Question

Prove that √2 + √3 is an irrational number

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Answer 1

Step-by-step explanation:

Let us assume that √2 + √3 is rational

So it can be written as $\frac{a}{b}$

→ √2 + √3 = $\frac{ a }{ b}$

Where a and b are co-primes and b ≠ 0

→ √2 = $\frac{a}{b}$ - √3

Now, squaring on both sides

→ (√2)² = ($\frac{a}{b}$ - √3)²

We know that

• (a - b)² = a² + b² - 2ab

Substitute above formula in above equation

→ 2 = $\frac{ a² }{ b² }$ + 3 - 2*√4 ($\frac{a}{b}$)

→ $\frac{ a² }{ b² }$ + 3 - 2 = 2 × √3 ($\frac{ a }{ b }$)

→ $\frac{ a² }{ b² }$ + 1 = 2 × √3 ($\frac{ a }{ b }$)

→ $\frac{ (a² \: + \: b²)}{ b² }$ × $\frac{ b }{ 2a }$ = √3

→ $\frac{ (a² \: + \: b²)}{ 2ab }$ = √3

Where,

• a, b are integers and $\frac{ (a² \: + \: b²)}{ 2ab }$ is a rational number

$\Huge\green{ √3 }$ is a rational number

It is contradiction to our assumption that √3 is irrational

:. Our assumption is wrong.

Thus, √2 + √3 is irrational.