 0.
So, |z|2 = |a|2+|b|2
|z|2 = a2 + b2
|z| = √(a2+b2)
Sample Questions
Question 1: Find the absolute value of the following complex number. z = 2-4i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 2-4i
we have : |z| = √(a2+b2)
here a = 2, b = -4
|z| = √(a2+b2)
= √(22+(-4)2)
= √(4 +16)
= √20
hence the absolute value of complex number. z = 3-4i is 5
Question 2: Find the absolute value of the following complex number. z = 3-9i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 3 – 9i
we have: |z| = √(a2+b2)
here a = 3, b = -9
|z| = √(a2+b2)
= √(32+(-9)2)
= √(9 +81)
= √90
hence the absolute value of complex number. z = 5 – 9i is √90
Question 3: Find the absolute value of the following complex number. z = 2- 7i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given: z = 2 – 7i
we have: |z| = √(a2+b2)
here a = 2, b = -7
|z| = √(a2+b2)
= √(22+(-7)2)
= √(4 +49)
= √53
hence the absolute value of complex number. z = 2 – 7i is √53
Question 4: Perform the indicated operation and write the answer in standard form: (2 + 4i) × (3 – 4i).and find its absolute value?
Solution:
(2 + 4i) × (3 – 4i)
= (6 – 8i + 12i – 16i2)
= 6 + 4i +16
= 22 – 4i
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 22 – 4i
we have : |z| = √(a2+b2)
here a = 22, b = -4
|z| = √(a2+b2)
= √(22)2+(-4)2)
= √(484+ 16)
= √500
hence the absolute value of complex number. z = 22 – 4i is √500
Question 5: Find the absolute value of the following complex number. z = 3 – 3i
Solution:
The absolute value of a real number is the number itself and represented by modulus,
To find the absolute value of complex number,
Given : z = 3 – 3i
we have : |z| = √(a2+b2)
here a = 3, b = -3
|z| = √(a2+b2)
= √(32+(-3)2)
= √(9 +9)
= √18
hence the absolute value of complex number. z = 3 – 3i is √18
Question 6: If z1, z2 are (1 – i), (-2 + 2i) respectively, find Im(z1z2/z1).
Solution:
Given: z1 = (1 – i)
z2 = (-2 + 2i)
Now to find Im(z1z2/z1),
Put values of z1 and z2
Im(z1z2/z1) = {(1 – i) (-2 + 2i)} / (1 – i)
= {( -2 +2i +2i -2i2)} / (1-i)
= {(-2 + 4i + 2) / (1 – i)
= {(4i) /(1 – i)}
= {(0+4i) (1 + i)} / {(1 + i)(1- i)}
= {(4i + 4i2) / (1 + 1)
= 4i -4 / 2
=(-4 + 4i) / 2
= -4/2 + 4/2 i
= -2 + 2i
Therefore, Im(z1z2/z1) = 2
Question 7: Perform the indicated operation and write the answer in standard form: (2 – 7i)(3 + 7i)
Solution:
Given: (2 – 7i)(3 + 7i)
= {6+ 14i – 21i – 49i2}
= (-7i +55)
= 55 -7i
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