Rational numbers can be written in the form of $$\frac { a }{ b }$$ where a, b are two integers and b is not equal to zero. The terminating or non-repeating decimal numbers are rational numbers. And irrational numbers can’t be written in $$\frac { a }{ b }$$ form. Here, we are giving the steps to arrange both rational and irrational numbers in ascending and descending orders. Check out the solved questions for a better understanding.

## Comparison between Rational and Irrational Numbers

Irrational numbers are those that are non-terminating and non-repeating decimal that can’t be expressed in the form of $$\frac { a }{ b }$$. Rational numbers are in the form of $$\frac { a }{ b }$$. We can easily compare rational numbers by comparing numerators of rational fractions. We need to take L.C.M and compare rational and irrational numbers.

We know that if ‘p’ and ‘q’ are two irrational square root numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers. In the same way to compare the combination of rational and irrational numbers, find the square of the numbers and then compare.

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### Questions on Comparing Rational and Irrational Numbers

Question 1:
Compare 5, √5

Solution:
Given two numbers are 5, √5
To compare the given numbers, let us first find out the square of both the numbers and then proceed with the comparison. So,
5 = 5 x 5 = 5² = 25
√5 = √5 x √5 = (√5)² = 5
Since 5 is less than 25
So, 5 > √5.

Question 2:
Compare ∛2, $$\frac { 5 }{ 8 }$$

Solution:
Given numbers are ∛2, $$\frac { 5 }{ 2 }$$
In the given numbers for comparison, one of them is rational $$\frac { 5 }{ 2 }$$ while another one is irrational number ∛2. To make the comparison between them, firstly we’ll make both numbers rational numbers and then a comparison process will be carried out. So, to make both the numbers rational, let us find the cube of both the numbers. So,
(∛2)³ = ∛2 x ∛2 x ∛2 = 2
($$\frac { 5 }{ 2 }$$)³ = $$\frac { 5 }{ 2 }$$ x $$\frac { 5 }{ 2 }$$ x $$\frac { 5 }{ 2 }$$ = $$\frac { 125 }{ 8 }$$
Now, L.C.M. of 1 and 8 is 8. So, the two numbers to be compared are $$\frac { 125 }{ 8 }$$ and $$\frac { 16 }{ 8 }$$. Now, the rational fractions have become like rational fractions. So, we just need to compare their numerators. Since, $$\frac { 125 }{ 8 }$$ is greater than $$\frac { 16 }{ 8 }$$.
So, $$\frac { 5 }{ 2 }$$ is greater than ∛2.

Question 3:
Arrange the following in ascending order.
6, $$\frac { 5 }{ 4 }$$, ∛6, ∛3

Solution:
We have to arrange the given series in ascending order. To do so, let us first of all find cube times of all the elements of the given series.
6³ = 6 x 6 x 6 = 216
($$\frac { 5 }{ 4 }$$)³ = $$\frac { 5 }{ 4 }$$ x $$\frac { 5 }{ 4 }$$ x $$\frac { 5 }{ 4 }$$ = $$\frac { 125 }{ 64 }$$
(∛6)³ = √6 x √6 x √6 = 6
(∛3)³ = √3 x √3 x √3 = 3
Now we have to make the comparison between $$\frac { 125 }{ 64 }$$, 216, 6, 3
This could be done by converting the series into like fractions and then proceeding.
So, the series becomes:
$$\frac { 125 }{ 64 }$$, $$\frac { 13824 }{ 64 }$$, $$\frac { 384 }{ 64 }$$, $$\frac { 192 }{ 64 }$$
Arranging the above series in ascending order we get;
$$\frac { 125 }{ 64 }$$ < $$\frac { 192 }{ 64 }$$ < $$\frac { 384 }{ 64 }$$ < $$\frac { 13824 }{ 64 }$$
So, the required series is:
$$\frac { 5 }{ 4 }$$, ∛6, ∛3, 6.