# Dividing Complex Numbers

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## Building on Binomial Multiplication

Dividing complex numbers usually requires multiplying both the numerator and denominator by the complex conjugate of the denominator. It is similar to multiplying binomials in algebra. The primary difference is that in dividing complex numbers, they divide real and imaginary parts separately. The main goal of finding the quotient of complex numbers is to eliminate the imaginary portion of the denominator.

We can use complex conjugates to perform division in the complex number system. If we want to find the quotient of a +bi / c = di where a,b,c, and d are real numbers, we simply multiply the numerator and the denominator by the conjugate of the denominator as follows:

a + bi / c +di = [a + bi / c + di] x [c - di / c - di]

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

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

If z = a + bi, then the conjugate (or complex conjugate) of z is the complex number a - bi denoted by z bar = a + bi or a - bi. Note that the conjugate of a + bi is a - bi, and the conjugate of a - bi is a + bi. Below are some illustrations of conjugates.

Complex Number | Conjugate |
---|---|

5 + 8i | 5 - 8i |

5 - 8i | 5 + 8i |

9i | -9i |

4 | 4 |

The following two properties are consequences of the definitions of the sum and the product of complex numbers.

Properties of Conjugate | Illustration |
---|---|

(a + bi) + (a - bi) = 2a | (4 + 3i) + (4 - 3i) = 4 + 4 = 2 x 4 = 8 |

(a + bi) (a - bi) = a^2 + b^2 | (4 + 3i) (4 - 3i) = 42 - (3i)^2 = 42 - (3^2) (i^2) = 42 + 32 |

The complex conjugate is a simple concept of reversing signs. Always remember that when multiplying a complex number by its complex conjugate, the result is always a Real number. The same goes when a complex number adds up with its complex conjugate; a resulting answer is a Real number.

## How to Divide Complex Numbers in Polar Form

**Write the quotient problem as a fraction**. It is easier to start dividing complex numbers if the expression is in fraction form in preparation for the next step.**Determine the complex conjugate of the denominator term**. In finding the complex conjugate of a complex number, change the sign between the two terms in the denominator. The complex conjugates are numbers considered to be the opposite imaginary part.**In dividing complex numbers, multiply both the numerator and denominator with the obtained complex conjugate**. Use learnings from multiplying complex numbers. Recall that the product of a complex number with its conjugate will always yield a real number. The imaginary parts of the complex number cancel each other. Just make sure not to forget to multiply the numerator with the complex conjugate to keep the quotient the same. Along the process, multiply imaginary numbers. Imaginary numbers, when squared, give a negative result. Thus, i^{2}= -1.**Use the FOIL method and simplify**. Apply the FOIL method learned from early algebra lessons to eliminate the parenthesis in the numerator and denominator. Then, simplify the resulting long equation by combining like terms. Combine like terms by combining imaginary numbers with imaginary and real numbers with real numbers.**Write the answer in the form a + bi**. Once the result is a + bi, reduce the answer to the simplest terms for the final answer.

There is no better way to learn this than to show it through examples.

## Example 1: Dividing Complex Numbers Using Complex Conjugates

Divide (5 - 4i) by (2 + i).

**Solution**

Identify the conjugate of the denominator (2 + i). Then, multiply both the numerator and the denominator by the obtained conjugate. The conjugate of (2 + i) is (2 - i).

(5 - 4i) / (2 + i) = (5 - 4i) / (2 + i) x (2 - i) / (2 - i)

(5 - 4i) / (2 + i) = (10 - 4) + (-5 - 8)i / (4 + 1)i

(5 - 4i) / (2 + i) = (6 - 13i / 5)

(5 - 4i) / (2 + i) = 6/5 - 13i/5

**Final Answer**

The quotient of (5 - 4i) by (2 + i) is 6/5 - 13i/5.

## Example 2: Quotient of Complex Numbers

**Solution**

In dividing complex numbers in a fractional polar form, determine the complex conjugate of the denominator. The complex conjugate of the polar form (9 + 2i) is (9 - 2i). Then, multiply (9 - 2i) with both the numerator and the denominator of the given equation. Upon multiplying, simplify the equation and note that i^{2} = -1.

