Every fraction has a decimal equivalent that either terminates (for example,1/4=0.25 ) or repeats (for example,2/9=0.2 ). Work with a group to discover which fractions have terminating decimals and which have repeating decimals. You may assume that the numerator of each fraction you consider is and focus your attention on the denominator. As you complete the table below, you will find that the key to this question lies with the prime factorization of the denominator.
1/2
1/3
1/4
1/5
1/6
1/7
1/8
1/9
1/10
1/11
1/12
State a general rule describing which fractions have decimal forms that terminate and which have decimal forms that repeat.
Now test your rule on at least three new fractions. That is, be able to predict whether a fraction such as or has a terminating decimal or a repeating decimal. Then confirm your prediction.
Complete the table shown by converting each fraction to decimal form. The answer to the other questions should then become apparent.
The 1/x fractions that do NOT have terminating decimal forms are
1/3, 1/6, 1/7, 1/9, 1/11, 1/13, 1/14 etc. In all of these repeating cases, the denominator is either a prime number other than 5, or can be factored into a set of numbers that contain a prime number other than 5. My guess is that 5 is a special case because it is 1/2 of ten, the base of the decimal numbering system.
Show us your work and we will be glad to critique it.
To determine whether a fraction has a decimal form that terminates or repeats, we can look at the prime factorization of the denominator.
Let's work through the given fractions and analyze their decimal forms:
1/2: In decimal form, this fraction is 0.5, which terminates.
1/3: In decimal form, this fraction is 0.3333..., which repeats.
1/4: In decimal form, this fraction is 0.25, which terminates.
1/5: In decimal form, this fraction is 0.2, which terminates.
1/6: In decimal form, this fraction is 0.1666..., which repeats.
1/7: In decimal form, this fraction is 0.142857142857..., which repeats.
1/8: In decimal form, this fraction is 0.125, which terminates.
1/9: In decimal form, this fraction is 0.1111..., which repeats.
1/10: In decimal form, this fraction is 0.1, which terminates.
1/11: In decimal form, this fraction is 0.090909..., which repeats.
1/12: In decimal form, this fraction is 0.0833..., which repeats.
From these examples, we can observe a pattern. Fractions that have denominators which are powers of 2 (e.g., 2, 4, 8) or denominators which are powers of 5 (e.g., 5, 10) have decimal forms that terminate. Fractions with denominators containing prime factors other than 2 or 5 (e.g., 3, 6, 7, 9, 11, 12) have decimal forms that repeat.
Based on this pattern, we can state the general rule:
Fractions whose denominators consist only of factors of 2 and/or 5 (such as powers of 2 or 5) have decimal forms that terminate. Fractions with denominators that have factors other than 2 or 5 (including prime factors other than 2 or 5) have decimal forms that repeat.
Now, let's test this rule on three new fractions:
1/16: The denominator 16 is a power of 2, so according to our rule, the decimal form should terminate. Confirming this, 1/16 in decimal form is 0.0625, which terminates.
1/20: The denominator 20 has factors of 2 and 5, so according to our rule, the decimal form should terminate. Confirming this, 1/20 in decimal form is 0.05, which terminates.
1/15: The denominator 15 has factors of 3 and 5, both of which are not 2 or 5. According to our rule, the decimal form should repeat. Confirming this, 1/15 in decimal form is 0.0666..., which repeats.
Our rule holds true for these three fractions, further affirming its validity.