- Books Name
- Mathematics Book for CBSE Class 10

- Publication
- Carrier Point

- Course
- CBSE Class 10

- Subject
- Mathmatics

**Revisiting Rational Numbers and Their Decimal Expansions**

In Class IX, you studied that rational numbers have either a terminating decimal

expansion or a non-terminating repeating decimal expansion. In this section, we are

going to consider a rational number, say ( 0)

p

q

q

¹ , and explore exactly when the

decimal expansion of

p

q

is terminating and when it is non-terminating repeating

(or recurring). We do so by considering several examples.

Let us consider the following rational numbers :

(i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408.

Now (i)

3

375 375

0.375

1000 10

= = (ii)

3

104 104

0.104

1000 10

= =

(iii)

4

875 875

0.0875

10000 10

= = (iv)

4

233408 233408

23.3408

10000 10

= =

As one would expect, they can all be expressed as rational numbers whose

denominators are powers of 10. Let us try and cancel the common factors between

the numerator and denominator and see what we get :

(i)

3

3 3 3 3

375 3 5 3

0.375

10 2 5 2

×

= = =

×

(ii)

3

3 3 3 3

104 13 2 13

0.104

10 2 5 5

×

= = =

×

(iii)

4 4

875 7

0.0875

10 2 5

= =

×

(iv)

2

4 4

233408 2 7 521

23.3408

10 5

× ×

= =

Do you see any pattern? It appears that, we have converted a real number

whose decimal expansion terminates into a rational number of the form ,

p

q

where p

and q are coprime, and the prime factorisation of the denominator (that is, q) has only

powers of 2, or powers of 5, or both. We should expect the denominator to look like

this, since powers of 10 can only have powers of 2 and 5 as factors.

Even though, we have worked only with a few examples, you can see that any

real number which has a decimal expansion that terminates can be expressed as a

rational number whose denominator is a power of 10. Also the only prime factors of 10

are 2 and 5. So, cancelling out the common factors between the numerator and the

denominator, we find that this real number is a rational number of the form ,

p

q

where

the prime factorisation of q is of the form 2n5m, and n, m are some non-negative integers.

Let us write our result formally:

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16 MATHEMATICS

Theorem 1.5 : Let x be a rational number whose decimal expansion terminates.

Then x can be expressed in the form ,

p

q

where p and q are coprime, and the

prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

You are probably wondering what happens the other way round in Theorem 1.5.

That is, if we have a rational number of the form ,

p

q

and the prime factorisation of q

is of the form 2n5m, where n, m are non negative integers, then does

p

q

have a

terminating decimal expansion?

Let us see if there is some obvious reason why this is true. You will surely agree

that any rational number of the form ,

a

b

where b is a power of 10, will have a terminating

decimal expansion. So it seems to make sense to convert a rational number of the

form

p

q

, where q is of the form 2n5m, to an equivalent rational number of the form ,

a

b

where b is a power of 10. Let us go back to our examples above and work backwards.

(i)

3

3 3 3 3

3 3 3 5 375

0.375

8 2 2 5 10

×

= = = =

×

(ii)

3

3 3 3 3

13 13 13 2 104

0.104

125 5 2 5 10

×

= = = =

×

(iii)

3

4 4 4 4

7 7 7 5 875

0.0875

80 2 5 2 5 10

×

= = = =

× ×

(iv)

2 6

4 4 4 4

14588 2 7 521 2 7 521 233408

23.3408

625 5 2 5 10

× × × ×

= = = =

×

So, these examples show us how we can convert a rational number of the form

p

q

, where q is of the form 2n5m, to an equivalent rational number of the form ,

a

b

where b is a power of 10. Therefore, the decimal expansion of such a rational number

terminates. Let us write down our result formally.

Theorem 1.6 : Let x =

p

q

be a rational number, such that the prime factorisation

of q is of the form 2n5m, where n, m are non-negative integers. Then x has a

decimal expansion which terminates.

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REAL NUMBERS 17

We are now ready to move on to the rational numbers

whose decimal expansions are non-terminating and recurring.

Once again, let us look at an example to see what is going on.

We refer to Example 5, Chapter 1, from your Class IX

textbook, namely,

1

7

. Here, remainders are 3, 2, 6, 4, 5, 1, 3,

2, 6, 4, 5, 1, . . . and divisor is 7.

Notice that the denominator here, i.e., 7 is clearly not of

the form 2n5m. Therefore, from Theorems 1.5 and 1.6, we

know that

1

7

will not have a terminating decimal expansion.

Hence, 0 will not show up as a remainder (Why?), and the

remainders will start repeating after a certain stage. So, we

will have a block of digits, namely, 142857, repeating in the

quotient of

1

7

.

What we have seen, in the case of

1

7

, is true for any rational number not covered

by Theorems 1.5 and 1.6. For such numbers we have :

Theorem 1.7 : Let x =

p

q

be a rational number, such that the prime factorisation

of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a

decimal expansion which is non-terminating repeating (recurring).

From the discussion above, we can conclude that the decimal expansion of

every rational number is either terminating or non-terminating repeating.