Lesson 14

More Arithmetic with Complex Numbers

  • Let’s practice adding, subtracting, and multiplying complex numbers.

14.1: Which One Doesn’t Belong: Complex Expressions

Which one doesn’t belong?

A. \(i^2\)

B. \((1 + i) + (1 - i)\)

C. \((1 + i)^2\)

D. \((1 + i)(1 - i)\)

14.2: Powers of $i$

  1. Write each power of \(i\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. If \(a\) or \(b\) is zero, you can ignore that part of the number. For example, \(0+3i\) can simply be expressed as \(3i\).

    \(i^0\)

    \(i^1\)

    \(i^2\)

    \(i^3\)

    \(i^4\)

    \(i^5\)

    \(i^6\)

    \(i^7\)

    \(i^8\)

  2. What is \(i^{100}\)? Explain your reasoning.
  3. What is \(i^{38}\)? Explain your reasoning.


  1. Write each power of \(1+i\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. If \(a\) or \(b\) is zero, you can ignore that part of the number. For example, \(0+3i\) can simply be expressed as \(3i\).
    1. \((1+i)^0\)
    2. \((1+i)^1\)
    3. \((1+i)^2\)
    4. \((1+i)^3\)
    5. \((1+i)^4\)
    6. \((1+i)^5\)
    7. \((1+i)^6\)
    8. \((1+i)^7\)
    9. \((1+i)^8\)
  2. Compare and contrast the powers of \(1+i\) with the powers of \(i\). What is the same? What is different?

14.3: Add 'Em Up (or Subtract or Multiply)

For each row, your partner and you will each rewrite an expression so it has the form \(a+bi\), where \(a\) and \(b\) are real numbers. You and your partner should get the same answer. If you disagree, work to reach agreement.

partner A partner B
\((7 + 9i) + (3 - 4i)\) \(5i(1 - 2i)\)
\(2i(3 + 4i)\) \((1 + 2i) - (9 - 4i)\)
\((4 - 3i)(4 + 3i)\) \((5 + i) + (20 - i)\)
\((2i)^4\) \((3 + i \sqrt{7})(3 - i \sqrt{7})\)
\((1 + i \sqrt{5}) - (\text- 7 - i \sqrt{5})\) \((\text- 2i)(\text- \sqrt{5} + 4i) \)
\(\left( \frac12 i \right) \left( \frac13 i \right) \left( \frac34 i \right)\) \(\left( \frac12 i \right)^3\)

 

Summary

Suppose we want to write the product \((3+5i)(7-2i)\) in the form \(a+bi\), where \(a\) and \(b\) are real numbers. For example, we might want to compare our solution with a partner’s, and having answers in the same form makes that easier. Using the distributive property,

\(\displaystyle \begin{align} (3+5i)(7-2i) &= 21 - 6i +35i - 10i^2 \\ &= 21 + 29i - 10(\text-1) \\ &= 21+ 29i +10 \\ &= 31 +29i \end{align}\)

Keeping track of the negative signs is especially important since it is easy to mix up the fact that \(i^2=\text-1\) with the fact that \(\text-i^2=\text-(\text-1) = 1\).

Next, suppose we want to write the difference \((\text-6+3i) - (2-4i)\) as a single complex number in the form \(a+bi\). Distributing the negative and combining like terms, we get:

\(\displaystyle \begin{align} (\text-6+3i) - (2-4i) &= \text-6 +3i - 2 - (\text-4i) \\ &= \text-8 +3i +4i \\ &= \text-8 +7i \end{align}\)

Again, it is important to be precise with negative signs. It is a common mistake to just subtract \(4i\) rather than subtracting \(\text-4i\).

Glossary Entries

  • complex number

    A number in the complex plane. It can be written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i^2 = \text-1\).

     

  • imaginary number

    A number on the imaginary number line. It can be written as \(bi\), where \(b\) is a real number and \(i^2 = \text-1\).

  • real number

    A number on the number line.