If the rational number which is in the form of a/b then , by dividing a by b ,we can get two situations
a. If the remainder becomes zero
While dividing if we get zero as the remainder after some steps then the decimal expansion of such number is called terminating.
Example:
7/8 = 0.875
b. If the remainder does not become zero
While dividing if the decimal expansion continues and not becomes zero then it is called non-terminating or repeating expansion.
Example:
1/3 = 0.3333….
Hence, the decimal expansion of rational numbers could be terminating or non-terminating recurring and vice-versa.
Some Properties of Rational Numbers
a) Every rational number is either a terminating decimal or a repeating decimal. Example:
d) For any three rational numbers p, q and r, the following order properties are true:
- Either p > q or q > p or p = q
- If p > q and q > r, then p > r
- If p > q, then p + r > q + r
- If p > q and r > 0, then pr > qr
Real Numbers and their Decimal Expansions
The above videos show how to convert Rational number of the form p/q into decimals. Also, we will learn to identify if the rational number in the decimal form is Terminating or Non Terminating Reccurring decimalsWrite the following in decimal form and say what kind of decimal expansion each has:
(i) 15/100
(ii) 1/9
(iii) 2/11
(iv) 3/13
Answer
| i) | 15/100 | 0.15 (Terminating) |
| ii) | 1/9 | 0.111111... (Non terminating repeating) |
| iii) | 2/11 | .18181818....(Non terminating repeating) |
| iv) | 3/13 | 0.230769230769... = 0.230769 (Non terminating repeating) |
Example
Express the following in the form p/q where p and q are integers and q ≠ 0.
Example : Show that 3.142678 is a rational number.
In other words, express 3.142678 in the form p/q, where p and q are integers and q ≠ 0.
Solution :
We have 3.142678 = 3142678/1000000, and hence is a rational number.
Now, let us consider the case when the decimal expansion is non-terminating recurring.
Example : Show that 0.3333...can be expressed in the form of p/q, where p and q are integers and q ≠ 0.
Solution :
Since we do not know what 0.3333... is , let us call it ‘y’ and so
y = 0.3333...
10 y = 10 × (0.333...) = 3.333...
Now, 3.3333... = 3 + y, since y = 0.3333...
Therefore, 10 y= 3 +y
Solving for y, we get
9y = 3, i.e.,
y =1/3
Example : Show that 1.272727... can be expressed in the form p/q, where p and q are integers and q ≠0.
Solution : Let x = 1.272727... Since two digits are repeating, we multiply x by 100 to get
100 x = 127.2727...
So, 100 x = 126 + 1.272727... = 126 + x
Therefore, 100 x – x = 126, i.e., 99 x = 126
i.e., x =126/99= 42/33=14/11
Hence x= 14/11
Example : Show that 0.2353535... can be expressed in the form p/q ,where p and q are integers and q≠0.
Solution : Let x = 0.235 .
Over here, note that 2 does not repeat, but the block 35 repeats. Since two digits are repeating, we multiply x by 100 to get
100 x = 23.53535...
So, 100 x = 23.3 + 0.23535... = 23.3 + x
Therefore, 99 x = 23.3
i.e., 99 x =233/10 , which gives x =233/990
So, every number with a non-terminating recurring decimal expansion can be expressed in the form p/q (q ≠ 0), where p and q are integers.
Summary
- The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
- The decimal expansion of an irrational number is non-terminating non-recurring.Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.
