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Products and Quotients of Complex Numbers

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Multiplying and Dividing Complex Numbers
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Mr. Marchez draws a triangle on the board. He labels the height (2 + 3 i ) and the base (2 - 4 i ). "Find the area of the triangle," he says. (Recall that the area of a triangle is A = \frac{1}{2}bh , 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 a = 1 concept) and then combine like terms. At the end, there will be an i^2 term. Recall that i^2=-1 and continue to simplify.

Example A

Simplify:

a) 6i(1-4i)

b) (5-2i)(3+8i)

Solution:

a) Distribute the 6i to both parts inside the parenthesis.

6i(1-4i)=6i-24i^2

Substitute i^2 = -1 and simplify further.

&=6i-24(-1)\\&=24+6i

Remember to always put the real part first.

b) FOIL the two terms together.

(5-2i)(3+8i) &= 15+40i-6i-16i^2\\&= 15+34i-16i^2

Substitute i^2 = -1 and simplify further.

&= 15+34i-16(-1)\\&= 15+34i+16\\&= 31+34i

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 a + bi , then its complex conjugate is a-bi . For example, the complex conjugate of -6 + 5i would be -6-5i . Therefore, rather than dividing complex numbers, we multiply by the complex conjugate.

Example B

Simplify \frac{8-3i}{6i} .

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.

\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

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 \frac{3-5i}{2+9i} .

Solution: Now we are dividing by 2 + 9i , so we will need to multiply the top and bottom by the complex conjugate, 2-9i .

\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

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 i^2 = -1 to simplify the fraction further. Your final answer should never have any power of i greater than 1.

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

(2 + 3i)(2 - 4i) = 4 - 8i + 6i -12i^2\\&= 4 - 2i - 12i^2

Substitute i^2 = -1 and simplify further.

&= 4 - 2i -12(-1)\\&= 4 - 2i + 12\\&= 16 - 2i

Now divide this product by 2.

 \frac {16 - 2i}{2} = 8 - i

Therefore the area of the triangle is 8 -i .

Guided Practice

1. What is the complex conjugate of 7-5i ?

Simplify the following complex expressions.

2. (7-4i)(6+2i)

3. \frac{10-i}{5i}

4. \frac{8+i}{6-4i}

Answers

1. 7 + 5i

2. FOIL the two expressions.

(7-4i)(6+2i) &= 42+14i-24i-8i^2\\&= 42-10i+8\\&= 50-10i

3. Multiply the numerator and denominator by 5i .

\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

4. Multiply the numerator and denominator by the complex conjugate, 6 + 4i .

\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

Vocabulary

Complex Conjugate

The “opposite” of a complex number. If a complex number has the form a+bi , its complex conjugate is a-bi . When multiplied, these two complex numbers will produce a real number.

Practice

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

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

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