Lesson 12
Infinite Decimal Expansions
Let’s think about infinite decimals.
Problem 1
Elena and Han are discussing how to write the repeating decimal \(x = 0.13\overline{7}\) as a fraction. Han says that \(0.13\overline{7}\) equals \(\frac{13764}{99900}\). “I calculated \(1000x = 137.77\overline{7}\) 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 \(0.13\overline{7}\) 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.
Problem 2
How are the numbers \(0.444\) and \(0.\overline{4}\) the same? How are they different?
Problem 3
- Write each fraction as a decimal.
-
\(\frac{2}{3}\)
-
\(\frac{126}{37}\)
-
-
Write each decimal as a fraction.
-
\(0.\overline{75}\)
-
\(0.\overline{3}\)
-
Problem 4
Write each fraction as a decimal.
-
\(\frac{5}{9}\)
-
\(\frac{5}{4}\)
-
\(\frac{48}{99}\)
-
\(\frac{5}{99}\)
-
\(\frac{7}{100}\)
-
\(\frac{53}{90}\)
Problem 5
Write each decimal as a fraction.
-
\(0.\overline{7}\)
-
\(0.\overline{2}\)
-
\(0.1\overline{3}\)
-
\(0.\overline{14}\)
-
\(0.\overline{03}\)
-
\(0.6\overline{38}\)
-
\(0.52\overline{4}\)
-
\(0.1\overline{5}\)
Problem 6
\(2.2^2 = 4.84\) 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.
![A zooming number line that is composed of 3 number lines, aligned vertically, each with 11 evenly spaced tick marks.](https://staging-cms-im.s3.amazonaws.com/7Q2ZdeUtGRFUZe2UN2ttWmRK?response-content-disposition=inline%3B%20filename%3D%228-8.8.D.PP.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.D.PP.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T083350Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4a4c0d888e2a0bdc50b7dcda0ead8f5e2f56c0807576204afff5ff4d7083ed60)