Sample Problems on Absolute Value of a Complex Number

Q1: Find the absolute value of z = 4 + 8i

Solution:

Given complex number is z = 4 + 8i

As we know that the formula of absolute value is

|z| = √ (a² + b²)

So, a = 4, and b = 8, we get

|z| = √(4² + 8²)

|z| = √80

Q2: Find the absolute value of z = 2 + 4i

Solution:

Given complex number is z = 2 + 4i

As we know that the formula of absolute value is

|z| = √ (a² + b²)

So, a = 2, and b = 4, we get

|z| = √(2² + 4²)

|z| = √20

Q3: Find the angle of the complex number: z = √3 + i

Solution:

Given complex number is z = √3 + i

As we know, that 

θ = tan^-1(b/a)

So, a = √3 , and b = 1, we get

θ = tan^-1(1/ √3 )

θ = 30°

Q4: Find the angle of the complex number: z = 6 + 6i

Solution:

Given complex number is z = 6 + 6i

As we know, that 

θ = tan^-1(b/a)

So, a = 6 , and b = 6, we get

θ = tan^-1(6/6)

θ = 45°

Q5: Convert z =  5 + 5i into polar form

Solution:

Given complex number is z = 5 + 5i

As we know that 

Z = r(cos θ + isin θ) …(1)

Now, we find the value of r

r = √(5² + 5²)

r = √(25 + 25)

r = √50

Now we find the value of θ

θ = tan^-1(5/5)

θ = tan^-1(1)

θ = 45°

Now put all these values in eq(1), we get

Z = √50(cos 45° + isin 45°)

Q6: Convert z =  2 + √3i into polar form

Solution:

Given complex number is z = 2 + 2√3i

As we know that 

Z = r(cos θ + isin θ) …(1)

Now, we find the value of r

r = √(2² + (2√3)²)

r = √(4 + 12)

r = √16

r = 4

Now we find the value of θ

θ = tan^-1(2√3/2)

θ = tan^-1(√3)

θ = 60°

Now put all these values in eq(1), we get

Z = 4(cos 60° + isin 60°)

Absolute value, or modulus, of a complex number measures its distance from the origin on the complex plane. If you have a complex number z = a + ib, where a is the real part while ib is the imaginary part of the complex number in which i is known as iota and b is a real number. Then the absolute value of x is denoted as |z|.

• Real Number: A real number is a number that is present in the number system which can be positive, negative, integer, rational irrational, etc. For example, 23, -3, 3/6.

• Imaginary Number: Imaginary numbers are those numbers that are not real numbers. For example, √3, √11, etc.

• Zero Complex Number: A zero complex number is a number that has its real and imaginary parts both equal to zero. For example, 0+0i.