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.

A

Number line by ones from 16 to 25.

B

Number line with 11 evenly spaced tick marks. Labeled 4, blank, blank, blank, blank, 4 point 5, blank, blank, blank, blank, 5. Point on second tick mark.

C

Number line with 11 evenly spaced tick marks.

D

Number line with 11 evenly spaced tick marks. Labeled 4, blank, blank, blank, blank, 4 point 5, blank, blank, blank, blank, 5. Point on fifth tick mark.

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.

Summary