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Products and Quotients of Complex Numbers

Strategies based on multiplying binomials and conjugates.

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Multiplying and Dividing Complex Numbers

Mr. Marchez draws a triangle on the board. He labels the height (2 + 3i) and the base (2 - 4i). "Find the area of the triangle," he says. (Recall that the area of a triangle is \begin{align*}A = \frac{1}{2}bh\end{align*}, b is the length of the base and h is the length of the height.)

Watch This

First watch this video.

Khan Academy: Multiplying Complex Numbers

Then watch this video.

Khan Academy: Dividing Complex Numbers

Guidance

When multiplying complex numbers, FOIL the two numbers together (see Factoring When the Leading Coefficient Equals 1 concept) and then combine like terms. At the end, there will be an @$\begin{align*}i^2\end{align*}@$ term. Recall that @$\begin{align*}i^2=-1\end{align*}@$ and continue to simplify.

Example A

Simplify:

a) @$\begin{align*}6i(1-4i)\end{align*}@$

b) @$\begin{align*}(5-2i)(3+8i)\end{align*}@$

Solution:

a) Distribute the @$\begin{align*}6i\end{align*}@$ to both parts inside the parenthesis.

@$$\begin{align*}6i(1-4i)=6i-24i^2\end{align*}@$$

Substitute @$\begin{align*}i^2 = -1\end{align*}@$ and simplify further.

@$$\begin{align*}&=6i-24(-1)\\ &=24+6i\end{align*}@$$

Remember to always put the real part first.

b) FOIL the two terms together.

@$$\begin{align*}(5-2i)(3+8i) &= 15+40i-6i-16i^2\\ &= 15+34i-16i^2\end{align*}@$$

Substitute @$\begin{align*}i^2 = -1\end{align*}@$ and simplify further.

@$$\begin{align*}&= 15+34i-16(-1)\\ &= 15+34i+16\\ &= 31+34i\end{align*}@$$

More Guidance

Dividing complex numbers is a bit more complicated. Similar to irrational numbers, complex numbers cannot be in the denominator of a fraction. To get rid of the complex number in the denominator, we need to multiply by the complex conjugate. If a complex number has the form @$\begin{align*}a + bi\end{align*}@$, then its complex conjugate is @$\begin{align*}a-bi\end{align*}@$. For example, the complex conjugate of @$\begin{align*}-6 + 5i\end{align*}@$ would be @$\begin{align*}-6-5i\end{align*}@$. Therefore, rather than dividing complex numbers, we multiply by the complex conjugate.

Example B

Simplify @$\begin{align*}\frac{8-3i}{6i}\end{align*}@$.

Solution: In the case of dividing by a pure imaginary number, you only need to multiply the top and bottom by that number. Then, use multiplication to simplify.

@$$\begin{align*}\frac{8-3i}{6i}\cdot \frac{6i}{6i} &= \frac{48i-18i^2}{36i^2}\\ &= \frac{18+48i}{-36}\\ &= \frac{18}{-36}+\frac{48}{-36}i\\ &= -\frac{1}{2}-\frac{4}{3}i\end{align*}@$$

When the complex number contains fractions, write the number in standard form, keeping the real and imaginary parts separate. Reduce both fractions separately.

Example C

Simplify @$\begin{align*}\frac{3-5i}{2+9i}\end{align*}@$.

Solution: Now we are dividing by @$\begin{align*}2 + 9i\end{align*}@$, so we will need to multiply the top and bottom by the complex conjugate, @$\begin{align*}2-9i\end{align*}@$.

@$$\begin{align*}\frac{3-5i}{2+9i}\cdot \frac{2-9i}{2-9i} &= \frac{6-27i-10i+45i^2}{4-18i+18i-81i^2}\\ &= \frac{6-37i-45}{4+81}\\ &= \frac{-39-37i}{85}\\ &= - \frac{39}{85}-\frac{37}{85}i\end{align*}@$$

Notice, by multiplying by the complex conjugate, the denominator becomes a real number and you can split the fraction into its real and imaginary parts.