1 / (9 + 2i) = 1 / (9 + 2i) x (9 - 2i) / (9 - 2i)

1 / (9 + 2i) = (9 - 2i) / (81 + 4)

1 / (9 + 2i) = (9 / 85) - (2i / 85)

**Final Answer**

The quotient of 1 / (9 + 2i) is (9 - 2i) / 85.

## Example 3: Quotient of Complex Numbers

Divide (7 - i) / (3 - 5i).

**Solution**

Divide the given complex numbers by identifying the complex conjugate of the denominator. For this case, the complex conjugate of the denominator (3 - 5i) is (3 + 5i). Multiply (3 + 5i) with both the numerator and the denominator of the given equation and simplify the equation.

(7 - i) / (3 - 5i) = (7 - i) / (3 - 5i) x (3 + 5i) / (3 + 5i)

(7 - i) / (3 - 5i) = (21 + 35i - 3i - 5i^{2}) / (9 + 25)

(7 - i) / (3 - 5i) = (26 + 32i) / 34

(7 - i) / (3 - 5i) = (13 / 17) + (16i / 17)

**Final Answer**

The quotient of (7 - i) / (3 - 5i) is (13 / 17) + (16i / 17).

## Example 4: Finding the Complex Conjugates

Identify the complex conjugates of the following complex equations.

- (1 +i)
- -10i
- 69-69i

**Solution and Answer**

Solve for the complex conjugate of a complex equation in polar form by changing the sign between the two terms. Thus, change the sign of the imaginary part alone.

For letter a, the given equation is (1 + i). The complex conjugate of the (1 + i) is (1 - i).

For letter b, changing the sign of the imaginary part and the term alone, the result would be +10i.

Lastly, for letter c, change the sign of the imaginary term -69 to a positive one. Thus, the complex conjugate of (69 - 69i) is (69 + 69i).

## Example 5: Division of Two Complex Numbers

Find the quotient of two complex numbers (20 - 3i) and (4 + 2i).

**Solution**

Rewrite the given complex numbers in fractional form. Then, find the conjugate of the denominator 4 + 2i. Change the sign of the imaginary term to get the conjugate; thus, the conjugate results to 4 - 2i. Multiply the obtained conjugate to the numerator and denominator of the given fraction. Take note that i^{2} = -1.

(20 - 3i) / (4 + 2i)

(20 - 3i) / (4 + 2i) = (20 - 3i) / (4 + 2i) x (4 - 2i) / (4 - 2i)

Apply the FOIL method when multiplying the complex binomials and simplify right after.

(20 - 3i) / (4 + 2i) = (20 - 3i) / (4 + 2i) x (4 - 2i) / (4 - 2i)

(20 - 3i) / (4 + 2i) = 80 - 40i - 12i + 6i^{2} / 16 - 8i + 8i -4i^{2}

(20 - 3i) / (4 + 2i) = 80 - 52i + 6(-1) / 16 - 4i^{2}

(20 - 3i) / (4 + 2i) = 80 - 52i - 6 / 16 - 4(-1)

(20 - 3i) / (4 + 2i) = -52i + 74 / 20

(20 - 3i) / (4 + 2i) = -13i/5 + 37/10

**Final Answer**

Dividing (20 - 3i) with (4 + 2i), the resulting quotient is -13i/5 + 37/10.

## Example 6: Finding the Quotient of Complex Numbers

Divide the complex numbers (1 - 2i) and (1 + 3i).

**Solution**

Rewrite the given complex numbers in fractional form. Then, find the conjugate of the denominator 1 + 3i. Change the sign of the imaginary term to get the conjugate. Multiply the obtained conjugate to the numerator and denominator of the given fraction. Take note that i^{2} = -1.

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

(1 - 2i) / (1 + 3i) = (1 - 2i) / (1 + 3i) x (1 - 3i) / (1 - 3i)

Use the FOIL method when multiplying the complex binomials. Just do not forget to simplify all terms.

(1 - 2i) / (1 + 3i) = (1 - 2i) / (1 + 3i) x (1 - 3i) / (1 - 3i)

(1 - 2i) / (1 + 3i) = (1 - 3i - 2i + 6i^{2}) / (1 - 3i + 3i - 9i^{2})

(1 - 2i) / (1 + 3i) = (1 - 5i + 6(-1)) / (1 - 9(-1))

(1 - 2i) / (1 + 3i) = (1 - 5i - 6) / (1 + 9)

(1 - 2i) / (1 + 3i) = (-5i - 5) / 10

(1 - 2i) / (1 + 3i) = (-5i/10) - (5/10)

(1 - 2i) / (1 + 3i) = (-i/2) - ½

**Final Answer**

The quotient of (1 - 2i) and (1 + 3i) is (-i/2) - ½.

