Show that 2-3 is an irrational number-

Let us assume that is rational .Then, there exist positive co primes a and b such that2 3=ab3=2 #160; abThis implies, 3 is a rational number, which is a cont ...

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Standard X

Mathematics

Irrational Numbers

Show that 2-3...

Question

Show that $2 - \sqrt{3}$ is an irrational number.

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Solution

Let us assume that is rational .Then, there exist positive co primes a and b such that
$2 - \sqrt{3} = \frac{a}{b} \sqrt{3} = 2 - \frac{a}{b}$
This implies, $\sqrt{3}$ is a rational number, which is a contradiction.
Hence, is irrational number.

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Show that 2-3 is an irrational number.

Let us assume that is rational .Then, there exist positive co primes a and b such that2-3=ab3=2 -abThis implies, 3 is a rational number, which is a contradiction.Hence, is irrational number.

Show that $2 - \sqrt{3}$ is an irrational number.

Let us assume that is rational .Then, there exist positive co primes a and b such that
$2 - \sqrt{3} = \frac{a}{b} \sqrt{3} = 2 - \frac{a}{b}$
This implies, $\sqrt{3}$ is a rational number, which is a contradiction.
Hence, is irrational number.