Lesson 15
Infinite Decimal Expansions
Let’s think about infinite decimals.
15.1: Searching for Digits
The first 3 digits after the decimal for the decimal expansion of \frac37 have been calculated. Find the next 4 digits.

15.2: 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.
-
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.
-
Use Noah’s method to calculate the fraction representation of:
- 0.1\overline{86}
- 0.7\overline{88}
Use this technique to find fractional representations for 0.\overline{3} and 0.\overline{9}.
15.3: Some Numbers Are Not Rational
-
- Why is \sqrt{2} between 1 and 2 on the number line?
- Why is \sqrt{2} between 1.4 and 1.5 on the number line?
- How can you figure out an approximation for \sqrt{2} accurate to 3 decimal places?
-
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.
-
- 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?
- 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?
-
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.
- 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 \frac75). 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 . . . 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 \frac2{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.
Video Summary
Glossary Entries
- 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 \frac13 is 0.\overline{3}, which means 0.3333333 . . . The decimal representation for \frac{25}{22} is 1.1\overline{36} which means 1.136363636 . . .