Question

prove that √2 is an irrational number
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Answer 1

Answer:

Let √2 be a rational number

Therefore, √2= p/q [ p and q are in their least terms i.e., HCF of (p,q)=1 and q ≠ 0

On squaring both sides, we get

p²= 2q² ...(1)

Clearly, 2 is a factor of 2q²

⇒ 2 is a factor of p² [since, 2q²=p²]

⇒ 2 is a factor of p

Let p =2 m for all m ( where m is a positive integer)

Squaring both sides, we get

p²= 4 m² ...(2)

From (1) and (2), we get

2q² = 4m² ⇒ q²= 2m²

Clearly, 2 is a factor of 2m²

⇒ 2 is a factor of q² [since, q² = 2m²]

⇒ 2 is a factor of q

Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1

Therefore, Our supposition is wrong

Hence √2 is not a rational number i.e., irrational number.

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Answer 2

Answer:

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Explanation:

Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.

√2=p/q

Squaring both sides,

2=p²/q²

The equation can be rewritten as

2q²=p²

From this equation, we know p² must be even (since it is 2 multiplied by some number). Since p² is an even number, it can be inferred that p is also an even number.

Since p is even, it can be written as 2m where m is some other whole number. This is because the definition of an even number is it can be written as 2 multiplied by a whole number. Substituting p=2m in the above equation:

2q²=p²=(2m)²=4m²

or

2q²=4m²

Dividing both sides of the equation by 2:

q²=2m²

By the same reasoning as before, q² is an even number (since it is written as 2 multiplied by some number). So q is an even number. Let q=2n where n is some whole number. We had assumed √2 to be equal to p/q. So:

√2=p/q=2m/2n

By canceling 2 in the numerator and the denominator of the Right hand side,

√2=m/n

We now have a fraction m/n simpler than p/q.

However, we now find ourselves in a position whereby we can repeat exactly the same process on m/n, and at the end of it, we can generate a simpler function, say g/h. This fraction can be put through the same process again, and the new fraction, say, e/f will be simpler again.

But we know that rational number cannot be simplified indefinitely. There must always be a simplest rational number and the original assumption that √2 is equal to p/q does not obey this rule.

So it can be stated that a contradiction has been reached.

If √2 could be written as a rational number, the consequence would be absurd. So it is true to say that √2 cannot be written in the form p/q. Hence √2 is not a rational number.

Thus, Euclid succeeded in proving that √2 is an Irrational number.