## 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:

1. How is "Google" related to "powers of 10"?
2. 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) $\frac{36}{100}$     (ii) $\frac{1}{11}$    (iii) $4\frac{1}{8}$    (iv) $\frac{3}{13}$    (v) $\frac{2}{11}$    (vi) $\frac{329}{400}$

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 $\frac{1}{7}=0.\overline{142857}$. Can you predict what the decimal expansion of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ 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 $\frac{1}{17}$? 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) $\frac{1}{13}$     b) $\frac{1}{15}$    c) $\frac{1}{21}$

EXERCISE: Look at several examples of rational numbers in the form $\frac{p}{q}$ (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) $\sqrt{23}$   (ii) $\sqrt{225}$   (iii) 0.3796   (iv) 7.478478...   (v) 1.101001000100001...

SELF PRACTICE: Classify the following numbers as rational or irrational:

(i) $\sqrt{625}$
(ii) $\sqrt{89}$
(iii) 0.05896
(iv) 9.0621621621...
(v) 3.121221222122221...

Discussion