Question

Prove that√2 is irrational no

Answer 1

Answer:

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Step-by-step explanation:

Let's suppose √2 is a rational number. Then we can write it √2  = a/b where a, b are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.

From the equality √2  = a/b it follows that 2 = a2/b2,  or  a2 = 2 · b2.  So the square of a is an even number since it is two times something.

From this we know that a itself is also an even number. Why? Because it can't be odd; if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd. Check it if you don't believe me!

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number. We don't need to know what k is; it won't matter. Soon comes the contradiction.

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:

2 = (2k)2/b2

2 = 4k2/b2

2*b2 = 4k2

b2 = 2k2

This means that b2 is even, from which follows again that b itself is even. And that is a contradiction!!!

WHY is that a contradiction? Because we started the whole process assuming that a/b was simplified to lowest terms, and now it turns out that a and b both would be even. We ended at a contradiction; thus our original assumption (that √2 is rational) is not correct. Therefore √2 cannot be rational.

Answer 2

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By Contradiction Method

Let √2 be a rational number.

So √2=a/b (since rational number can be expressed in form of a/b where bnot equal to 0)

2=a²/b² (Squaring both sides)

a²=2b²

So a² is divisible by 2.

Hence a² is divisible by 2..............(eq.1)

Let a=2c

a²=4c² (Squaring Both sides)

According to eq.1

2b²=4c²

b²=2c²

So b² is divisible by 2.

Hence b Is divisible by 2.........(eq.2)

ACCORDING to eq.1 and eq.2

Both a and b have common factor 2

But a and b cannot have common factor other than 1.

So √2 is not a rational number.

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