Find the real and imaginary parts of e^i/2

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

 e^ix = cos(x) + i * sin(x)

or 

e^iπ as cos π + i * sin π

Or we can say that if any complex number is in the form e^ix, 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 e^i/2

Solution:

Let the expression e^i/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 e^iπ

Solution:

From Euler’s formula, we can write e^iπ 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 e^iπ +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 9i  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 + 0.5i 

Therefore the real part of the complex number is 2.5

and the imaginary part of the complex number is 0.5