# 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 + *i*0*. *For example, 0.

2. Purely real number: Here, a ≠ 0, b = 0 so z = a + *i*0*.* 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 + *i*b. 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

^{i}^{x }= cos(x) +i* sin(x)or

e

^{i}^{π}as cos π +i* sin π

Or we can say that if any complex number is in the form e^{i}^{x}, 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 *i*sin 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**^{i}^{π}

**Solution:**

From Euler’s formula, we can write e

^{i}^{π}as cos π +i* sin πcos π = -1

sin π = 0

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

So the equation becomes e

^{i}^{π }+1 = 0, this beautiful equation is called Euler’s identity.

**Question 2: Find the imaginary and real part of 5 + ****i****6.9**

**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 9****i.**

**Solution:**

If a complex number is given in the form x

then it doesn’t have a real part.iThat is real part of the complex number 9

iis 0Imaginary par is 9

i

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

**Solution:**

In this type of problem, we need to remove the

ifrom 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 + 3

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

i) * (1 –i)) / (1 –i^{2})= ((2 + 3

i) * (1 –i)) / (1 – (-1))= ((2 + 3

i) * (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

iTherefore the real part of the complex number is 2.5

and the imaginary part of the complex number is 0.5