## 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...

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