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.

Decimal Expansion Of Rational Numbers

Before going into a representation of the decimal expansion of rational numbers, let us understand what rational numbers are. Any number that can be represented in the form of p/q, such that p and q are integers and q ≠ 0 are known as rational numbers. So when these numbers have been simplified further, they result in decimals. Let us learn how to expand such decimals here.

Examples: 6 , -8.1,\frac{4}{5} etc. are all examples of rational numbers.

 

How to Expand Rational Numbers in Decimals?

The real numbers which are recurring or terminating in nature are generally rational numbers.

Decimal - Expansion Of Rational Numbers

For example, consider the number 33.33333……. It is a rational number as it can be represented in the form of 100/3. It can be seen that the decimal part .333…… is the non-terminating repeating part, i .e. it is a recurring decimal number.

Also the terminating decimals such as 0.375, 0.6 etc. which satisfy the condition of being rational (0.375 = \frac{3}{2^3} ,0.6 = \frac{3}{5}).

Consider any decimal number. For e.g. 0.567. It can be written as 567/1000 or \frac{567}{10^3} . Similarly, the numbers 0.6689,0.032 and .45 can be written as \frac{6689}{10^4} ,\frac{32}{10^3} and \frac{45}{10^2} respectively in fractional form.

Thus, it can be seen that any decimal number can be represented as a fraction which has denominator in powers of 10. We know that prime factors of 10 are 2 and 5, it can be concluded that any decimal rational number can be easily represented in the form of \frac{p}{q}, such that p and q are integers and the prime factorization of q is of the form 2^x~ 5^y, where x and y are non-negative integers.

This statement gives rise to a very important theorem.

Theorems

Theorem 1: If m be any rational number whose decimal expansion is terminating in nature, then m can be expressed in form of \frac{p}{q}, where p and q are co-primes and the prime factorization of q is of the form 2^x~ 5^y, where x and y are non-negative integers.

The converse of this theorem is also true and it can be stated as follows:

Theorem 2: If m is a rational number, which can be represented as the ratio of two integers i.e. \frac{p}{q} and the prime factorization of q takes the form 2^x~ 5^y, where x and y are non-negative integers then, then it can be said that m has a decimal expansion which is terminating.

Consider the following examples:

  1. \frac{7}{8} = \frac{7}{2^3} = \frac{7~×~5^3}{2^3~×~5^3} = \frac{875}{10^3}
  2. \frac{3}{80} = \frac{3}{2^4~×~5} = \frac{3~×~5^3}{2^4~×~5^4} = \frac{375}{10^4}

Moving on, to decimal expansion of rational numbers which are recurring, the following theorem can be stated:

Theorem 3: If m is a rational number, which can be represented as the ratio of two integers i.e. \frac{p}{q} and the prime factorization of q does not takes the form 2^x~ 5^y, where x and y are non-negative integers. Then, it can be said that m has a decimal expansion which is non-terminating repeating (recurring).

Consider the following examples:

  1. \frac{1}{6} = 0.1666…. = 0.1\overline{6}
  2. \frac{7}{12} = 0.58333… = 0.58\overline{3}
  3. \frac{9}{11} = 0.8181… = 0.\overline{81}

Rational Number to decimal Examples

Case 1: Remainder equal to zero

Example: Find the decimal expansion of 3/6.

Decimal Expansion of Rational Numbers - Example 1

Here, the quotient is 0.5 and the remainder is 0. Rational number 3/6 results in a terminating decimal.

Case 2: Remainder not equal to zero 

Example: Express 5/13 in decimal form.

Decimal Expansion of Rational Numbers Example -2

Here, the quotient is 0.384615384 and the remainder is not zero. Notice that the number…384 after the decimal is repeating. Hence, 5/13 gives us a non-terminating recurring decimal expansion. And this can be written as 5/13=recurring

A rational number gives either terminating or non-terminating recurring decimal expansion. Thus, we can say that a number whose decimal expansion is terminating or non-terminating recurring is rational. 

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