Division of complex numbers

Complex numbers- Introduction

Complex numbers are made up of a real part and an imaginary part. Complex numbers are denoted as a + bi, where a and b are real numbers. The real part is written before the imaginary part.

Continue reading this to understand the meaning of complex numbers, learn the Division of a complex number by examples.

Complex numbers definition

A Complex number is a sum of a real number and an imaginary number.

Complex Number = a + bi

The real part (a) in the complex number is any tangible value whose square is always positive. Real numbers comprise positive and negative integers, fractions, decimals, irrational and rational numbers.

All real numbers are complex numbers, with their imaginary part having a value of zero.

The imaginary number (bi) in the complex number is the number whose square is always negative. All imaginary numbers (bi) have two parts. In the imaginary number bi, b is a non-zero real number, and “i” , the imaginary unit, is known as iota. The squared value of iota is -1.

  √ (-1), √ (-40)   √ (-4), √ (-81) are all examples of imaginary numbers, as their squares are negative. Note that the square of the imaginary unit will always be negative.

5 + 6i, 27 + 3i, and 8 + 9i are all examples of complex numbers.

Division of complex numbers

The division of two complex numbers is different from other mathematical operations on complex numbers because of one reason. An imaginary number cannot be divided, and a fraction must always have a real number as its denominator to divide it.

Hence, when dividing two complex numbers, we need to convert the denominator into a real number by eliminating the imaginary unit (i).

Complex number division formula

Consider two complex numbers, a + bi, and c + di, then the formula to divide these two complex numbers is given as,

(a + bi) /( c + di) = {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

To understand this formula of dividing two complex numbers, we need to see how it is derived. 

In the above-mentioned complex numbers, we can see that the denominator has an imaginary part that needs to be eliminated. We need to find a term that eliminates this imaginary part when multiplied and divided by it.

Such a term is called the complex conjugate of the denominator and is made by changing the sign of the imaginary unit in the denominator. The real part remains the same.

For example, if (a + bi) and (c + di) are two complex numbers such that they are required to be divided, then the complex conjugate of the denominator (c +di) would be (c – di). The complex conjugate is always the reciprocal of the sign of the imaginary part in the denominator.

The complex conjugate of 5 + 7i is 5 – 7i and that of 8 – 9i is 8 + 9i. 

So, to divide (a + bi) by (c + di) such that the value of c and d is not zero, we need to multiply and divide it by the complex conjugate of the denominator (c +di) 

(a + bi)/(c+ di) * (c – di)/ (c – di) 

={(ac- adi + bci – bdi^2)/( c^2 – cdi + cdi – d^2i^2)}

 Since i^2= -1

This becomes, 

= {(ac- adi + bci + bd)/ (c^2 + d^2)}

= {(ac + bd) + (bc- ad)i/ (c^2 + d^2)}

= {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

Division of a complex number examples

Let us see a few examples to further understand this concept by applying the complex number division formula.

1)  Express the complex number (8+6i)/(2−3i) in the form of a+ib using the dividing complex numbers formula.

Solution: Let a = 8, b = 6, c = 2, and d = -3.

Using the dividing complex numbers formula,

(a + bi) /( c + di) = {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

=(8 + 6i) /( 2 + (-3)i) = {( 8*2 + 6*(-3) / 2^2 + (-3)^2 ) + ( 6*2- 8*(-3)/ 2^2 + (-3)^2) i }

= {( 16- 18 / 4+ 9) + ( 12 + 24 /4 + 9) i }

= {( -2/13) + ( 36/13) i }

Answer: (8+6i)/(2−3i) = ( -2/13) + ( 36/13) i 

2) Simplify 3+4i/ 8- 2i using the dividing complex numbers formula.

Solution: Let a = 3, b = 4, c = 8, and d = -2.

Using the division of complex numbers calculator,

(a + bi) /( c + di) = {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

= {( 3*8 + 4*(-2) / 8^2 + (-2)^2 ) + ( 4*8- 3*(-2)/ 8^2 + (-2)^2) i }

= {( 24 – 8 / 64+ 4) +[ ( 32 + 6)/ 64+ 4]i }

= {( 24 – 8 / 64+ 4) +[ ( 32 + 6)/ 64+ 4]i }

= {( 16/ 68) + (38/ 68) i}

Answer: (3+4i)/( 8- 2i ) = 4/17 + 19/34 i

3) Find the value of  12+4i/ 4 + 2i using the dividing complex numbers formula.

Solution: Let a = 12, b = 4, c = 4, and d = 2.

Using the dividing complex numbers calculator,

(a + bi) /( c + di) = {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

= {( 12*4 + 4*2 / 4^2 + 2^2 ) + ( 4*4- 12*2/ 4^2 + 2^2) i }

= {( 48 +  8/ 16+ 4) +[ ( 16 – 24)/ 16+ 4]i }

= {( 56/ 16+ 4) + (-8/ 16+ 4)i }

= {( 56/ 20) + (-8/ 20) i}

Answer: (12+4i)/( 4 + 2i ) = 14/5 – 2/5 i

4) Calculate the value of  10+4i/ 6 + 2i using the dividing complex numbers formula.

   Also, state the complex conjugate.

Solution: Let a = 10, b = 4, c = 6, and d = 2.

As we know, the complex conjugate of the denominator is made by changing the sign of the imaginary unit in the denominator.

Hence, the complex conjugate of the above equation is 6 – 2i.

Using the dividing complex numbers calculator,

(a + bi) /( c + di) = {( ac + bd / c^2 + d^2 ) + ( bc- ad/ c^2 + d^2) i }

= {( 10*6+ 4*2 / 6^2 + 2^2 ) + ( 4*6- 10*2/ 6^2 + 2^2) i }

= {( 60 +  8/ 36+ 4) +[ ( 24 – 20)/ 36+ 4]i }

= {( 68/ 36 + 4) + (4/36 + 4)i }

= {(68/ 40) + (4/ 40) i}

Answer: (10+4i)/( 6 + 2i ) = 17/10 + 1/10 i

CONCLUSIONS

We saw the basic structure and definition of complex numbers. Then, we proceeded to see the division of a complex by a real followed by the division of a complex with another complex.