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∣.

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

FAQs on Finding Absolute Value of a Complex Number

What is the absolute value of a complex number?

Absolute value (or modulus) of a complex number z = a + bi is a measure of its distance from the origin in the complex plane. It is calculated using the formula ∣z∣ = a² + b², where ‘a‘ is the real part and ‘b‘ is the imaginary part.

What is the Absolute Value of 3+4i?

Absolute value of 3 + 4i is, √(3² + 4²) = √(25) = 5.

What is the Absolute Value of -1-i?

Absolute value of -1 – i is, √(1² + 1²) = √(2).

How do you Find the Absolute Value of a Complex Number in Polar Form?

If a complex number is given in polar form as z = r(cos⁡θ + isin⁡θ), the absolute value ∣z∣ simply the magnitude ‘r’. This is because the polar form represents the number’s distance r from the origin directly.