Skip to content
Related Articles

Related Articles

Find the real and imaginary parts of ei/2

View Discussion
Improve Article
Save Article
  • Last Updated : 28 Nov, 2021
View Discussion
Improve Article
Save Article

Complex numbers are the superset of real numbers. Or we can say that complex numbers are part of the number system in mathematic. In 1799 a mathematician Caspar Wessel first discovered complex numbers. Much later Euler introduced the concept of naming i to √-1. Complex numbers can be represented in the following way:

z = a + ib

where a and b are the real numbers and i is an imaginary number which is also known as iota and its value is√-1. For example, consider the number 2/5. This number can be written as 2/5 + i*0, where a = 2/5 and b = 0. One interesting thing about complex numbers is that graphically multiplying i to a vector will make the vector rotate anticlockwise by 90°. 

Classification of Complex Numbers

Complex numbers are classified into the following types:

1. Zero complex number: Here, a = 0, b = 0 so z = 0 + i0. For example, 0.

2. Purely real number: Here, a ≠ 0, b = 0 so z = a + i0. For example, 5, 7, 8.

3. Purely imaginary number: Here, a = 0 , b ≠ 0 so z = 0 + ib. For example, 9i, -3i, 2i.

4. Imaginary number: Here, a ≠ 0, b ≠ 0 so z = a + ib. For example, 2 + 3i, 3 – 13i.

Euler’s Formula

This formula is used to establish the relationship between trigonometric function and exponent function. The Euler formula is

 eix = cos(x) + i * sin(x)

or 

eiπ as cos π + i * sin π

Or we can say that if any complex number is in the form eix, then it can be written as cos(x) + i * sin(x). This is called the Euler formula. Here the real part is cos x and the imaginary part is isin x.

Find the real and imaginary parts of ei/2

Solution:

Let the expression ei/2 be y.

Therefore t can be written as exp(i/2)

or, t = exp(i * 1/2)

or, t = cos(1/2) + i sin(1/2)

or, t = 0.87758256189 + i * 0.4794255386

Therefore real part is 0.87758256189 and the imaginary part is 0.4794255386.

Sample Problems

Question 1: Find the imaginary and real part of eiπ

Solution:

From Euler’s formula, we can write eiπ as cos π + i * sin π

cos π = -1 

sin π = 0

Therefore imaginary part is 0 and the real part is -1

So the equation becomes eiπ +1 = 0, this beautiful equation is called Euler’s identity.

Question 2: Find the imaginary and real part of 5 + i6.9

Solution:

This problem is fairly straightforward.  When we are given a complex number like this, 

it is very easy to write the real and imaginary part of it.

imaginary part of the complex number = 6.9

real part of the complex number= 5

Question 3: Find the real and imaginary part of the complex number 50.

Solution:

If a real number is given as a complex number then it is clear that the complex number does not have an imaginary part.

So the imaginary part of the complex number is 0

And, the real part of the complex number is 50.

Question 4: Find the real and imaginary part of the complex number 9i.

Solution:

If a complex number is given in the form xi then it doesn’t have a real part.

That is real part of the complex number 9 is 0

Imaginary par is 9i 

Question 5: Find the real and imaginary part of the complex number (2 + 3i)/(1 + i)

Solution:

In this type of problem, we need to remove the i from the denominator.

If a complex number is given as the ratio of two different complex numbers, then multiply the numerator and 

denominator with the conjugate

The complex conjugate of a complex number is the number itself but with opposite sign. 

For example, there complex conjugate of a number a + ib is a – ib.

So the complex conjugate of the denominator is 1 – i.  

Multiplying this with numerator and denominator we will get, 

((2 + 3i) * (1 – i)) / (1 + i) * (1 – i)

= ((2 + 3i) * (1 – i)) / (1 – i2)

= ((2 + 3i) * (1 – i)) / (1 – (-1)) 

= ((2 + 3i) * (1 – i)) / 2 

= (2(1 – i) ) / 2 + (3i * (1 – i))/2

= 1 – i + 3i/2 + 3/2

= 5/2 + i/2

 = 2.5 + 0.5

Therefore the real part of the complex number is 2.5

and the imaginary part of the complex number is 0.5

My Personal Notes arrow_drop_up
Recommended Articles
Page :

Start Your Coding Journey Now!