# Lesson 20

Quadratics and Irrationals

- 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.

- 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 \(b\).
- 6.28
- \(\text{-}\sqrt{81}\)
- \(\sqrt{\frac{4}{121}}\)
- -7.1234
- \(0.\overline{3}\)
- \(\frac{1.1}{13}\)

- All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
- \(\frac{47}{1,000}\)
- \(\text{-}\frac{12}{5}\)
- \(\frac{\sqrt{9}}{6}\)
- \(\frac{53}{9}\)
- \(\frac{1}{7}\)

- 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.

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