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Compare Irrational Numbers

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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.

compare irrational numbers

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.

  1. 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}

 

  1. 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}.

 

compare irrational numbers
compare irrational numbers

 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

  1. Which of the two numbers is greater? \sqrt{12} or \sqrt{17}
  1. Which of the two numbers is smaller? \sqrt{91} or \sqrt{97}
  1. Insert appropriate symbol > or < between the given numbers: \sqrt[{3}]{73}\sqrt[{3}]{82}
  1. Arrange the following numbers in ascending order of their magnitudes:

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

  1. Arrange the following numbers in descending order of their magnitudes:

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

Answer key
  1. \sqrt{17}
  2. \sqrt{91}
  3. \sqrt[3]{73} < \sqrt[3]{82}.
  4. \sqrt[4]{7} < \sqrt[4]{12} < \sqrt[4]{35}.
  5. \sqrt[3]{34} > \sqrt[3]{29} > \sqrt[3]{17}.

 

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