Lesson 15: Infinite Decimal Expansions

15.1: Searching for Digits

The first 3 digits after the decimal for the decimal expansion of $\frac37$ have been calculated. Find the next 4 digits.

15.2: Some Numbers Are Rational

Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.

1. The cards show Noah’s work calculating the fraction representation of $0.4\overline{85}$. Arrange these in order to see how he figured out that $0.4\overline{85} = \frac{481}{990}$ without needing a calculator.

2. Use Noah’s method to calculate the fraction representation of:

1. $0.1\overline{86}$
2. $0.7\overline{88}$

15.3: Some Numbers Are Not Rational

1. Why is $\sqrt{2}$ between 1 and 2 on the number line?
2. Why is $\sqrt{2}$ between 1.4 and 1.5 on the number line?
3. How can you figure out an approximation for $\sqrt{2}$ accurate to 3 decimal places?
4. Label all of the tick marks. Plot $\sqrt{2}$ on all three number lines. Make sure to add arrows from the second to the third number lines.

1. Elena notices a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. What value do you get for $\pi$ using these values and the equation for circumference, $C=2\pi r$?
2. Diego learned that one of the space shuttle fuel tanks had a diameter of 840 cm and a circumference of 2,639 cm. What value do you get for $\pi$ using these values and the equation for circumference, $C=2\pi r$?
3. Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of $\pi$ and plot that number on all three number lines.

4. How can you explain the differences between these calculations of $\pi$?

Summary

Not every number is rational. Earlier we tried to find a fraction whose square is equal to 2. That turns out to be impossible, although we can get pretty close (try squaring $\frac75$). Since there is no fraction equal to $\sqrt{2}$ it is not a rational number, which is why we call it an irrational number. Another well-known irrational number is $\pi$.

Any number, rational or irrational, has a decimal expansion. Sometimes it goes on forever. For example, the rational number $\frac{2}{11}$ has the decimal expansion $0.181818 . . .$ with the 18s repeating forever. Every rational number has a decimal expansion that either stops at some point or ends up in a repeating pattern like $\frac2{11}$. Irrational numbers also have infinite decimal expansions, but they don't end up in a repeating pattern. From the decimal point of view we can see that rational numbers are pretty special. Most numbers are irrational, even though the numbers we use on a daily basis are more frequently rational.