Lesson 11
Introducing the Number $i$
11.1: Math Talk: Squared (5 minutes)
Warmup
The purpose of this Math Talk is to elicit strategies and understandings students have for squaring numbers that involve square roots, including the imaginary number \(\sqrt{\text1}\) that was introduced in the previous lesson. These understandings help students develop fluency and will be helpful later in this lesson when students will multiply imaginary numbers and decide how to write square roots of negative numbers as real multiples of \(i\).
In this activity, students have an opportunity to look for repeated reasoning when squaring expressions that involve square roots (MP8).
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Find the value of each expression mentally.
\(\left(2\sqrt{3}\right)^2\)
\(\left(\frac12 \sqrt{3}\right)^2\)
\(\left(2\sqrt{\text 1}\right)^2\)
\(\left(\frac12 \sqrt{\text 1}\right)^2\)
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
 “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
 “Did anyone have the same strategy but would explain it differently?”
 “Did anyone solve the problem in a different way?”
 “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
 “Do you agree or disagree? Why?”
Design Principle(s): Optimize output (for explanation)
11.2: It is $i$ (10 minutes)
Activity
In the previous lesson, students used the notation \(\sqrt{\text1}\) to represent a number that was a solution to \(x^2=\text1\) as an introduction to imaginary and complex numbers. In this activity, students are formally introduced to the symbol \(i\) used to represent the imaginary unit. Students solve quadratic equations of the form \(x^2=k\) for various integers \(k\) and plot their solutions in the complex plane. The key idea is that, just as positive numbers have two real square roots, negative numbers have two imaginary square roots.
Launch
Remind students that in the previous lesson, they wrote \(\sqrt{\text1}\) to describe an imaginary solution to the equation \(x^2=\text1\). But just as \(x^2=4\) has two solutions, 2 and 2, \(x^2=\text1\) also has two solutions. \(\sqrt{\text1}\) is one of them. Explain to students that the \(\sqrt{}\) symbol is supposed to mean the positive square root of a real number, so mathematicians decided to use a different symbol for the two imaginary solutions to the equation \(x^2=\text1\). The two square roots of 1 are written \(i\) and \(\text i\). That means that \(i^2 = \text1\) and \((\text i)^2 = \text1\). Display this graph for all to see:
Then display these equations for all to see:
\(\displaystyle \begin{array}{l} (2i)^2 &= 2^2 \boldcdot (i)^2 \\ &= 4 \boldcdot \text 1 \\ &=\text 4 \end{array}\)
\(\displaystyle \begin{array}{l} (\text 2i)^2 &= (\text 2)^2 \boldcdot (i)^2 \\ &= 4 \boldcdot \text 1 \\ &=\text 4 \end{array}\)
Supports accessibility for: Conceptual processing; Language
Student Facing
Find the solutions to these equations, then plot the solutions to each equation on the imaginary or real number line.
 \(a^2 = 16\)
 \(b^2 = \text 9\)
 \(c^2 = \text 5\)
Student Response
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Activity Synthesis
The purpose of the discussion is to highlight that just as positive real numbers have two square roots, one positive and one negative, it is also true that negative real numbers have two square roots, one on the positive imaginary axis and one on the negative imaginary axis. Display the complex plane from the task for all to see and select students to share how they reasoned about the solutions to the equations and plotted their solutions, recording their thinking for all to see. Ask students, “What do you notice about solutions to \(x^2=k\) if \(k\) is positive? What if \(k\) is negative?” (If \(k\) is positive, the two solutions are on the positive and negative real axis. If \(k\) is negative, the solutions are on the positive and negative imaginary axis.)
Design Principle(s): Support sensemaking; Optimize output (for justification)
11.3: The $i$’s Have It (5 minutes)
Activity
In the previous activity, students found that negative real numbers have two imaginary square roots. In future lessons, students will use the quadratic formula to find complex solutions, which involves interpreting expressions like \(\sqrt{\text36}\). In this activity, students learn and apply the convention that for any positive real number \(a\), \(\sqrt{\texta}=i\sqrt{a}\).
Launch
Explain to students that sometimes negative numbers end up inside of the square root symbol as a result of the steps used to solve equations. In the previous activity, we saw that negative numbers have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. By convention, the \(\sqrt{}\) symbol with a negative number inside refers to the square root on the positive imaginary axis. For example, \(\sqrt{\text4}=2i\).
Student Facing
Write these imaginary numbers using the number \(i\).
 \(\sqrt{\text 36}\)
 \(\sqrt{\text 10}\)
 \(\text \sqrt{\text 100}\)
 \(\text \sqrt{\text 17}\)
Student Response
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Anticipated Misconceptions
Students may need to work through some intermediate steps in order to express the square roots in terms of \(i\). For example, students could break a square root apart like this: \(\sqrt{\text36} = \sqrt{\text1}\boldcdot \sqrt{36} = i \boldcdot 6 = 6i\).
Activity Synthesis
Select students to share their responses and encourage students to show that their answers make sense by squaring. Mention that for answers that involve \(i\) multiplied by a radical, it is customary to write the \(i\) first so it doesn’t look like it might be part of the radicand. For example, \(\sqrt{\text15}\) is usually written as \(i\sqrt{15}\) rather than \(\sqrt{15}i\).
Supports accessibility for: Language; Socialemotional skills
11.4: Complex Numbers (15 minutes)
Activity
In an earlier lesson, students represented the imaginary number line as an axis perpendicular to the real number line. In this lesson, students used the real and imaginary number lines together to represent purely real and purely imaginary numbers. In this activity, students combine real and imaginary numbers with addition in order to plot complex numbers in the complex plane. The crucial difference between the complex plane and coordinate planes students have used before is that each point in the complex plane represents a single complex number rather than a pair of real numbers as they have seen before.
Launch
Display this image for all to see:
Ask students, “What would happen if we tried to add a real number to an imaginary number? How could we represent \(5+i\) visually?” Select 1–2 students to give brief responses before giving students time to work on the task. Follow with a wholeclass discussion.
Supports accessibility for: Visualspatial processing
Student Facing
 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.

