# Lesson 12

Infinite Decimal Expansions

### 12.1: Searching for Digits

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

### 12.2: 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. $$0.7\overline{88}$$

Use this technique to find fractional representations for $$0.\overline{3}$$ and $$0.\overline{9}$$.

### 12.3: 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.

### Glossary Entries

• cube root

The cube root of a number $$n$$ is the number whose cube is $$n$$. It is also the edge length of a cube with a volume of $$n$$. We write the cube root of $$n$$ as $$\sqrt[3]{n}$$.

For example, the cube root of 64, written as $$\sqrt[3]{64}$$, is 4 because $$4^3$$ is 64. $$\sqrt[3]{64}$$ is also the edge length of a cube that has a volume of 64.

• repeating decimal

A repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them.

For example, the decimal representation for $$\frac13$$ is $$0.\overline{3}$$, which means 0.3333333 . . . The decimal representation for $$\frac{25}{22}$$ is $$1.1\overline{36}$$ which means 1.136363636 . . .