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You can find several such proofs in the Wikipedia article "Square root of 2". Many of these proofs (or perhaps all, but I didn't check carefully) apply to the square root of ANY positive integer, assuming the integer is not a perfect square.
Wiki User
∙ 9y ago
Wiki User
∙ 9y agoThe proof is by the method of reductio ad absurdum. We start by assuming that sqrt(2) is rational. That means that it can be expressed in the form p/q where p and q are co-prime integers. Thus sqrt(2) = p/q. This can be simplified to 2*q^2 = p^2 Now 2 divides the left hand side (LHS) so it must divide the right hand side (RHS). That is, 2 must divide p^2 and since 2 is a prime, 2 must divide p. That is p = 2*r for some integer r. Then substituting for p gives, 2*q^2 = (2*r)^2 = 4*r^2 Dividing both sides by 2 gives q^2 = 2*r^2. But now 2 divides the RHS so it must divide the LHS. That is, 2 must divide q^2 and since 2 is a prime, 2 must divide q. But then we have 2 dividing p as well as q which contradicts the requirement that p and q are co-prime. The contradiction implies that sqrt(2) cannot be rational.
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Prove the square root of 2 not irrational?
This is impossible to prove, as the square root of 2 is irrational.
Is the square root of 6 rational?
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.
Prove that root 2 is irrational number?
I linked a good resource that explains what you asked below.
Is a square root of 2 rational or irrational?
The square root of 2 is an irrational number
Are negative square roots rational or 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.
Prove the square root of 2 not irrational?
This is impossible to prove, as the square root of 2 is irrational.
Prove that square root of 2 is an irrational number?
The square root of 2 is 1.141..... is an irrational number
How do you prove that root 2 root 3 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.
Is the square root of 6 rational?
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.
Prove that root 2 is irrational number?
I linked a good resource that explains what you asked below.
Is the square root of a 143 a 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.
Prove that square of 2 is irrational?
It is not possible to prove something that is not true. The square of 2 is rational, not irrational.
Is a square root of 2 rational or irrational?
The square root of 2 is an irrational number
Is the square root a 2 an irrational number?
Yes, the square root of 2 is an irrational number.
Are root 2 and root 3 rational or irrational?
The square roots of 2 and 3 are irrational but not transcendent.
Are negative square roots rational or 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.
How is root 2 irrational?
2 is a prime number and its square root is an irrational number that cannot be expressed as a fraction
