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Number Systems

Short Answer Type

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

3 over 13

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

2 over 11

177 Views

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

329 over 400

158 Views

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.]

281 Views

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

364 Views

36. Express the following in the form

p over q

where p and q are integers and q ≠ 0.

0. top enclose 47

320 Views

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

234 Views

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.

266 Views

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.

232 Views

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?

Examples

left parenthesis straight i right parenthesis space 1 half equals fraction numerator 1 cross times 5 over denominator 2 cross times 5 end fraction equals 5 over 10 equals 0.5 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose left parenthesis ii right parenthesis space 1 fourth equals fraction numerator 1 cross times 25 over denominator 4 cross times 25 end fraction equals 25 over 100 equals 0.25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose squared left parenthesis iii right parenthesis space 3 over 8 equals fraction numerator 1 cross times 125 over denominator 8 cross times 125 end fraction equals 375 over 1000 equals 0.375 space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose cubed left parenthesis iv right parenthesis space 31 over 25 equals fraction numerator 31 cross times 4 over denominator 25 cross times 4 end fraction equals 124 over 100 equals 1.24 space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 5 squared end enclose left parenthesis straight v right parenthesis space 7 over 125 equals fraction numerator 7 cross times 8 over denominator 125 cross times 8 end fraction equals 56 over 1000 equals 0.056 space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 5 cubed end enclose left parenthesis vi right parenthesis space 13 over 20 equals fraction numerator 13 cross times 5 over denominator 20 cross times 5 end fraction equals 65 over 100 equals 0.65 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose squared cross times 5 left parenthesis vii right parenthesis space 29 over 16 equals fraction numerator 29 cross times 625 over denominator 16 cross times 625 end fraction equals 18125 over 10000 equals 1.8125 space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose to the power of 4

The property that q must satisfy is that the prime factorisation of q must have only powers of 2 or powers of 5 or both.

430 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

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