How to find the Absolute Value of a Complex Number? (original) (raw)

Last Updated : 23 Jul, 2025

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|.

Absolute-Value-of-Complex-Number

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 = a 2 + b 2

****|z| = √(a** 2 +b 2 )

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,

**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 z 1 , z 2 are (1 - i), (-2 + 2i) respectively, find Im(z 1 z 2 /z 1 ).

**Solution:

Given:

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:

= {6+ 14i - 21i - 49i2}

= (-7i +55) = 55 -7i