Prove that 2-3 is irrational-

Let 2+√(3) be a rational number equals to r ∴ 2+√(3) = r √(3) = r 2 Here L.H.S is an irrational number while R.H.S. r 2 is rational ∴ L.H.S ≠ R.H.S Hence it con ...

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Standard X

Mathematics

Number Theory: Interesting Results

Prove that 2+...

Question

Prove that $2 + \sqrt{3}$ is irrational.

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Solution

Let $2 + \sqrt{3}$ be a rational number equals to r
$∴ 2 + \sqrt{3} = r$
$\sqrt{3} = r - 2$
Here L.H.S is an irrational number while R.H.S. r-2 is rational $∴ L . H . S \neq R . H . S$
Hence it contradicts our assumption that $2 + \sqrt{3}$ is rational.
$∴ 2 + \sqrt{3}$ is irrational.

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Prove that 2+√3 is irrational.

Let 2+√3 be a rational number equals to r ∴2+√3=r √3=r−2 Here L.H.S is an irrational number while R.H.S. r-2 is rational ∴L.H.S≠R.H.S Hence it contradicts our assumption that 2+√3 is rational. ∴2+√3 is irrational.

Prove that $2 + \sqrt{3}$ is irrational.

Let $2 + \sqrt{3}$ be a rational number equals to r
$∴ 2 + \sqrt{3} = r$
$\sqrt{3} = r - 2$
Here L.H.S is an irrational number while R.H.S. r-2 is rational $∴ L . H . S \neq R . H . S$
Hence it contradicts our assumption that $2 + \sqrt{3}$ is rational.
$∴ 2 + \sqrt{3}$ is irrational.