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It is proved by contradiction (reductio ad absurdum), a powerful type of proof in mathematics.

Assume that the square root of 2 is rational and is equal to a/b where a and b are integers.

Squaring both sides gives:

2=a2/b2

a2=2b2

Since a2 is even, it implies that a is even

So, replacing a by 2k where k is an integer, we have:

(2k)2=2b2

b2=2k2

Since b2 is even, it implies that b is even, which is in contradiction to our first statement: a/b is in lowest terms.

Thus, the square root of 2 is irrational. (Q.E.D)

Note: It can be proved that if a2 is even, then a is also even.

Proof:

Odd numbers are of the form 2n+1, where n is an integer.

When an odd number is squared, we have:

(2n+1)2 = 4n2+4n+1 = 2(2n2+2n) +1 = 2y+1

where y = 2n2+2n

y is also an integer; so, 2y+1 is an odd number, which in turn means that the square of any odd number is also odd.

Therefore, if the square of a number is even, the number cannot be odd; it has to be even. (Q.E.D)

Q: How do you prove that square root of 2 is irrational?

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The square root of 2 is 1.141..... is an irrational number

It is not possible to prove something that is not true. The square of 2 is rational, not irrational.

Yes, the square root of 2 is an irrational number.

irrational

The same reasoning you may have seen in high school to prove that the square root of 2 is not rational can be applied to the square root of any natural number that is not a perfect square.

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This is impossible to prove, as the square root of 2 is irrational.

The square root of 2 is 1.141..... is an irrational number

It is known that the square root of an integer is either an integer or irrational. If we square root2 root3 we get 6. The square root of 6 is irrational. Therefore, root2 root3 is irrational.

No; you can prove the square root of any positive number that's not a perfect square is irrational, using a similar method to showing the square root of 2 is irrational.

The square root of 2 is an irrational number

It is not possible to prove something that is not true. The square of 2 is rational, not irrational.

If the positive square root (for example, square root of 2) is irrational, then the corresponding negative square root (for example, minus square root of 2) is also irrational.

Yes, the square root of 2 is an irrational number.

Yes. The square root of a positive integer can ONLY be either:* An integer (in this case, it isn't), OR * An irrational number. The proof is basically the same as the proof used in high school algebra, to prove that the square root of 2 is irrational.

irrational

sqrt(32) = 4sqrt(2) The square root of '2' is irrational, so the square root of '32' is irrational.

The same reasoning you may have seen in high school to prove that the square root of 2 is not rational can be applied to the square root of any natural number that is not a perfect square.