## Rational Numbers and their Decimal Expansions

There are about 300,000 spices of living green plants on the Earth. They provide oxygen and a lot more. As the plant grows, its leaves grow in size. Do you think that as the leaf grows in size, the ratio of its length to its breadth stays the same? To test this, you can:

• pick any plant
• find 3-4 leaves at different stages of their growth
• measure the lengths and widths of these leaves in millimeters using a scale.
• Find the ratio of width to length (Is this a rational number?)
• Express the ratio as a decimal number (to help comparing the ratios)

Do share the outcomes of your experiment in the discussion board below.

### DECIMAL FORM OF RATIONAL NUMBERS

The decimal form of a rational number can either be terminating or recurring. Is it possible to tell which rational number will have a terminating (or a recurring) decimal representation without actually dividing the numerator by the denominator.

It turns out that the prime factorisation of the denominator can tell us what kind of decimal representation a rational number will have.

And, you are about to learn how this works.

EXERCISE: Without actually performing the long division, state whether $\frac{13}{3125}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

EXERCISE: Without actually performing the long division, state whether $\frac{17}{8}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

EXERCISE: Without actually performing the long division, state whether $\frac{64}{445}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

SELF PRACTICE: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
i) $\frac{15}{1600}$

ii) $\frac{29}{343}$

iii) $\frac{23}{2^{3}5^{2}}$

iv) $\frac{129}{2^{3}5^{2}7^{5}}$

v) $\frac{6}{15}$

vi) $\frac{13}{6250}$

vii) $\frac{59}{1331}$

viii) $\frac{23}{2^{6}5^{5}}$

ix) $\frac{258}{2^{3}5^{5}9^{7}}$

x) $\frac{21}{35}$

EXERCISE: Write down the decimal expansion of:
i) $\frac{13}{3125}$

ii) $\frac{17}{8}$

SELF PRACTICE: Write down the decimal expansions of those rational numbers given below which have terminating decimal expansions.
i) $\frac{13}{6250}$

ii) $\frac{27}{16}$

iii) $\frac{128}{566}$

iv) $\frac{45}{1800}$

v) $\frac{59}{1331}$

vi) $\frac{23}{2^{6}5^{5}}$

vii) $\frac{258}{2^{3}5^{5}9^{7}}$

viii) $\frac{21}{35}$

ix) $\frac{350}{500}$

x) $\frac{154}{280}$

EXERCISE: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form $\frac{p}{q}$, what can you say about the prime factor of q?
i) 43.123456789
ii) 0.120120012000120000...
iii) $4.3\overline{123456789}$

SELF PRACTICE: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form $\frac{p}{q}$, what can you say about the prime factor of q?
i) 54.987654321
ii) 0.540540054000540000...
iii) $54.\overline{987654321}$

Discussion