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 $i$

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 \(3i\).
\(i^0\)
\(i^1\)
\(i^2\)
\(i^3\)
\(i^4\)
\(i^5\)
\(i^6\)
\(i^7\)
\(i^8\)
 What is \(i^{100}\)? Explain your reasoning.
 What is \(i^{38}\)? Explain your reasoning.

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 \(3i\).
 \((1+i)^0\)
 \((1+i)^1\)
 \((1+i)^2\)
 \((1+i)^3\)
 \((1+i)^4\)
 \((1+i)^5\)
 \((1+i)^6\)
 \((1+i)^7\)
 \((1+i)^8\)
 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 \((3+5i)(72i)\) 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,
\(\displaystyle \begin{align} (3+5i)(72i) &= 21  6i +35i  10i^2 \\ &= 21 + 29i  10(\text1) \\ &= 21+ 29i +10 \\ &= 31 +29i \end{align}\)
Keeping track of the negative signs is especially important since it is easy to mix up the fact that \(i^2=\text1\) with the fact that \(\texti^2=\text(\text1) = 1\).
Next, suppose we want to write the difference \((\text6+3i)  (24i)\) as a single complex number in the form \(a+bi\). Distributing the negative and combining like terms, we get:
\(\displaystyle \begin{align} (\text6+3i)  (24i) &= \text6 +3i  2  (\text4i) \\ &= \text8 +3i +4i \\ &= \text8 +7i \end{align}\)
Again, it is important to be precise with negative signs. It is a common mistake to just subtract \(4i\) rather than subtracting \(\text4i\).
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 \(i^2 = \text1\).
 imaginary number
A number on the imaginary number line. It can be written as \(bi\), where \(b\) is a real number and \(i^2 = \text1\).
 real number
A number on the number line.