Compare Irrational Numbers

We have already discussed Rational and Irrational Numbers, so now let us take a look on how to compare irrational numbers.

Irrational numbers include numbers which are not perfect square roots or perfect cube roots and can’t be found out exactly.

For Example $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ , $\sqrt{6}$ , $\sqrt{7}$ , $\sqrt{8}$ , $\sqrt{10}$ are not perfect square roots.

Similarly $\sqrt[{3}]{2}$ , $\sqrt[{3}]{3}$ , $\sqrt[{3}]{4}$ , $\sqrt[{3}]{5}$ , $\sqrt[{3}]{6}$ , $\sqrt[{3}]{7}$ , $\sqrt[{3}]{9}$ are not perfect cube roots, and hence irrational.

To compare two irrational numbers which are of the first form , say $\sqrt{3}$ and $\sqrt{5}$ we find the square of both numbers and compare them

For Example: $(\sqrt{3}) ^{2}$ = $\sqrt{3}$ x $\sqrt{3}$ = 3,

$(\sqrt{5}) ^{2}$ = $\sqrt{5}$ x $\sqrt{5}$ = 5 and since 3 < 5 hence $\sqrt{3}$ < $\sqrt{5}$ .

To compare two irrational numbers which are of the second form, say $\sqrt[3]{2}$ and $\sqrt[3]{3}$ we find the cube of both numbers and compare them.

For Example: $(\sqrt[3]{2}) ^{3}$ = $\sqrt[3]{2}$ x $\sqrt[3]{2}$ x $\sqrt[3]{2}$ = 2,

$(\sqrt[3]{3}) ^{3}$ = $\sqrt[3]{3}$ x $\sqrt[3]{3}$ x $\sqrt[3]{3}$ = 3 and since 2 < 3 hence $\sqrt[3]{2}$ < $\sqrt[3]{3}$ .

For Example Consider $\sqrt[5]{15}$ and $\sqrt[5]{21}$

$(\sqrt[5]{15}) ^{5}$ = $\sqrt[5]{15}$ x $\sqrt[5]{15}$ x $\sqrt[5]{15}$ x $\sqrt[5]{15}$ x $\sqrt[5]{15}$ = 15,

$(\sqrt[5]{21}) ^{5}$ = $\sqrt[5]{21}$ x $\sqrt[5]{21}$ x $\sqrt[5]{21}$ x $\sqrt[5]{21}$ x $\sqrt[5]{21}$ = 21 and 15 < 21 hence $\sqrt[5]{15}$ < $\sqrt[5]{21}$ .

Check Point

Which of the two numbers is greater? $\sqrt{12}$ or $\sqrt{17}$

Which of the two numbers is smaller? $\sqrt{91}$ or $\sqrt{97}$

Insert appropriate symbol > or < between the given numbers: $\sqrt[{3}]{73}$ , $\sqrt[{3}]{82}$

Arrange the following numbers in ascending order of their magnitudes:

$\sqrt[{4}]{12}$ , $\sqrt[{4}]{35}$ , $\sqrt[{4}]{7}$

Arrange the following numbers in descending order of their magnitudes:

$\sqrt[{3}]{29}$ , $\sqrt[{3}]{34}$ , $\sqrt[{3}]{17}$

Answer key

$\sqrt{17}$
$\sqrt{91}$
$\sqrt[3]{73}$ < $\sqrt[3]{82}$ .
$\sqrt[4]{7}$ < $\sqrt[4]{12}$ < $\sqrt[4]{35}$ .
$\sqrt[3]{34}$ > $\sqrt[3]{29}$ > $\sqrt[3]{17}$ .

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