Question

Write one rational and one irrational number lying between 0.25 and 0.32.

Answer 1

Hint: In the above problem, we are asked to find the rational and irrational number between 0.25 and 0.32. We know that a rational number is in the form of $\dfrac{p}{q}$ where p and q are natural numbers and q cannot be 0. Now, an irrational number is a number, which is not a terminating and non-recurring number. So, to find a rational and irrational number between 0.25 and 0.32 follow the definitions of rational and irrational numbers.

Complete step-by-step solution:
We have given two numbers 0.25 and 0.32 and we have to find a rational number and an irrational number.
Now, we know that a rational number is a number which can be written in the form of $\dfrac{p}{q}$ where p and q are natural numbers and q cannot be 0 so a rational number which can be present between 0.25 and 0.32 is 0.26. You can write 0.26 in the form of $\dfrac{p}{q}$ which we are shown in the below.
$0.26$
We can write 0.26 in the form of a fraction by removing the decimal and divide 26 by 100 as follows:
$\dfrac{26}{100}$
Dividing numerator and denominator by 2 we get,
$\dfrac{13}{50}$
As you can see that $\dfrac{13}{50}$ is in the form of $\dfrac{p}{q}$ where p is 13 and q is 50 which are natural numbers and q (i.e. 50) is not equal to 0.
Now, we are going to find an irrational number between 0.25 and 0.32. We know that an irrational number is a number which is a non-terminating and non-recurring number.
So, according to the definition of an irrational number, an irrational number between 0.25 and 0.32 is:
0.260260026000……
The number that we have written above is non-terminating meaning this number is not ending and non-recurring means the digits of this number are not repeating. You might think that 26 is repeating but after the decimal, there are no digits repeating continuously without a stop like here 26 then one zero, then 26 then two zeros then 26 then three zeros. If 26 is always repeating like 0.26262626… then this number is not an irrational number.
Hence, we got a rational number as 0.26 and an irrational number as 0.260260026000….

Note: There are other rational numbers that are also possible between 0.25 and 0.32 apart from the one which we have shown above. The rational numbers are 0.27, 0.28, and 0.29 like this. You can find other irrational numbers between 0.25 and 0.32 apart from 0.260260026000……
also like 0.261261126111261111…., 0.270270027000…..