Find the real and imaginary parts of ei/2
Last Updated :
28 Nov, 2021
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 eiÏ€ as cos Ï€ + i * sin Ï€
cos π = -1
sin π = 0
Therefore imaginary part is 0 and the real part is -1
So the equation becomes eiÏ€ +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
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