Gabriel and Lisbeth are having a discussion about rational and irrational numbers. Gabriel has been performing an experiment by measuring circles. He used a string and a ruler to measure the distance around several circles, as well as their diameters.

Question

Gabriel records the distance and diameters in the table shown.

diameter (in.). distance (in.)

7 22

3 9.5

6.25 19.75

8 25

2.5 8

Then, after he plots the points on a graph, he draws a line of best fit. This is a straight line that most closely follows a trend in the data. He then calculates its slope.
y=3.1434x

Gabriel claims that the slope of the line, approximately 3.14, represents the irrational number ㅠ. Lisbeth is confused and asks how dividing two rational numbers can result in an irrational number. How can Gabriel respond to her question? He can think of two possible claims:

Claim A: Maybe is actually a rational number.

Claim B: Maybe the circumference is actually irrational.

Choose one claim that you think could be true and assume that the other is false. Write a script for what Gabriel could say to Lisbeth to explain why one could be possible and the other cannot.

If you are not sure how to proceed, here are some tips you can use as you

work through each Try This question:

1. For Claim A, Gabriel could explain what the criteria are for a number to be considered rational and describe how does or does not meet those criteria.

Refer to the section on sums and products of rational numbers for further details.

2. For Claim B, Gabriel could use the properties of multiplying rational and irrational numbers together to support or refute it. Refer to the section on sums and products of rational and irrational numbers for further details,

Answer 1

Script:

Gabriel: Hey Lisbeth! I know you're confused about how I got the slope of approximately 3.14 and why I think it represents the irrational number π. Let’s break it down together.

First, I want to address Claim A: "Maybe π is actually a rational number." For a number to be considered rational, it must be able to be expressed as the quotient of two integers, which means you can write it as a fraction \( \frac{a}{b} \), where both a and b are integers and b is not zero.

The number π, however, has been proven to be irrational, meaning it cannot be expressed exactly as a fraction of two integers. In fact, its decimal representation goes on forever without repeating. Therefore, Claim A can't be true; π is not a rational number.

Now, let’s look at Claim B: "Maybe the circumference is actually irrational." This one can actually be true! When we calculate the circumference of a circle using the formula \( C = π \cdot d \), we are taking π, which is irrational, and multiplying it by a rational number (the diameter \( d \)).

According to mathematical principles, when you multiply a rational number by an irrational number, the product is always irrational. So, the circumference of a circle, derived from π and a diameter, is indeed irrational.

Therefore, even though the ratio of various circumferences to their diameters in our experiment approximately gives us the value of π, that doesn't mean the circumference itself is rational. It actually confirms that the circumference based on π is irrational!

So, to recap: Claim A isn’t true; π is irrational, but Claim B is true because the circumference of circles, calculated using this irrational number, also results in irrational numbers.

I hope that clears things up!