Dividing Complex Numbers

Dividing complex numbers means writing a quotient like \dfrac{3 + 2i}{1 - i} in standard form, a + bi. To do it, multiply the top and bottom by the complex conjugate of the denominator. The bottom becomes a real number. Then you can split the fraction into a real part and an imaginary part.

Before this lesson: You should know Complex Numbers, Operations with Complex Numbers, Multiplying Binomials and Rationalizing the Denominator.

Why Division Needs a Trick

Standard form a + bi has no i in a denominator. So an answer like \dfrac{3 + 2i}{1 - i} is not finished. You need a way to get rid of the i on the bottom.

The trick is the complex conjugate. The conjugate of c + di is c - di. Only the sign of the imaginary part changes. When you multiply a number by its conjugate, the i terms cancel and the result is real.

(c + di)(c - di) = c^2 + d^2

This is the same idea as rationalizing a denominator like \dfrac{1}{3 - \sqrt{2}}. There you multiply by 3 + \sqrt{2} to clear the square root. Here you multiply by the conjugate to clear the i.

How to Divide Complex Numbers

To find \dfrac{a + bi}{c + di}, follow these steps.

1. Find the conjugate of the denominator, c - di.
2. Multiply the top and the bottom by that conjugate. This is the same as multiplying by 1, so the value does not change.
3. Multiply out the top with FOIL. Replace i^2 with −1.
4. Multiply out the bottom. It always comes out to c^2 + d^2, a real number.
5. Split the fraction into a real part and an imaginary part. Simplify each fraction.

Here is the whole process in one line:

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

You don’t need to memorize this formula. The steps above give the same result every time.

Two quick special cases

• Dividing by a real number: divide each part. For example, \dfrac{8 - 6i}{2} = 4 - 3i.
• Dividing by a pure imaginary number like 2i: multiply the top and bottom by i. Then i \cdot i = i^2 = -1, so the bottom becomes real.

Worked Examples

Example 1: Divide by a real number

Simplify \dfrac{10 - 15i}{5}.

1. Split the fraction into two parts: \dfrac{10}{5} - \dfrac{15}{5}i
2. Divide the real part: \dfrac{10}{5} = 2
3. Divide the imaginary part: \dfrac{15}{5} = 3
4. Write in standard form: 2 - 3i

Example 2: Divide by a pure imaginary number

Simplify \dfrac{6 + 4i}{2i}.

1. Multiply the top and bottom by i: \dfrac{(6 + 4i) \cdot i}{2i \cdot i}
2. Distribute on the top (see distributive property): 6i + 4i^2
3. Replace i^2 with −1 on the top: 6i - 4, which is -4 + 6i
4. Multiply the bottom: 2i \cdot i = 2i^2
5. Replace i^2 with −1 on the bottom: 2(-1) = -2
6. Now the quotient is \dfrac{-4 + 6i}{-2}
7. Divide each part by −2 (see multiplying and dividing integers): \dfrac{-4}{-2} + \dfrac{6}{-2}i
8. Simplify: 2 - 3i
9. Check: (2 - 3i)(2i) = 4i - 6i^2 = 4i + 6, which is 6 + 4i. ✓

Example 3: Use the conjugate

Simplify \dfrac{3 + 2i}{1 - i}. This is the example in the diagram.

1. The denominator is 1 - i, so its conjugate is 1 + i
2. Multiply the top and bottom by the conjugate: \dfrac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)}
3. FOIL the top (see multiplying binomials): 3 + 3i + 2i + 2i^2
4. Combine the i terms (see combining like terms): 3 + 5i + 2i^2
5. Replace i^2 with −1: 3 + 5i - 2
6. Combine the real parts: 1 + 5i
7. FOIL the bottom: 1 + i - i - i^2
8. The i terms cancel: 1 - i^2
9. Replace i^2 with −1: 1 - (-1)
10. Simplify: 1 + 1 = 2
11. Now the quotient is \dfrac{1 + 5i}{2}
12. Split into two parts: \dfrac{1}{2} + \dfrac{5}{2}i

Example 4: An answer with fractions

Simplify \dfrac{5 + i}{3 - 4i}.