In both Examples B and C, substitute @$\begin{align*}i^2 = -1\end{align*}@$ to simplify the fraction further. Your final answer should never have any power of @$\begin{align*}i\end{align*}@$ greater than 1.

Intro Problem Revisit The area of the triangle is @$\begin{align*} \frac{(2 + 3i)(2 - 4i)}{2}\end{align*}@$ so FOIL the two terms together and divide by 2.

@$$\begin{align*}(2 + 3i)(2 - 4i) = 4 - 8i + 6i -12i^2\\ &= 4 - 2i - 12i^2\end{align*}@$$

Substitute @$\begin{align*}i^2 = -1\end{align*}@$ and simplify further.

@$$\begin{align*}&= 4 - 2i -12(-1)\\ &= 4 - 2i + 12\\ &= 16 - 2i\end{align*}@$$

Now divide this product by 2.

@$\begin{align*} \frac {16 - 2i}{2} = 8 - i\end{align*}@$

Therefore the area of the triangle is @$\begin{align*}8 -i\end{align*}@$.

Guided Practice

1. What is the complex conjugate of @$\begin{align*}7-5i\end{align*}@$?

Simplify the following complex expressions.

2. @$\begin{align*}(7-4i)(6+2i)\end{align*}@$

3. @$\begin{align*}\frac{10-i}{5i}\end{align*}@$

4. @$\begin{align*}\frac{8+i}{6-4i}\end{align*}@$

Answers

1. @$\begin{align*}7 + 5i\end{align*}@$

2. FOIL the two expressions.

@$$\begin{align*}(7-4i)(6+2i) &= 42+14i-24i-8i^2\\ &= 42-10i+8\\ &= 50-10i\end{align*}@$$

3. Multiply the numerator and denominator by @$\begin{align*}5i\end{align*}@$.

@$$\begin{align*}\frac{10-i}{5i} \cdot \frac{5i}{5i} &= \frac{50i-5i^2}{25i^2}\\ &= \frac{5+50i}{-25}\\ &= \frac{5}{-25}+\frac{50}{-25}i\\ &= -\frac{1}{5}-2i\end{align*}@$$

4. Multiply the numerator and denominator by the complex conjugate, @$\begin{align*}6 + 4i\end{align*}@$.

@$$\begin{align*}\frac{8+i}{6-4i} \cdot \frac{6+4i}{6+4i} &= \frac{48+32i+6i+4i^2}{36+24i-24i-16i^2}\\ &= \frac{48+38i-4}{36+16}\\ &= \frac{44+38i}{52}\\ &= \frac{44}{52} + \frac{38}{52}i\\ &= \frac{11}{13}+\frac{19}{26}i\end{align*}@$$

Explore More

Simplify the following expressions. Write your answers in standard form.

  1. @$\begin{align*}i(2-7i)\end{align*}@$
  2. @$\begin{align*}8i(6+3i)\end{align*}@$
  3. @$\begin{align*}-2i(11-4i)\end{align*}@$
  4. @$\begin{align*}(9+i)(8-12i)\end{align*}@$
  5. @$\begin{align*}(4+5i)(3+16i)\end{align*}@$
  6. @$\begin{align*}(1-i)(2-4i)\end{align*}@$
  7. @$\begin{align*}4i(2-3i)(7+3i)\end{align*}@$
  8. @$\begin{align*}(8-5i)(8+5i)\end{align*}@$
  9. @$\begin{align*}\frac{4+9i}{3i}\end{align*}@$
  10. @$\begin{align*}\frac{6-i}{12i}\end{align*}@$
  11. @$\begin{align*}\frac{7+12i}{-5i}\end{align*}@$
  12. @$\begin{align*}\frac{4-2i}{6-6i}\end{align*}@$
  13. @$\begin{align*}\frac{2-i}{2+i}\end{align*}@$
  14. @$\begin{align*}\frac{10+8i}{2+4i}\end{align*}@$
  15. @$\begin{align*}\frac{14+9i}{7-20i}\end{align*}@$

Vocabulary

complex number

complex number

A complex number is the sum of a real number and an imaginary number, written in the form a + bi.

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