# Lesson 15

Infinite Decimal Expansions

### Problem 1

Elena and Han are discussing how to write the repeating decimal $$x = 0.13\overline{7}$$ as a fraction. Han says that $$0.13\overline{7}$$ equals $$\frac{13764}{99900}$$. “I calculated $$1000x = 137.77\overline{7}$$ because the decimal begins repeating after 3 digits. Then I subtracted to get $$999x = 137.64$$. Then I multiplied by $$100$$ to get rid of the decimal: $$99900x = 13764$$. And finally I divided to get $$x = \frac{13764}{99900}$$.” Elena says that $$0.13\overline{7}$$ equals $$\frac{124}{900}$$. “I calculated $$10x = 1.37\overline{7}$$ because one digit repeats. Then I subtracted to get $$9x = 1.24$$. Then I did what Han did to get $$900x = 124$$ and $$x = \frac{124}{900}$$.”

Do you agree with either of them? Explain your reasoning.

### Problem 2

How are the numbers $$0.444$$ and $$0.\overline{4}$$ the same? How are they different?

### Problem 3

1. Write each fraction as a decimal.
1. $$\frac{2}{3}$$

2. $$\frac{126}{37}$$

2. Write each decimal as a fraction.

1. $$0.\overline{75}$$

2. $$0.\overline{3}$$

### Problem 4

Write each fraction as a decimal.

1. $$\frac{5}{9}$$

2. $$\frac{5}{4}$$

3. $$\frac{48}{99}$$

4. $$\frac{5}{99}$$

5. $$\frac{7}{100}$$

6. $$\frac{53}{90}$$

### Problem 5

Write each decimal as a fraction.

1. $$0.\overline{7}$$

2. $$0.\overline{2}$$

3. $$0.1\overline{3}$$

4. $$0.\overline{14}$$

5. $$0.\overline{03}$$

6. $$0.6\overline{38}$$

7. $$0.52\overline{4}$$

8. $$0.1\overline{5}$$

### Solution

$$2.2^2 = 4.84$$ and $$2.3^2 = 5.29$$. This gives some information about $$\sqrt 5$$.
Without directly calculating the square root, plot $$\sqrt{5}$$ on all three number lines using successive approximation.