## Example 7: Dividing Complex Numbers

Divide the complex numbers (8 + 5i) and (-4 - i).

**Solution**

The first step is to rewrite the problem as a fraction. The denominator is (-4 - i), then its conjugate is (-4 + i). Multiply the obtained complex conjugate with the top and bottom terms of the fraction.

(8 + 5i) / (-4 - i)

(8 + 5i) / (-4 - i) = (8 + 5i) / (-4 - i) x (-4 + i) / (-4 + i)

Use the FOIL method and simplify the complex equation.

(8 + 5i) / (-4 - i) = (8 + 5i) / (-4 - i) x (-4 + i) / (-4 + i)

(8 + 5i) / (-4 - i) = (-32 + 8i - 20i + 5i^{2}) / (16 -4i + 4i -i^{2})

(8 + 5i) / (-4 - i) = (-32 -12i + 5(-1)) / (16 - (-1))

(8 + 5i) / (-4 - i) = (-32 -12i - 5) / (16 + 1)

(8 + 5i) / (-4 - i) = (-12i - 37) / 17

(8 + 5i) / (-4 - i) = (-12i/17) - 37/17

**Final Answer**

The quotient is (-12i/17) - 37/17 when you divide the complex numbers (8 + 5i) and (-4 - i).

## Example 8: Finding Complex Conjugates

Find the complex conjugate of each.

- 2 + i√6
- -3i/4
- 4 - i

**Solution and Answer**

- The given complex equation is already in the form of a + bi. Recall that the complex conjugate of a + bi is a - bi. Therefore, the complex conjugate of 2+i√6 is 2-i√6.
- Rewrite the given imaginary number in the form of a + bi. By rewriting in the form a + bi, the result will be 0 - 3i/4. The complex conjugate of 0 - 3i/4 is 0 + 3i/4 or 3i/4.
- In finding the complex conjugate of a complex equation, change the sign of the imaginary number. For the case of 4-i, the complex conjugate will be 4+i.

## Example 9: Converting Complex Equations in Polar Form to Rectangular Form

Convert (6 + 2i)/2i to rectangular form.

**Solution**

In converting the given complex equation (6 + 2i)/2i into a rectangular form, we need to remove the "i" in the denominator. Multiply both the numerator (6 + 2i) and denominator 2i with i. It is an excellent technique for getting rid of unwanted "i" in the denominator of a complex fraction. It is a simple multiplication of both the numerator and the denominator with "i."

(6 + 2i) / 2i = (6 + 2i) / 2i x i / i

(6 + 2i) / 2i = (6 + 2i)i / (2i) (i)

(6 + 2i) / 2i = (6i + 2i^{2}) / 2i^{2}

(6 + 2i) / 2i = (6i + 2(-1)) / 2(-1)

(6 + 2i) / 2i = (6i - 2) / -2

(6 + 2i) / 2i = -6i/2 + 1

**Final Answer**

The rectangular form of the complex equation (6 + 2i)/2i is equal to -6i/2 + 1.

## Example 10: Simplifying Complex Equations Using Division

Simplify the following imaginary equations.

- 20/4i
- 12/6i

**Solution**

In simplifying expressions with "i" in the denominator, multiply both the numerator and denominator by i. It eliminates the i's in the denominator and transfers it into the numerator.

20/4i = 20/4i x i / i

20/4i = 20i / 4i^{2}

20/4i = 20i / 4 (-1)

20/4i = 20i / -4

20/4i = -5i

12/6i = 12/6i x i / i

12/6i = 12i / 6i^{2}

12/6i = 12i / 6(-1)

12/6i = 12i / -6

12/6i = -2i

**Final Answer**

The simplified forms of 20/4i and 12/6i are -5i and -2i.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**

## Comments

**Umesh Chandra Bhatt** from Kharghar, Navi Mumbai, India on May 30, 2021:

Nicely explained.