Lesson 10

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.

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.

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.

Tell students, “Once people invented negative numbers, that meant that any addition or subtraction equation would always have a solution on the number line. The numbers on the number line are called the real numbers. It turns out there are numbers that are not on the number line. We are going to learn about those numbers today.”

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.

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

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.

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.”

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.

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).

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.

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.)