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.
Solution
Rational 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}}
Solution
Rational 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}}
Solution
Rational 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 04
Arrange the below numbers in ascending order.
\mathtt{\ \ 6,\ 3\sqrt{6} ,\ 10\sqrt{32} \ and\ 55}
Solution
Two 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}}
Solution
Two 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