Absolute value (or modulus) of a complex number z = a + bi, where a and b are real numbers, is a measure of its distance from the origin in the complex plane. The absolute value is denoted by ∣z∣.
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Absolute Value of Complex Number
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, the mod of the complex number, z is extended from 0 to z and the 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)
How to Find Absolute Value of a Complex Number?
Steps to Calculate the Absolute Value of a Complex Number are:
Step 1: Identify the real part (a) and the imaginary part (b) of the complex number z = a + bi.
Step 2: Square both the real part and the imaginary part: a2 and b2.
Step 3: Add the squares: a2 + b2.
Step 4: Take the square root of the sum: √(a2 + b2)
Read More,
- What is Z Bar in Complex Numbers?
- Polar and Exponential Forms of Complex Numbers
- Is Every Real Number a Complex Number?
Examples on Absolute Value of a Complex Number
Example 1: Find the absolute value of the following complex number. z = 2-4i
Solution:
Absolute value of a real number is the number itself and is represented by modulus,
To find the absolute value of the 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
Example 2: Find the absolute value of the following complex number. z = 3-9i
Solution:
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, absolute value of complex number. z = 5 - 9i is √90
Example 3: Find the absolute value of the following complex number. z = 2- 7i
Solution:
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, absolute value of complex number. z = 2 - 7i is √53
Example 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
Absolute value of a real number is the number itself and represented by modulus,
To find 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 absolute value of complex number. z = 22 - 4i is √500
Example 5: Find the absolute value of the following complex number. z = 3 - 3i
Solution:
Absolute value of a real number is the number itself and represented by modulus,
To find 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, absolute value of complex number. z = 3 - 3i is √18
Example 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
Example 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