Comparison between Two Irrational Numbers

The definition of irrational numbers is can not written in the form of a fraction i.e \(\frac { a }{ b } \) and b is not equal to 0. Comparing two irrational numbers means deciding which is the greatest one and which is the smallest one among the two numbers. As the irrational numbers are square roots and cube roots it is difficult to compare them. So, have a look at the below sections to find the ascending order or descending order of irrational numbers.

Irrational numbers are the numbers in mathematics that can not be represented in the form of a simple fraction. Examples of irrational numbers are square root and cube root of the numbers. If you want to compare two irrational numbers with different orders, then we have to convert them into the same order and compare. Follow these guidelines to convert irrational numbers having different orders to the same order.

At first, write the given irrational numbers order.
Find the least common multiple.
Make the order of irrational numbers the same using the least common multiple values.
Now compare the radicands.

Also, check:

How to Write Ascending, Descending Order of Irrational Numbers?

It is not easy to compare two irrational numbers. One important parameter for the comparison between two irrational numbers is if the square or cube of two numbers be ‘p’ and ‘q’ such that ‘q’ is greater than ‘p’, then q² is also greater than p² and q³ also greater than p³ and so on.

Arranging the irrational numbers having the same order from least to greatest is called the ascending order.
Arranging irrational numbers having the same order from greatest to least is called the descending order.

Solved Problems on Comparing Irrational Numbers

Problem 1:
Compare √7, √19

Solution:
Given two irrational numbers are √7, √19
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
√7 = √7 x √7 = (√7)² = 7
√19 = √19 x √19 = (√19)² = 19
7 is lesser than 19.
So, √7 is less than √19

Problem 2:
Arrange the following irrational numbers in descending order.
2√2, √5, √17, √11, √21

Solution:
Given irrational numbers are 2√2, √5, √17, √11, √21
We know that if ‘p’ and ‘q’ are two numbers such that ‘q’ is greater than ‘p’, then q² will be greater than p². So, square the given numbers.
2√2 = 2√2 x 2√2 = (2√2)² = 8
√5 = √5 x √5 = (√5)² = 5
√17 = √17 x √17 = (√7)² = 17
√11 = √11 x √11 = (√11)² = 11
√21 = √21 x √21 = (√21)² = 21
Arranging the descending order means placing the numbers from the greatest to the smallest.
So, 21 > 17 > 11 > 5 > 8
Hence, the descending order of numbers is √21 > √17 > √11 > √5 > 2√2.

Problem 3:
Write the irrational numbers ∜5, √3 and ∛4 in ascending and descending orders.

Solution:
Given irrational numbers are ∜5, √3 and ∛4
Order of the irrational numbers are 4, 2, 3
The least common multiple of (4, 2, 3) = 12
So, we have to change the order of each number as 12
Change ∜5 as 12th root
∜5 = (4 x 3) √5³
= 12 √125
√3 = (2 x 6) √3⁶
= 12 √729
∛4 = (3 x 4) √4⁴
= 12 √256
125 < 256 < 729.
Therefore, ascending order is ∜5, ∛4, √3 and descending order is √3, ∛4 and ∜5.

FAQ’s on Comparision between Irrational Numbers

1. How to compare two irrational numbers?

To compare two square root numbers, find the square of the given numbers and then compare.

2. Which is the smallest number out of √2, √3?

Find the square of two numbers.
√2 = √2 x √2 = (√2)² = 2
√3 = √3 x √3 = (√3)² = 3
So, √2 is the smallest number.

3. How to compare irrational numbers on the number line?

To compare irrational numbers that are square root, simply find the number that we are taking a square root of. And compare the numbers as whole numbers on the number line.