# Lesson 10

A New Kind of Number

## 10.1: Numbers Are Inventions (10 minutes)

### Warm-up

The goal of this activity is to prepare students for extending the real numbers to the complex numbers in the next few activities. The point of this activity is not for students to come up with an appropriate explanation for why \(5-8\) is well-defined, but to grapple with the cognitive dissonance that many people experience when extending the positive integers to the integers.

### Launch

Arrange students in groups of 2. Give a few minutes of quiet think time before asking students to share their responses with their partners. Follow with a whole-class discussion.

### Student Facing

Jada was helping her cousin with his math homework. He was supposed to solve the equation \(8 + x = 5\). He said, “If I subtract 8 from both sides, I get \(x = 5 - 8\). This doesn’t make sense. You can’t subtract a bigger number from a smaller number. If I have 5 grapes, I can’t eat 8 of them!”

What do you think Jada could say to her cousin to help him understand why \(5 - 8\) actually does make sense?

### Student Response

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### Activity Synthesis

Select 1–2 groups to share their ideas about how to explain negative numbers and to discuss any interesting differences between explanations.

Display a number line showing only non-negative numbers, like the one shown here.

Tell students, “For a long time, people only worked with positive numbers. They didn’t think you could subtract a bigger number from a smaller number. But then people invented negative numbers. We can think of this as expanding the number line from just the positive half to include the negative half.” Display a number line showing both positive and negative numbers, like the one shown here.

## 10.2: The Square Root of Negative One (15 minutes)

### Activity

In this activity, students use the graph of \(y = x^2\) to explain why \(x^2 = \text- 1\) does not have any real solutions, revisiting what they have done in previous lessons from the perspective of the number line. In the synthesis, students should make connections between the graph of \(y=x^2\) and the fact that real numbers can’t square to make negative numbers.

### Launch

Tell students that sometimes people call the number line the *real number line*.

*Representation: Internalize Comprehension.*Activate or supply background knowledge. Provide students with access to additional blank number lines to support students who benefit from additional processing time.

*Supports accessibility for: Visual-spatial processing; Organization*

### Student Facing

Numbers on the number line are often called **real numbers**.

- The equation \(x^2 = 9\) has 2 real solutions. How can you see this on the graph of \(y = x^2\)? Draw points on this real number line to represent these 2 solutions.
- How many real solutions does \(x^2 = 0\) have? Explain how you can see this on the graph of \(y = x^2\). Draw the solution(s) on a real number line.
- How many real solutions does \(x^2 = \text{-} 1\) have? Explain how you can see this on the graph of \(y = x^2\). Draw the solution(s) on a real number line.

### Student Response

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### Anticipated Misconceptions

If students do not draw a separate real number line, but instead try to represent the values of \(x\) that satisfy the quadratic equations as points on the coordinate plane, remind them that the horizontal axis of the coordinate plane is a very different thing from the real number line. All points on the coordinate plane represent pairs of numbers, not single numbers. When students begin representing imaginary and real numbers together in the next activity, and use the complex plane to represent complex numbers in the next few lessons, they may be tempted to think of the complex plane as the same kind of thing as the coordinate plane. But just like the real number line, the complex plane only represents individual numbers, not pairs of them.

### Activity Synthesis

Select students to share their explanations. Display a graph of \(y = x^2\) and draw the lines \(y = 9\), \(y = 0\), and \(y = \text- 1\) for all to see. Then make sure the real number line that shows the solutions is drawn separately.

Tell students that whenever we square a number, we multiply it by itself. When squaring a number on the number line, otherwise known as a *real* number, the result is either

- Positive, because a positive times a positive is positive, or
- 0, because 0 times 0 is 0, or
- Positive, because a negative times a negative is positive.

So squaring a *real* number never results in a negative number. We can see this in the graph of \(y = x^2\) because none of the points on the graph are below the \(x\)-axis. That tells us that the equation \(x^2 = \text{-} 1\) does not have any real solutions, which means none of the numbers on the number line make this equation true.

Let’s invent a new number that is *not* on the number line that does satisfy this equation. Let’s write it as \(\sqrt{\text- 1}\) and draw a point to represent this number. We can put it anywhere we want as long as we don’t put it on the real number line. For example, let’s put it here, right above 0 on the real number line. Display this image for all to see:

Explain that \(\sqrt{\text- 1}\) is defined to be a solution to the equation \(x^2 = \text{-} 1\), so

\(\left(\sqrt{\text- 1}\right)^2 = \text{-}1\)

Lastly, tell students, “Even though it isn’t on the real number line and therefore isn’t a real number, it really is a number—it is just a different kind of number called an** imaginary number**. It could have been named a “blue number” or a “fish number.” The word “imaginary” shouldn’t be taken literally.”

*Speaking: MLR8 Discussion Supports.*As students share their responses to the first question, press for details by asking how they know that the intersection of the horizontal line \(y=9\) and the graph of \(y=x^2\) represents the solution to the equation \(x^2=9\). This will help students make connections between the graphs of \(y=x^2\) and \(y=9\) and the equation \(x^2=9\).

*Design Principle(s): Support sense-making*

## 10.3: Imaginary Numbers (10 minutes)

### Activity

In this activity, students build the imaginary axis using what they know about the real number line. Students know that real numbers can be represented as horizontal arrows that start at 0 on the number line, and apply this same thinking to draw imaginary numbers using vertical arrows that start at 0.

