Mathematical Operations on Complex Numbers

Addition, subtraction, multiplication, and division are the four fundamental arithmetic operations in mathematics. Algebraic operations on complex numbers are given by these four fundamental arithmetic operations. The combination of a real number with an imaginary number makes up what is known as a complex number.

The operations that can be performed on complex numbers using algebraic methods are defined entirely by those methods. The connection between the various numbers of operations can be understood by the application of several fundamental algebraic laws, such as the associative, commutative, and distributive laws. The solution to the algebraic equations can be found in a straightforward manner by applying these rules. As a result of the fact that algebra is an idea that is dependent on both known and unknown values (variables), its very own set of rules has been devised to solve difficulties. 

Complex numbers: 

In mathematics, a complex number is defined as the combination of a real number and an imaginary number. This is the most basic explanation for what a complex number is. When doing mathematical computations, real numbers are the numbers that are typically used as the basis for our work. However, imaginary numbers are only utilised for mathematical purposes when dealing with complex numbers; this is not the case in most other situations. 

Equality of complex numbers: 

Assume that the two complex numbers are z1 and z2 respectively. 

Here z₁ = a₁+i b₁ and z₂ = a₂+ib₂ 

We can say that the real part of the first complex number is equal to the real part of the second complex number if the two complex numbers, z1 and z2, are equal to one another. On the other hand, we can say that the imaginary part of the first complex number is equal to the imaginary part of the second complex number if the two complex numbers are equal to one another. 

(i.e) Re(z₁) = Re(z₂) and Im(z₁) = Im(Z₂)

Thus, the equality of complex number states that, 

If a₁+ib₁ = a₂+ib₂, then a₁ = a₂ and b₁ = b₂. 

Operations on Complex Numbers: 

The following elementary algebraic operations on complex numbers are covered in this article: 

• Addition of Two Complex Numbers
• Subtraction(Difference) of Two Complex Numbers
• Multiplication of Two Complex Numbers
• Division of Two Complex Numbers. 

Addition of Two Complex Numbers: 

It is well known that the formula for a complex number is z=a+ib, where a and b are two real values.

Consider the following two complicated numbers: z1 = a₁ + ib₁ and z₂ = a₂ + ib₂

After that, the definition of the sum of the complex numbers z₁ and z₂ is as follows: 

z₁+z₂ =( a₁+a₂ )+i( b₁+b₂ ) 

It is clear to see that the real component of the resultant complex number is equal to the sum of the real components of each of the complex numbers, but the imaginary component of the resulting complex number is equivalent to the sum of the imaginary components of each of the complex numbers. 

That is, Re(z₁+z₂ )= Re( z₁ )+Re( z₂ )

Im( z₁+z₂ )=Im( z₁)+Im(z₂)

For the complex numbers,

z₁ = a₁+ib₁

z₂ = a₁+ib₂

z₃ = a₃+ib₃

………..

………..

zₙ = aₙ+ibₙ

a₁+a₂+a₃+….+aₙ = (a₁+a₂+a₃+….+aₙ)+i(b₁+b₂+b₃+….+bₙ) 

Difference of Two Complex Numbers: 

Taking into account the complex numbers z₁ = a₁+ib₁ and z₂ = a₂+ib₂, the difference between z₁ and z₂, denoted by the symbol z₁-z₂, may be defined as follows: 

z₁-z₂ = (a₁-a₂)+i(b₁-b₂) 

From the definition, it is understood that,

Re(z₁-z₂)=Re(z₁)-Re(z₂)

Im(z₁-z₂)=Im(z₁)-ImRe(z₂) 

Every single one of the real numbers is a complicated number that has an imaginary component represented by zero. 

Multiplication of two complex numbers: 

We know the expansion of (a+b)(c+d)=ac+ad+bc+bd

Similarly, consider the complex numbers z₁ = a₁+ib₁ and z₂ = a₂+ib₂

Then, the product of z₁ and z₂ is defined as:

z₁ z₂=(a₁+ib₁)(a₂+ib₂)

z₁ z₂ = a₁ a₂+a₁ b₂ i+b₁ a₂ i+b₁ b₂ i₂

Since,  i2 = -1, therefore, 

z₁ z₂ = (a₁ a₂ – b₁ b₂ ) + i(a₁ b₂ + a₂ b₁ ) 

Division of Complex Numbers: 

If we take into consideration the complex numbers z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂, then the definition of the quotient z₁/z₂ is as follows: 

Z₁/Z₂ = Z₁x 1/Z₂ 

As a result, in order to determine z₁/z₂, we need to perform the operation of multiplying z₁ by the multiplicative inverse of z₂.

Let us now go to a more in-depth discussion of the division of complex numbers: 

Let z₁ = a₁+ib₁ and z₂ = a₂+ib₂, then z₁/z₂ is given as:

Z₁/z₂ = (a₁+ib₁)/(a₂+ib₂)

Hence, (a₁+ib₁)/(a₂+ib₂) = [(a₁+ib₁)(a₂-ib₂)]/[(a₂+ib₂)(a₂-ib₂)]

(a₁ +ib₁)/(a₂+ib₂) = [(a₁a₂)-(a₁b₂i)+(a₂b₁i)+b₁b₂)]/[(a2

₂²+b₂²)]

(a₁+ib₁)/(a₂+ib₂) = [(a₁a₂)+(b₁b₂) +i(a₂b₁-a₁b₂)]/(a₂²+b₂²) 

Solved examples: 

Q1. Add 2+4i and -1+3i.

Solution:

Given the two complex numbers are:

2+4i and -1+3i

(2+4i)+(-1+3i)

⇒ (2-1)+(4i+3i)

⇒ 1 + 7i 

Q2. Simplify: 7 + i + 4 + 4.

Solution:

7 + i + 4 + 4

⇒ (7+4+4) + i

⇒ 15 + i 

Q3. Multiply the complex numbers: (5+3i). (3+4i)

Solution:

Given (5+3i). (3+4i)

(5+3i). (3+4i) = 15+20i+9i-12

(5+3i). (3+4i) = (15-12) + i(20+9)

(5+3i). (3+4i) = 3+29i

Hence, the product of (5+3i) and (3+4i) is 3+29i. 

Conclusion: 

Addition and multiplication are the two arithmetic operations that may be technically described as being performed on complex numbers. Because of this, it will be necessary, in some fashion, to define both subtraction and division in terms of these two operations.