Complex number is the sum of a real number and an imaginary number. These are the numbers that can be written in the form of a+ib, where a and b both are real numbers. It is denoted by z.

Here the value ‘a’ is called the real part which is denoted by Re(z), and ‘b’ is called the imaginary part Im(z). In complex numbers form a +bi, ‘i’ is an imaginary number called “iota”.

The value of i is (√-1) or we can write as i² = -1.

For example:

• 7+15i is a complex number, where 7 is a real number (Re) and 15i is an imaginary number (Im).
• 8 + 5i is a complex number where 8 is a real number (Re) and  5i is an imaginary number (im)

Absolute value of a complex number

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;

|5| = 5

|-5| = 5

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)

Find the absolute value of the complex number z = 3 – 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 = 3-4i 

We have : |z| = √(a²+b²)

Here a = 3, b = -4

 |z| = √(a²+b²)

     = √(3²+(-4)²)

     = √(9 +16)

     = √25

     = 5

Hence the absolute value of complex number z = 3-4i is 5.

Similar Questions

Question 1: Find the absolute value of the following complex number. z = 5-9i

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 = 5 – 9i

We have: |z| = √(a²+b²)

Here a = 5, b = -9

|z| = √(a²+b²)

    = √(5²+(-9)²)

    = √(25 +81)

    = √106

Hence the absolute value of complex number z = 5 – 9i is √106.

Question 2: Find the absolute value of the following complex number z = 2- 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 = 2 – 3i

We have: |z| = √(a²+b²)

here a = 2, b = -3

|z| = √(a²+b²)

   = √(2²+(-3)²)

   = √(4 +9)

   = √13

hence the absolute value of complex number z = 2 – 3i is √13.

Question 3: Perform the indicated operation and write the answer in standard form: (5 + 4i) × (6 – 4i).and find its absolute value?

Solution:

(5 + 4i) × (6 – 4i)

= (30 -20i +24i – 16i²)

= 30 + 4i +16

= 46 – 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 = 46 – 4i

we have : |z| = √(a²+b²)

here a = 46 , b = -4

|z| = √(a²+b²)

    = √(46)²+(-4)²)

    = √(2116+ 16)

    = √2132

Hence the absolute value of complex number. z = 46 – 4i is √2132

Question 4: Find the absolute value of the following complex number. z = 3 – 5i

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 – 5i

We have : |z| = √(a²+b²)

here a = 3, b = -5

|z| = √(a²+b²)

    = √(3²+(-5)²)

    = √(9 +25)

    = √34

hence the absolute value of complex number z = 3 – 5i is √34

Question 5: 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 + 2i / 2

= (-4 + 2i) / 2 

= -4/2 + 2/2 i

= -2 + i

Therefore, Im(z1z2/z1) = 1

Question 6: Perform the indicated operation and write the answer in standard form: (2 – 7i)(2 + 7i)  

Solution: 

Given: (2 – 7i)(2 + 7i)  

= {4 + 14i – 14i – 49i²}

= (4 +49)

= 53 + 0i