\(\text 2  i\)

\(\text 6 + 3i\)

\(5+4i\)

\(1  3i\)

Student Response
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Student Facing
Are you ready for more?
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.
Student Response
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Activity Synthesis
Select 1–2 students to share their thinking for the last question and display their responses for all to see.
Tell students that together, the real number line and the imaginary number line form a coordinate system, and this complex number plane helps to visualize complex numbers. People call the real number line the real axis and the imaginary number line the imaginary axis. One important distinction to make is that points in coordinate planes students have seen before have been pairs of real numbers, like \((\text8, 7)\), but in the complex plane, each point represents a single complex number. Consider asking students:
 “How is the complex plane the same as coordinate planes you have used before? How is it different?” (Before, we needed two real numbers to make a point in the coordinate plane, but in the complex plane, each point represents a single complex number that results from adding a real number to an imaginary number. They are the same because they have two axes, they have positive and negative directions on those axes, and it’s possible to make gridlines that show different coordinates.)
 “In your own words, how would you plot \(\text 5i+7\)?” (This one has the real part and imaginary part switched from the task, but it should be the same process. Start at 0 and more 7 spaces along the positive real axis, and then move down 5 spaces to represent \(\text 5i\).)
Design Principle(s): Support sensemaking
Lesson Synthesis
Lesson Synthesis
In this lesson, students have investigated solutions to \(x^2=\texta\) for positive \(a\) and learned conventions for using the symbol \(i\). Here are some questions for discussion:
 “What is the number \(i\)?” (The number \(i\) is one of two imaginary solutions to the equation \(x^2=\text1\). It is plotted on the positive imaginary axis in the complex plane while the other solution, \(\texti\), is plotted on the negative imaginary axis.)
 “How are \(\sqrt{25}\) and \(\sqrt{\text25}\) different? How are they alike?” (\(\sqrt{25}\) is a real number on the positive real axis, while \(\sqrt{\text25}\) is an imaginary number on the positive imaginary axis. They are alike because they each represent one of two solutions to a quadratic equation.)
 “How are \(\sqrt{25}\) and \(\sqrt{\text25}\) related to the equations \(x^2=25\) and \(x^2=\text25\)?” (One of the solutions of \(x^2=25\) is \(\sqrt{25}\). One of the solutions of \(x^2=\text25\) is \(\sqrt{\text25}\).)
 “How can you write \(\sqrt{\text25}\) using the number \(i\)?” (\(\sqrt{\text25} = 5i\) because \(5i\) is the number on the positive imaginary axis that squares to make 25. Squaring \(5i\) gives \(5i \boldcdot 5i = 25i^2 = \text25\).)
11.5: Cooldown  Squaring Imaginary Numbers (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
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 \(\text 3 + 2i\).