8.4.2: Infinite Decimal Expansions

Lesson

Let's think about infinite decimals.

Exercise \(\PageIndex{1}\): Searching for Digits

The first 3 digits after the decimal for the decimal expansion of \(\frac{3}{7}\) have been calculated. Find the next 4 digits.

Figure \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\): Some Numbers are Rational

Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.

1. The cards show Noah’s work calculating the fraction representation of \(0.4\overline{85}\). Arrange these in order to see how he figured out that \(0.4\overline{85}=\frac{481}{990}\) without needing a calculator.
2. Use Noah’s method to calculate the fraction representation of:
   1. \(0.1\overline{86}\)
   2. \(0.7\overline{88}\)

Are you ready for more?

Use this technique to find fractional representations for \(0.\overline{3}\) and \(0.\overline{9}\).

Exercise \(\PageIndex{3}\): Some Numbers are Not Rational

1.

1. Why is \(\sqrt{2}\) between 1 and 2 on the number line?
2. Why is \(\sqrt{2}\) between 1.4 and 1.5 on the number line?
3. How can you figure out an approximation for \(\sqrt{2}\) accurate to 3 decimal places?
4. Label all of the tick marks. Plot \(\sqrt{2}\) on all three number lines. Make sure to add arrows from the second to the third number lines.

Figure \(\PageIndex{2}\)

2.

1. Elena notices a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. What value do you get for \(\pi\) using these values and the equation for circumference, \(C=2\pi r\)?
2. Diego learned that one of the space shuttle fuel tanks had a diameter of 840 cm and a circumference of 2,639 cm. What value do you get for \(\pi\) using these values and the equation for circumference, \(C=2\pi r\)?
3. Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of \(\pi\) and plot that number on all three number lines.

Figure \(\PageIndex{3}\)

d. How can you explain the differences between these calculations of \(\pi\)?

Summary

Not every number is rational. Earlier we tried to find a fraction whose square is equal to 2. That turns out to be impossible, although we can get pretty close (try squaring \(\frac{7}{5}\)). Since there is no fraction equal to \(\sqrt{2}\) it is not a rational number, which is why we call it an irrational number. Another well-known irrational number is \(\pi\).

Any number, rational or irrational, has a decimal expansion. Sometimes it goes on forever. For example, the rational number \(\frac{2}{11}\) has the decimal expansion \(0.181818\ldots\) with the 18s repeating forever. Every rational number has a decimal expansion that either stops at some point or ends up in a repeating pattern like \(\frac{2}{11}\). Irrational numbers also have infinite decimal expansions, but they don't end up in a repeating pattern. From the decimal point of view we can see that rational numbers are pretty special. Most numbers are irrational, even though the numbers we use on a daily basis are more frequently rational.

Glossary Entries

Definition: Repeating Decimal

A repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them.

For example, the decimal representation for \(\frac{1}{3}\) is \(0.\overline{3}\), which means 0.3333333 . . . The decimal representation for \(\frac{25}{22}\) is \(1.1\overline{36}\) which means 1.136363636 . . .

Practice

Exercise \(\PageIndex{4}\)

Elena and Han are discussing how to write the repeating decimal \(x=0.1\overline{37}\) as a fraction. Han says that \(0.1\overline{37}\) equals \(\frac{13764}{99900}\). “I calculated \(1000x=137.7\overline{77}\) 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 \(0.1\overline{37}\) 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.

Exercise \(\PageIndex{5}\)

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

Exercise \(\PageIndex{6}\)

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. \(0.\overline{3}\)

Exercise \(\PageIndex{7}\)

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. \(\frac{53}{90}\)

Exercise \(\PageIndex{8}\)

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. \(0.1\overline{5}\)

Exercise \(\PageIndex{9}\)

\(2.2^{2}=4.84\) 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.

Figure \(\PageIndex{4}\)