Question

Help me to do this Kalpesh Prabhakar help me

Answer 1

Hi,

To prove that square root of 5 is irrational, we will use a proof by contradiction. What is a proof by contradiction?.

Suppose we want to prove that a math statement is true. Simply put, we assume that the math statement is false and then show that this will lead to a contradiction.

If it leads to a contradiction, then the statement must be true.

To show that √5 is an irrational number, we will assume that it is rational.
Then, we need to find a contradiction when we make this assumption.

If we are going to assume that √5 is rational, then we need to understand what it means for a number to be rational.

Basically, if square root of 5 is rational, it can be written as the ratio of two numbers as shown below:


$\sqrt{5} =x/y$

Square both sides of the equation above

$( \sqrt{5} )^2=x^2/y^2$

Multiply both sides by y^2

$5*y^2=x^2/y^2*y^2$

We get 5 × y^2 = x^2

Since 5 × y^2 is equal to x^2, 5 × y^2 and x^2 must have the same number of prime factors

We just showed that

x^2 has an even number of prime factors

y^2 has also an even number of prime factors

5 × y^2 will then have an odd number of prime factors

The number 5 counts as 1 prime factor, so 1 + an even number of prime factors is an odd number of prime factors

5 × y^2 is the same number as x^2. However, 5 × y^2 gives an odd number of prime factor while x^2gives an even number of prime factors

This is a contradiction since a number cannot have an odd number of prime factors and an even number of prime factors at the same time

The assumption that square root of 5 is rational is wrong. Therefore, square of 5 is irrational

Hoping you understand from this 

and to do this directly, I am attaching screenshots of method.

:D

Answer 2

Hey mate !!

Here's your answer !!

To prove : 

√ 5 is irrational.

Proof :

Let us assume √ 5 to be a rational number.

We know that any rational number is of the form p / q where q ≠ 0.

=> √ 5 = p / q   ( Fact -  Here p and q have a HCF of 1 )

=> √ 5 * q = p 

=> q√5 = p

Squaring on both sides we get,

= 5q² = p   -----( 1 )

=> q² = p² / 5 

Since p² divides 5, p also divides 5 - Fundamental Theorem Of Arithmetic

=> P has a factor 5

So let the quotient of p / 5 be r

=> p / 5 = r

=> p = 5 * r

=> p = 5r   -----( 2 )

Substitute ( 2 ) in ( 1 )

We get,

= 5q² = ( 5p )²

=> 5q² = 25p²

=> q² = 5p²

=> q² / 5 = p

=> q also divides 5. Hence 5 is a factor of q

So now p and q have a common factor 5.

But this contradicts the fact that p and q only have a HCF of 1.

Hence our assumption was wrong.

Hence √ 5 is irrational.

Hope my answer helped !!

Cheers !!