Lesson 20

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

A

B

C

D

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?

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.