Lesson 20

• Let’s explore irrational numbers.

20.1: Where is $\sqrt{21}$?

Which number line accurately plots the value of $$\sqrt{21}$$? Explain your reasoning.

20.2: Some Rational Properties

Rational numbers are fractions and their opposites.

1. All of these numbers are rational numbers. Show that they are rational by writing them in the form $$\frac{a}{b}$$ or $$\text{-}\frac{a}{b}$$ for integers $$a$$ and $$b$$.
1. 6.28
2. $$\text{-}\sqrt{81}$$
3. $$\sqrt{\frac{4}{121}}$$
4. -7.1234
5. $$0.\overline{3}$$
6. $$\frac{1.1}{13}$$
2. All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
1. $$\frac{47}{1,000}$$
2. $$\text{-}\frac{12}{5}$$
3. $$\frac{\sqrt{9}}{6}$$
4. $$\frac{53}{9}$$
5. $$\frac{1}{7}$$
3. What do you notice about the decimal representations of rational numbers?

20.3: Approximating Irrational Values

Although $$\sqrt{2}$$ is irrational, we can approximate its value by considering values near it.

1. How can we know that $$\sqrt{2}$$ is between 1 and 2?
2. How can we know that $$\sqrt{2}$$ is between 1.4 and 1.5?
3. Approximate the next decimal place for $$\sqrt{2}$$.
4. Use a similar process to approximate the $$\sqrt{5}$$ to the thousandths place.