How to compare two irrational numbers

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Comparing rational and irrational numbers
Examples of rational and irrational number comparison
Comparison between Two Irrational Numbers
How to Write Ascending, Descending Order of Irrational Numbers?
Solved Problems on Comparing Irrational Numbers
FAQ’s on Comparision between Irrational Numbers

In this chapter we will learn to compare numbers in different format; rational and irrational numbers.

After completing the chapter, you will able to arrange mix of rational and irrational number in ascending or descending order.

Comparing rational and irrational numbers

We know that irrational numbers are generally present in the form of square root or cube root.

Finding exact value of each irrational number for comparison purpose will be complex and time confusing.

The best way to compare rational number with square root number is by taking square of all numbers so they we get all numbers in integer form and then do the comparison.

For example;

Let rational number ” a ” and irrational number

\mathtt{\sqrt{b}}

are given for comparison.

Since the irrational number is in square root, the comparison is not possible.So, take square of both numbers.

$\mathtt{\Longrightarrow \ ( a)^{2} \ \ \&\ \left(\sqrt{b}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ a^{2} \ \&\ b}$

Now we have got both integer value, so the comparison is easily possible.

I hope you understood the above process. Let us consider some examples for further reference.

Examples of rational and irrational number comparison

Example 01
Compare the number 15 and $\mathtt{2\sqrt{19}}$ . Find which one is greater number.

SolutionRational number = 15

Irrational number = $\mathtt{2\sqrt{19}}$

Here one of the number is in square root, so comparison is not possible.

Square both the numbers and then compare.

Squaring number 15

$\mathtt{\Longrightarrow \ 15^{2}}\\\ \\ \mathtt{\Longrightarrow \ 225}$

Squaring $\mathtt{2\sqrt{19}}$

$\mathtt{\Longrightarrow \ \left( 2\sqrt{19}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 4\ \times \ 19)}\\\ \\ \mathtt{\Longrightarrow \ 76}$ Now we have two numbers 225 and 76.

We know that 225 > 76

Hence, 15 > $\mathtt{2\sqrt{19}}$

Example 02
Compare the numbers 11 and $\mathtt{3\sqrt[3]{21}}$

SolutionRational number = 11

Irrational number = $\mathtt{3\sqrt[3]{21}}$

Here one of the number is in form of cube root.

Comparing simple number 11 with cube root number is very difficult.

Take cube of both the numbers for better comparison.

Cubing number 11.

$\mathtt{\Longrightarrow \ ( 11)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 11\times 11\times 11)}\\\ \\ \mathtt{\Longrightarrow \ 1331}$

Cubing number $\mathtt{3\sqrt[3]{21}}$

$\mathtt{\Longrightarrow \ \left( 3\sqrt[3]{21}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 27\times 21) \ }\\\ \\ \mathtt{\Longrightarrow \ 567}$ Now compare numbers 1331 and 567.

We know that 1331 > 567.

Hence, 11 > $\mathtt{3\sqrt[3]{21}}$

Example 03
Compare the numbers $\mathtt{\ \frac{8}{3} \ and\ \ 4\sqrt[3]{5}}$

SolutionRational number = 8 / 3

Irrational number = $\mathtt{\ 4\sqrt[3]{5}}$

Here one of the given number is in the form of cube root.For effective comparison, take cube of both the given numbers.

Cubing 8 / 3

$\mathtt{\Longrightarrow \ \left(\frac{8}{3}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8\times 8\times 8}{3\times 3\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{512}{27}} \\\ \\ \mathtt{\Longrightarrow \ \frac{512}{27} =18.96}$

Cubing $\mathtt{\ 4\sqrt[3]{5}}$

$\mathtt{\Longrightarrow \ \left( 4\sqrt[3]{5}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 64\ \times 5}\\\ \\ \mathtt{\Longrightarrow \ 320}$

Now we have two numbers 18.96 and 320.We know that 18.96 < 320.

Hence, $\mathtt{\frac{8}{3} \ < \ \ 4\sqrt[3]{5}}$

Example 04Arrange the below numbers in ascending order.

$\mathtt{\ \ 6,\ 3\sqrt{6} ,\ 10\sqrt{32} \ and\ 55}$

SolutionTwo numbers are present in the form of square root.To compare the numbers, we must square all the numbers.

Squaring number 6.

$\mathtt{\Longrightarrow \ 6^{2}}\\\ \\ \mathtt{\Longrightarrow \ 36}$

Squaring $\mathtt{3\sqrt{6}}$

$\mathtt{\Longrightarrow \left( 3\sqrt{6}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 9\ \times 6}\\\ \\ \mathtt{\Longrightarrow \ 54}$

Squaring $\mathtt{10\sqrt{32}}$

$\mathtt{\Longrightarrow \left( 10\sqrt{32}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 100\ \times 32}\\\ \\ \mathtt{\Longrightarrow \ 3200}$

Squaring 55

$\mathtt{\Longrightarrow ( 55)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 3025}$

Now compare the numbers 36, 54, 3200 & 3025. Also arrange these numbers in ascending order.

Numbers arranged in ascending order are; 36 < 54 < 3025 < 3200.

Hence, the proper sequence is $\mathtt{6\ < \ 3\sqrt{6} \ < \ 55\ < 10\sqrt{32}}$

Example 05
Arrange the following numbers in descending order.

$\mathtt{10,\ \frac{13}{6} ,\ 3\sqrt{7} \ and\ \sqrt{5}}$

SolutionTwo of the numbers are in square root.For effective number comparison, square all the given numbers.

Squaring number 10

$\mathtt{\Longrightarrow ( 10)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 100}$

Squaring number 13/6

$\mathtt{\Longrightarrow \left(\frac{13}{6}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{169}{36}}\\\ \\ \mathtt{\Longrightarrow \ 4.69}$

Squaring $\mathtt{3\sqrt{7}}$

$\mathtt{\Longrightarrow \left( 3\sqrt{7}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 9\times 7}\\\ \\ \mathtt{\Longrightarrow \ 63}$

Squaring $\mathtt{\sqrt{5}}$

$\mathtt{\Longrightarrow \left(\sqrt{5}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 5}$

Now we have got the numbers 100, 4.69, 63 and 5.

Arranging these numbers in descending orders.

100 > 63 > 5 > 4.69

Hence, we get 10 > $\mathtt{3\sqrt{7}}$ > $\mathtt{\sqrt{5}}$ > 13/6

Next chapter : Finding irrational numbers between two integers

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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.

Comparison between Two 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.