Lesson 15: Infinite Decimal Expansions
Let’s think about infinite decimals.
Illustrative Math Unit 8.8, Lesson 15 (printable worksheets)
Lesson 15 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 7/5). Since there is no fraction equal to √2 it is not a rational number, which is why we call it an irrational number. Another well-known irrational number is π.
Any number, rational or irrational, has a decimal expansion. Sometimes it goes on forever. For example, the rational number 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 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.
Lesson 15.1 Searching for Digits
The first 3 digits after the decimal for the decimal expansion of 3/7 have been calculated. Find the next 4 digits.
Lesson 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 . Arrange these in order to see how he figured out that without needing a calculator.
- Use Noah’s method to calculate the fraction representation of:
Are you ready for more?
Use this technique to find fractional representations for
Lesson 15.3 Some Numbers Are Not Rational
- a. Why is √2 between 1 and 2 on the number line?
b. Why is √2 between 1.4 and 1.5 on the number line?
c. How can you figure out an approximation for √2 accurate to 3 decimal places?
d. Label all of the tick marks. Plot √2 on all three number lines. Make sure to add arrows from the second to the third number lines.
- a. 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 using these values and the equation for circumference, C = 2πr?
b. 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 using these values and the equation for circumference, C = 2πr?
c. Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of and plot that number π on all three number lines.
d. How can you explain the differences between these calculations of π?
Lesson 15 Practice Problems
- Elena and Han are discussing how to write the repeating decimal as a fraction. Han says that equals . “I calculated because the decimal begins repeating after 3 digits. Then I subtracted to get . Then I multiplied by to get rid of the decimal: . And finally I divided to get .” Elena says that equals . “I calculated because one digit repeats. Then I subtracted to get . Then I did what Han did to get and .”
Do you agree with either of them? Explain your reasoning.
- How are the numbers and the same? How are they different?
- a. Write each fraction as a decimal.
b. Write each decimal as a fraction.
- Write each fraction as a decimal.
- Write each decimal as a fraction.
- 2.22 = 4.84 and 2.32 = 5.29. This gives some information about √5.
Without directly calculating the square root, plot √5 on all three number lines using successive approximation.
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