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How to find the Absolute Value of a Complex Number?

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  • Last Updated : 21 Mar, 2022

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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