1.4 Decimal Representation of Rational Numbers [35m]
You definitely know about "Google" and have used its search engine, gmail, and other services. So here are two questions for you to think about:
- How is "Google" related to "powers of 10"?
- How are powers of 10 related to decimal numbers?
If you know the answers do write them in the discussion board at the end of this lesson.
EXERCISE: Write each of these numbers in decimal form and say what kind of decimal expansion it has.
(i)     (ii)     (iii)     (iv)     (v)     (vi)
If a recurring decimal is multiplied by an integer, will the resulting product also be a recurring decimal?
A recurring decimal is a rational number. If that rational number is multiplied by an integer, the product will again be a rational number. The decimal representation of the product can either be terminating or recurring. Can the decimal representation of the product be a terminating decimal?
EXERCISE: You know that . Can you predict what the decimal expansion of are, without actually doing the long division? If so, how?
FACT: The maximum number of digits in the repeating block of digits in a decimal expansion is always less than the divisor.
EXERCISE: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.
SELF PRACTICE: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of these numbers:
a) b) c)
Perform the division to check your answer.
EXERCISE: Look at several examples of rational numbers in the form (q ≠ 0), 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?
EXERCISE: Classify the following numbers as rational or irrational:
(i) (ii) (iii) 0.3796 (iv) 7.478478... (v) 1.101001000100001...
SELF PRACTICE: Classify the following numbers as rational or irrational:
(i) (ii) (iii) 0.05896 (iv) 9.0621621621... (v) 3.121221222122221...