Lesson 11
Introducing the Number $i$
 Let’s meet \(i\).
11.1: Math Talk: Squared
Find the value of each expression mentally.
\(\left(2\sqrt{3}\right)^2\)
\(\left(\frac12 \sqrt{3}\right)^2\)
\(\left(2\sqrt{\text 1}\right)^2\)
\(\left(\frac12 \sqrt{\text 1}\right)^2\)
11.2: It is $i$
Find the solutions to these equations, then plot the solutions to each equation on the imaginary or real number line.
 \(a^2 = 16\)
 \(b^2 = \text 9\)
 \(c^2 = \text 5\)
11.3: The $i$’s Have It
Write these imaginary numbers using the number \(i\).
 \(\sqrt{\text 36}\)
 \(\sqrt{\text 10}\)
 \(\text \sqrt{\text 100}\)
 \(\text \sqrt{\text 17}\)
11.4: Complex Numbers
 Label at least 8 different imaginary numbers on the imaginary number line.

When we add a real number and an imaginary number, we get a complex number. The diagram shows where \(2 + i\) is in the complex number plane. What complex number is represented by point \(A\)?

Plot these complex numbers in the complex number plane and label them.

\(\text 2  i\)

\(\text 6 + 3i\)

\(5+4i\)

\(1  3i\)

Diego says that all real numbers and all imaginary numbers are complex numbers but not all complex numbers are imaginary or real. Do you agree with Diego? Explain your reasoning.
Summary
A square root of a number \(a\) is a number whose square is \(a\). In other words, it is a solution to the equation \(x^2 = a\). Every positive real number has two real square roots. For example, look at the number 35. Its square roots are \(\sqrt{35}\) and \(\text\sqrt{35}\), because those are the two numbers that square to make 35 (remember, the \(\sqrt{}\) symbol is defined to indicate the positive square root). In other words, \(\left(\sqrt{35}\right)^2=35\) and \(\left(\text\sqrt{35}\right)^2=35\).
Similarly, every negative real number has two imaginary square roots. The two square roots of 1 are written \(i\) and \(\text i\). That means that
\(\displaystyle i^2 = \text1\)
and
\(\displaystyle (\text i)^2 = \text1\)
Another example would be the number 17. Its square roots are \(i\sqrt{17}\) and \(\texti\sqrt{17}\), because
\(\displaystyle \begin{array}{} \left(i\sqrt{17}\right)^2 &= 17i^2 \\ &=\text17 \end{array}\)
and
\(\displaystyle \begin{array}{} \left(\texti\sqrt{17}\right)^2 &=17(\texti)^2 \\ &= 17i^2 \\ & =\text17 \end{array}\)
In general, if \(a\) is a positive real number, then the square roots of \(\text a\) are \(i \sqrt{a}\) and \(\text i \sqrt{a}\).
Rarely, we might see something like \(\sqrt{\text17}\). It’s not immediately clear which of the two square roots it is supposed to represent. By convention, \(\sqrt{\text17}\) is defined to indicate the square root on the positive imaginary axis, so \(\sqrt{\text 17}=i\sqrt{17}\).
When we add a real number and an imaginary number, we get a complex number. Together, the real number line and the imaginary number line form a coordinate system that can be used to represent any complex number as a point in the complex plane. For example, the point shown represents the complex number \(\text 3 + 2i\).
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.