A complex number is a number consisting of real and imaginary part. It can be written in the form of a + ib that is a and b refers to the real part and i refers to the imaginary part. Division is one of the arithmetic operation that is used to find the value of complex numbers. Division operation contains dividend, remainder and quotient. Quotient is the one of the part of the division method.Quotient ring is also known as factor ring.

**Division format :**

16 / 2 = 8.

16 is dividend , 2 is divider, 8 is quotient and remainder is 0.

**Division of integers:**

45 / 10 = 4.5

In decimal form, we get the value of 4.5

In fraction form or mixed form we get the value of 4`(5)/(10)`

**Division of rational numbers:**

(a/b) /( c/d )= a/b *d/c = ad / bc

**Division of complex numbers:**

`(a +ib)/(c + id)`

= `(a + ib)/(c + id)` *`(c - id)/(c - id)`

= `(a + ib)/(c - id)` / `(c + id)/(c - id)`

=ac - iad +ibc -`i^2` bd / `i^2` - icd +icd - `i^2d^2`

=`(ac - iad + ibc +bd)/(c^2 + d^2)`

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

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**Conjugate of quotient of complex numbers:**

Let x and y be two complex numbers, and let the conjugates be x* and y*

We have to prove the theorem of quotient of complex numbers

(a) (x / y)* = x* / y*

(b) (x - y)* = x* - y*

(c) (xy)* = x*y*

Let us prove the theorem of quotient of complex numbers

(a) (x / y)* = x* / y*

**Proof:**

Let x = a + ib and y = c + id

LHS= (x / y)*

x / y = (a + ib) / (c + id)

= (a + ib)(c - id) / (c + id)(c - id)

= (ac + ibc - aid - `i^2` bd) / [`c^2` - `i^2` `d^2` ]

= (ac + ibc - aid + bd) / (`c^2` + `d^2` ), [ `i^2` = -1]

= [(ac + bd) + i(bc - ad)] / (`c^2 ` + `d^2` )

So, LHS (x / y)* = [(ac + bd) - i(bc - ad)] / (`c^2` + `d^2` )

RHS = x* /y *

= (a - ib) / (c - id)

= (a - ib)(c + id) / (c - id)(c + id)

= (ac - ibc + aid - `i^2` bd) / [`c^2` - `(id)^2` ]

= (ac - ibc + aid + bd) / (`c^2` + `d^2` ), [ `i^2 ` = -1]

= [(ac + bd) - i(bc - ad)] / (`c^2` + `d^2` )

so, RHS= LHS

Hence the theorem is proved.

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**Example 1 **

Using rational numbers, Find the value of complex numbers of 12 / 5 / 3 / 25

**Solution:**

=12/ 5 *25 /3

=20

**Example 2**

Find the quotient of complex numbers 3 + i / 2 + i

**Solution:**

=3 + i /2 + i * 2 – i / 2 – i

=(3 + i)(2 – i) / (2 + i )( 2 – i)

=6 – 3i + 2i –`i^2` / 4 – 2i + 2i –`i^2`

=6 – i +1 / 4+1

=7 – i / 5

**Practicing problem1**

Find the quotient of complex numbers are 2 + i / 1+ i

Answer: (3 – i) / 2

**Practicing problem2**

Using rational numbers, find the value of complex numbers are (4/5) / (8/5)

Answer: 1 / 2