Algebra of Complex Numbers

A number that can be written in the form x + i.y, where x, y are real numbers and i = (−1)1/2 is called a complex number.  If A = x + i.y, is a complex number, then x and y are called real and imaginary parts of the complex number, respectively, and written as Re(A) = x, Im.(A) = y.

Order relations “greater than” and “less than” do not apply to complex numbers calculations. If the imaginary part of a complex number Im. (A) = y, is zero, then the complex number is known as a purely real number, and if the real part  Re(A) = x, is zero, then it is called the purely imaginary number. In this article, we will learn about the algebra of complex numbers. 

Addition of two complex numbers

Let A1 = (x + i.y) and A2 = (s + i.t) 

Be any two complex numbers. 

Then, the sum A1 + A2 is defined as follows: 

A1 + A2 = (x + s) + i (y + t), 

which is also a complex number,

For example, (3 + i2) + (– 9 +i5),

(3 – 9) + i (2 + 5),

 – 6 + i 7.

To know the algebra of complex numbers meaning, the following are the properties of the addition of two complex numbers,

1. As the result of the sum of two given complex numbers is also a complex number, the given set of the complex numbers is closed with respect to addition. 
2. The addition of two complex numbers is commutative, which means A1 + A2 = A2 + A1  
3. The addition of two complex numbers is associative, which means (A1 + A2 ) + A3 = A1 + (A2 + A3)  
4. For any complex number A = a + i,b, there exist 0, i.e., (0 + 0i) complex number such that A + 0 = 0 + A = A, 

This is known as the identity element for the addition of complex numbers. 

1. For any complex number A = a + ib, there always exists a number – A = – x – iy such that A + (– A) = (– A) + A = 0. This is known as the additive inverse of A. 

Multiplication of two complex numbers

Let A1 = x + iy and A2 = s + it, be two complex numbers. 

Then A1 . A2 = (x + iy).(s + it),

(x.s – y.t) + i (xt + y.s).

The following are the properties of the multiplication of two complex numbers,

1. The result of the product of two complex numbers is also a complex number, the given set of complex numbers is closed with respect to multiplication. 
2. Multiplication of complex numbers is commutative, which means A1.A2 = A2.A1 
3. Multiplication of complex numbers is associative, which means (A1.A2).A3 = A1.(A2.A3) 
4. For any given complex number A = a + ib, there exists a complex number 1, which is, (1+ 0i) such that A.1 = 1.A = A, 

This is known as the identity element for multiplication.

1. For any non zero complex number A = a + i.b, there exists a complex number 1/A such that A.1/A =1/ A. A, it is the multiplicative inverse of a+i.b
2. For any three complex numbers A1 , A2 and A3

A1 . (A2 + A3 ) = A1 . A2 + A1 . A3 

And, (A1 + A2 ) . A3 = A1 . A3 + A2 . A3 

That is, for complex numbers, multiplication is distributive over addition.

Solved Examples

Algebra of complex numbers examples:

Example1: Multiply complex numbers A = 3 − 1i and B = − 7 + 5i.

Solution: For multiplying the given complex numbers A and B, 

we will use the formula, 

(x + i.y).(s + i.t) = (x.s – y.t) + i.(x.t + y.s). 

Here, x = 3, y = -1, s = -7, t = 5,

(3 − 1.i).(− 7+ 5i),

[3 × (-7) – (-1) × 5) + i{(3 × 5) + (-1) × (-7)}],

= (-21 – 5) + i (15 + 7)

= -26 + i.22.

Example 2: Find the addition of two complex numbers (4 + 6i) and (-3 – 4i).

Solution: The given complex numbers are,

(4 + 6i), and (-3-4.i)

Implying the addition of two complex numbers,

(4 + 6.i) + (-3 – 4.i),

= 4 + 6.i – 3 – 4.i,

= 4 – 3 + 6.i – 4.i,

= 1 + 2.i.

Example 3: Multiply complex numbers A = (6 − 1.i) and B = (− 4 + 5i).

Solution: For multiplying the given complex numbers A and B, 

we will use the formula, 

(x + i.y).(s + i.t) = (x.s – y.t) + i.(x.t + y.s). 

Here, x = 6, y = -1, s = -4, t = 5,

(6 − 1.i).(− 4+ 5i),

[6 × (-4) – (-1) × 5) + i{(6 × 5) + (-1) × (-4)}],

= (-24 – 5) + i (30 + 4)

= -29 + i.34.

Example 4: Multiply complex numbers A = 3 + 3i and B =  7 + 2i.

Solution: For multiplying the given complex numbers A and B, 

we will use the formula, 

(x + i.y).(s + i.t) = (x.s – y.t) + i.(x.t + y.s). 

Here, x = 3, y = 3, s = 7, t = 2,

(3 + 3.i).( 7+ 2.i),

[3 × (7) – (3) × 2) + i{(3 × 2) + (3) × (7)}],

= (21 – 6) + i (6 + 21)

= 15 + i.27.

Conclusion

This article discussed the algebra of complex number meaning with the solutions of additions and multiplications of two complex numbers. We also read about the meaning and importance of algebra of complex numbers. Algebra of Complex Numbers is defined by the following,

1. Two complex numbers A1 = (x + iy) and A2 = (s + it) are said to be equal if x = s and y = t.

Let A1 = x + iy and A2 = s + it be two complex numbers then A1 + A2 = (x + s) + i (y + t).