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Number Systems

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31. Write the following in decimal form and say what kind of decimal expansion each has:

$3 over 13$

3 over 13

13 space right parenthesis space 3.00000000000 space left parenthesis 0.23076923079...... space space space space bottom enclose 26 space end enclose space space space space space 40 space space space space space bottom enclose space 39 thin space space space end enclose space space space space space space 100 space space space space space space space bottom enclose thin space thin space space 91 thin space space space end enclose space space space space space space space space 90 space space space space space space thin space bottom enclose thin space space 78 space space space thin space thin space end enclose space space space space space space space space space space space 120 space space space space space space space space space bottom enclose space space space 117 space space space end enclose space space space space space space space space space space space space space space 30 space space space space space space space space space space space space bottom enclose space space 26 space space space space end enclose space space space space space space space space space space space space space space space space space space 40 space space space space space space space space space space space space space space space bottom enclose space space 39 space space space space end enclose space space space space space space space space space space space space space space space space space space space 100 space space space space space space space space space space space space space space space space space bottom enclose space space space space 91 space space space end enclose space space space space space space space space space space space space space space space space space space space space space 90 space space space space space space space space space space space space space space space space space space bottom enclose space space space 78 space space space end enclose space space space space space space space space space space space space space space space space space space space space 120 space space space space space space space space space space space space space space space space space space space bottom enclose space 117 space space end enclose space space space space space space space space space space space space space space space space space space space space space space space 3 therefore space space 3 over 13 equals space 0.230769230769 space...... space equals space 0. top enclose 230769

The decimal expansion is non-terminating repeating.

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32. Write the following in decimal form and say what kind of decimal expansion each has:

2 over 11

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33. Write the following in decimal form and say what kind of decimal expansion each has:

329 over 400

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34. You know that

1 over 7 equals 0. top enclose 142857

Can you predict what the decimal expansions of

2 over 7 comma space 3 over 7 comma space 4 over 7 comma space 5 over 7 comma space 6 over 7

are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of

1 over 7

carefully.]

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35. Express the following in the form

straight p over straight q

where p and q are integers and q ≠ 0.

left parenthesis straight i right parenthesis space 0. top enclose 6

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36. Express the following in the form

p over q

where p and q are integers and q ≠ 0.

0. top enclose 47

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37. Express the following in the form

straight p over straight q

where p and q are integers and q ≠ 0.

0. top enclose 001

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38. Express 0.99999...... in the form

straight p over straight q comma

Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

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39. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of

1 over 17 ?

Perform the division to check your answer.

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40. Look at several examples of rational numbers in the form

straight p over straight q left parenthesis q not equal to 0 right parenthesis comma

where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

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