Chapter 1: Real Numbers

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Q
Real Numbers Solutions

Question:

Prove that โˆš5 is irrational

Answer:

Proof: To prove that √5 is an irrational number 

Let us assume that √5 is rational. Then √5  =  frac{a}{b} 

Since a and b are co primes with only 1 common factor and b≠0.  

   √5  =  frac{a}{b}    

a = √5b 

On squaring both sides, we get

a² = 5b²      ------- (i)

b² =   frac{a^2}{5}       [By theorem if p divides q then p can also divide q²] 

So, 5 can also divide a.   ……….. (ii)

a = 5c 

On squaring on both sides, we get 

a² = 25c²     ……… (iii)

From equations (i) and (iii)

5b² = 25c²

b² = 5c² 

c² =   frac{b^2}{5}     [By theorem if p divides q then p can also divide q²]  

So, 5 can also divide b.  ……….. (iv)

We know that a and b are co-primes having only 1 common factor but from (ii) and (iv) we can see a and b are not co-primes.

This contradiction arises because we assumed that 5 is a rational number.

our assumption is wrong.

 √5 is an irrational number.

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