Unlike natural numbers, real numbers include all rational and irrational numbers. Thus, the real number consists of positive and negative numbers from the number line. A set of real numbers consists of various numerical categories. The imaginary number definition is that they do not exist for physical demonstration or confirmation. The combination of such real and imaginary numbers form complex numbers. The imaginary part of a complex number was introduced to solve problems with no real solution, like the square root of -1. 

Real numbers

• They comprise numbers that come under the categories of the whole, natural, rational, irrational numbers and integers. 
• The real numbers exhibit the basic properties of association, Commutation, distribution and identity. 

Imaginary numbers

• Imaginary numbers are not present for any physical confirmations.
• The imaginary number is represented as ‘i’.
• The imaginary number was introduced to solve the problems that did not have a real answer.
• The imaginary numbers definition is that they are so unreal and are valued as the square root of -1.
• Imaginary roots are half the set of complex numbers.
• All imaginary numbers are complex, but not all complex numbers are imaginary.

Complex numbers

• Complex numbers are the combo of both imaginary and real numbers. 
• It helps know the square root of the minus signed numerical.
• The general format of a complex number is a+bi, where a and b are real numbers, and ‘i’ is the imaginary part of a complex number.

Imaginary parts of a complex number

• The imaginary part, ‘i’ that is in a complex number, can be expressed as √-1.
• An imaginary number is distinct from real numbers because the square of a positive or negative number gives a positive number, anyway.
• But, in imaginary numbers, the square root of -1 is -1.
• They are the negative square of the square roots.
• The ‘i’, denoting the imaginary part of the complex number, is called iota.

Values of i

‘i’ is the imaginary part of the complex number that has a square root value of -1. From this value, the values of ‘i’ with other powers can be found as follows.

• i = √-1
• i2 = -1
• i3 = i.i2 = i(-1) = -i
• i4 = (i2)2 = (-1)2 = 1
• i4n = 1
• i4n + 1 = i
• i4n + 2 = -1
• i4n + 3 = -i

Mathematical Operations on imaginary numbers

The basic mathematical operations on an imaginary are similar to that of the other real numbers. Consider two complex numbers (a+bi) and (c+di)

Addition

When the complex numbers add up,

a+bi + c+di 

Group the real and imaginary parts separately.

a + c + bi + di

Group them under parentheses.

(a + c) + (bi + di)

(a + c) + i(b + d)

Subtraction 

When the complex numbers are subtracted,

(a+bi) – (c+di) 

Simplify and Group the real and imaginary parts separately.

a + bi – c – di

a – c + bi – di

Group them under parentheses.

(a – c) + (bi – di)

(a – c) + i(b – d)

Multiplication 

When the complex numbers are multiplied,

(a+bi)(c+di) 

Simplify and Group the real and imaginary parts separately.

ac+bci+adi+bdi2

ac+bci+adi-bd

(ac-bd) + i(bc+ad)

Division 

When the complex numbers are divided,

(a+bi) / (c+di) 

Multiply with the conjugate of the imaginary part.

(a+bi) (c-di) / (c+di) (c-di)

Simplify and Group the real and imaginary parts separately.

[(ac+bd)+ i(bc-ad)] / (c2+d2) 

Solved examples on the imaginary part of a complex number

Here are some solved examples for your better understanding and clarity.

1. Add and subtract the complex numbers (4+2i) and (5-6i).

Addition 

(4+2i) + (5-6i)

= 4 + 2i + 5 – 6i

= 4 + 5 + 2i – 6i

(4+2i) + (5-6i) = 9 – 4i

Subtraction 

(4+2i) – (5-6i)

= 4 + 2i – 5 + 6i

= 4 – 5 + 2i + 6i

(4+2i) – (5-6i) = -1 + 8i or 8i + 1

2. Multiply the complex numbers (1+i) and (2+2i).

(1+i)(2+2i)

= 2(1+i) + 2i(1+i)

= 2 + 2i + 2i + 2i2

= 2 + 2i + 2i + 2(-1)

= 2 + 2i + 2i – 2

= 2 – 2 +2i + 2i

= 0 + 4i

(1+i)(2+2i) = 4i

3. Find the value of i8.

Method 1:

Knowing i2 = -1,

Split up the given i8

i8 = i2 × i2 × i2 × i2 

i8 = -1 × -1 × -1 × -1

i8 = +1

Method 2:

Knowing i2 = -1, 

Using i4n = 1,

i8 = i4(2)

i8 = +1

Conclusion

The complex number is a combo of imaginary and real numbers, which helps solve the root of negative numbers. The real numbers consist of all the whole, natural, rationals, irrationals and integers. On the other hand, imaginary numbers definition is those that do not exist in reality and cannot be proved by any physical apparatus. The imaginary part of a complex number is denoted by ‘i’, known as iota. ‘i’ is the square root of -1, which is also -1. The basic mathematical operations employed on the complex numbers are similar to the normal ones. This article tells you about imaginary numbers and imaginary parts of a complex.