### Launch

Explain that sometimes we represent numbers on the real number line with arrows. Positive numbers are represented with arrows that point to the right, and negative numbers are represented with arrows that point to the left. The real number 1 is represented on the number line shown. Ask, “How would we represent -1 on the real number line?” (Draw an arrow starting at 0 and ending at -1).

### Student Facing

- On the real number line:
- Draw an arrow starting at 0 that represents 3.
- Draw an arrow starting at 0 that represents -5.

- This diagram shows an arrow that represents \(\sqrt{\text- 1}\).
- Draw an arrow starting at 0 that represents \(3\sqrt{\text- 1}\).
- Draw an arrow starting at 0 that represents \(\text- \sqrt{\text- 1}\).
- Draw an arrow starting at 0 that represents \(\text- 5\sqrt{\text- 1}\).

### Student Response

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### Student Facing

#### Are you ready for more?

The absolute value of a real number is the length of the arrow that represents it.

- What is the relationship between the absolute value of a real number and the absolute value of the square of that number?
- If we want \(\sqrt{\text- 1}\) and its square to have this same relationship, then what should the absolute value of \(\sqrt{\text-1}\) be?
- What should the absolute value of \(3\sqrt{\text- 1}\) be?

### Student Response

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### Anticipated Misconceptions

If students aren't sure how to represent multiples of \(\sqrt{\text-1}\) on the imaginary number line, it may be helpful for them to think of taking a certain number of steps of distance \(\sqrt{\text-1}\) along the imaginary number line, in the same way that we can represent multiples of 1 on the real number line by going left or right in steps of distance 1. For example, to represent 3 on the real number line, you can go 3 times farther to the right than 1 is; similarly, to represent \(3\sqrt{\text-1}\) on the imaginary number line, you would go 3 times farther up than \(\sqrt{\text-1}\) is.

### Activity Synthesis

Tell students that they have just built the *imaginary number line*. Points on the imaginary number line are numbers that are a real number times the imaginary number \(\sqrt{\text-1}\). Here are some questions for discussion:

- “How are the real number line and imaginary number line the same? How are they different?” (They are the same because they are both lines, and points on the lines represent different numbers. Both number lines have the number 0 on them. They are different because all imaginary numbers have a factor that is \(\sqrt{\text-1}\).)
- “What happens when you square an imaginary number? Choose an imaginary number and square it. How is that different from what happens with real numbers?” (For example, \((2\sqrt{\text-1})^2=4 \boldcdot \text-1 = \text-4\). Imaginary numbers square to make negative numbers. Real numbers don’t do that.)

*Conversing: MLR2 Collect and Display.*During the discussion, listen for and collect the language students use to compare characteristics of the real number line and the imaginary number line. Call students’ attention to words and phrases such as “both lines have the number 0” or “real numbers square to make real numbers.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Consider using a Venn diagram to organize the words and phrases. Remind students to borrow language from the display as needed. This will help students use mathematical language in describing similarities and differences between real and imaginary numbers.

*Design Principle(s): Maximize meta-awareness*

*Representation: Internalize Comprehension.*Use color and annotations to illustrate student thinking. As students share their reasoning on how they drew their arrows to create an imaginary number line, scribe their thinking on a visible display. Invite students to identify how the the real number line and imaginary number line are the same and different.

*Supports accessibility for: Visual-spatial processing; Conceptual processing*

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students defined a new number that is a solution to the equation \(x^2=\text-1\) and used that new number to construct the imaginary number line and complex plane. Here are some possible questions for discussion:

- “What is a real number?” (A real number is a number on the real number line. Real numbers can be positive, negative, or 0. Real numbers can’t square to make negative numbers because positive real numbers square to make positive real numbers, negative real numbers square to make positive real numbers, and 0 squares to make 0.)
- “What is an imaginary number?” (The imaginary number \(\sqrt{\text-1}\) is defined to be a number that squares to make -1, so we already know it can’t be a real number. Multiplying \(\sqrt{\text-1}\) by real numbers creates the imaginary number line.)
- “What is a complex number?” (Putting the real and imaginary number lines together so they intersect at 0 makes a coordinate plane called the complex plane. Adding real and imaginary numbers together makes a complex number, and complex numbers can be plotted in the complex plane.)
- “One common misconception is that imaginary numbers are
*made up*and real numbers aren’t. What would you say to someone with that misconception?” (An imaginary number is just a number whose square isn’t a positive number. The name has nothing to do with the ordinary English meaning of the word “imaginary.”)

## 10.4: Cool-down - What is Real? What is Imaginary? (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Sometimes people call the number line the *real number line*.

- A positive number times a positive number is always positive. So when we square a positive number, the result will always be positive.
- A negative number times a negative number is always positive. So when we square a negative number, the result will always be positive.
- 0 squared is 0.

So squaring a *real* number never results in a negative number. We can conclude that the equation \(x^2 = \text{-} 1\) does not have any real number solutions. In other words, none of the numbers on the real number line satisfy this equation.

Mathematicians invented a new number that is *not* on the real number line. This new number was invented as a solution to the equation \(x^2 = \text{-} 1\). For now, let’s write it \(\sqrt{\text- 1}\) and draw a point to represent this number. We can put it anywhere we want as long as we don’t put it on the real number line. For example, let’s put it here, right above 0 on the real number line:

This new number \(\sqrt{\text- 1}\) is a solution to the equation \(x^2 = \text{-} 1\), so \(\left(\sqrt{\text- 1}\right)^2 = \text-1\). If we draw a line that passes through 0 on the real number line and \(\sqrt{\text- 1}\), we get the *imaginary number line*. The numbers on the imaginary number line are called the **imaginary numbers**.