1. The conjugate of 3 - 4i is 3 + 4i
2. Multiply the top and bottom by the conjugate: \dfrac{(5 + i)(3 + 4i)}{(3 - 4i)(3 + 4i)}
3. FOIL the top: 15 + 20i + 3i + 4i^2
4. Combine the i terms: 15 + 23i + 4i^2
5. Replace i^2 with −1: 15 + 23i - 4
6. Combine the real parts: 11 + 23i
7. Use the conjugate rule on the bottom: 3^2 + 4^2
8. Square each number: 9 + 16
9. Add: 25
10. Now the quotient is \dfrac{11 + 23i}{25}
11. Split into two parts: \dfrac{11}{25} + \dfrac{23}{25}i
12. Neither fraction can be reduced, so the answer is \dfrac{11}{25} + \dfrac{23}{25}i

Example 5: Word problem (electric current)

In AC circuits, engineers use complex numbers. Current equals voltage divided by impedance, I = \dfrac{V}{Z}. A circuit has voltage V = 10 + 5i volts and impedance Z = 3 + 4i ohms. Find the current.

1. Set up the division: I = \dfrac{10 + 5i}{3 + 4i}
2. The conjugate of 3 + 4i is 3 - 4i
3. Multiply the top and bottom by the conjugate: \dfrac{(10 + 5i)(3 - 4i)}{(3 + 4i)(3 - 4i)}
4. FOIL the top: 30 - 40i + 15i - 20i^2
5. Combine the i terms: 30 - 25i - 20i^2
6. Replace i^2 with −1: 30 - 25i + 20
7. Combine the real parts: 50 - 25i
8. Use the conjugate rule on the bottom: 3^2 + 4^2 = 9 + 16 = 25
9. Now the quotient is \dfrac{50 - 25i}{25}
10. Divide each part by 25: \dfrac{50}{25} - \dfrac{25}{25}i
11. Simplify: 2 - i
12. Answer: the current is 2 - i amps.
13. Check: (2 - i)(3 + 4i) = 6 + 8i - 3i - 4i^2 = 6 + 5i + 4 = 10 + 5i. ✓

Common Mistakes

Checking Your Answer and Other Shortcuts

• Check by multiplying. Division undoes multiplication. If \dfrac{p}{q} = r, then r \cdot q = p. Examples 2 and 5 show this check.
• Reciprocal of a complex number: \dfrac{1}{c + di} = \dfrac{c - di}{c^2 + d^2}. For example, \dfrac{1}{2 + i} = \dfrac{2 - i}{5}.
• Dividing by i: since \dfrac{1}{i} = -i, dividing by i is the same as multiplying by -i.
• Square roots of negatives: write \sqrt{-9} as 3i before you divide (see complex numbers).

Practice Problems

Write each answer in standard form, a + bi.

1. Simplify \dfrac{18 - 6i}{6}.
2. Simplify \dfrac{8 + 12i}{4i}.
3. Simplify \dfrac{5}{i}.
4. Simplify \dfrac{2 + 3i}{1 + i}.
5. Simplify \dfrac{4 - 7i}{2 + 3i}.
6. Find the reciprocal of 2 + i. That is, simplify \dfrac{1}{2 + i}.
7. Simplify \dfrac{7 + i}{1 - 2i}.
8. Simplify \dfrac{6 - 2i}{3 + 4i}.
9. Simplify \dfrac{-3 + 5i}{-2 - i}.
10. A circuit has voltage V = 17 + i volts and current I = 3 - i amps. Impedance is Z = \dfrac{V}{I}. Find the impedance.

FAQ

How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the denominator. Then simplify the top, simplify the bottom to a real number, and split the result into a + bi form.

Why do you multiply by the conjugate when dividing complex numbers?

A number times its conjugate is always real: (c + di)(c - di) = c^2 + d^2. Multiplying by the conjugate removes i from the denominator without changing the value.

What is 1 divided by i?

It is -i. Multiply the top and bottom by i to get \dfrac{i}{i^2} = \dfrac{i}{-1} = -i.

How do you check a complex division answer?

Multiply your answer by the denominator. If you get the numerator back, the answer is right.

Summary

• To divide complex numbers, multiply the top and bottom by the conjugate of the denominator.
• The conjugate of c + di is c - di, and their product is c^2 + d^2, a real number.
• Use FOIL on the top and replace i^2 with −1.
• To divide by a pure imaginary number like 2i, multiply the top and bottom by i.
• Split the result into a real part and an imaginary part, so the answer is in a + bi form.
• Next, you will use complex numbers to solve quadratic equations that have no real solutions, starting from the quadratic formula.

Previous lesson: Operations with Complex Numbers

Next lesson: Complex Solutions of Quadratic Equations

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