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By definition an irrational number is one that cannot be expressed in a fraction x/y where x and y are integers (whole numbers). If the screen on your calculator went on forever and you typed in square root of 2, the result would run forever (just like pi). Because the square root of two does not end (the value after the decimal continues) it cannot be expressed in the form of a fraction x/y therefore it is an irrational number.
The 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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sqrt(32) = 4sqrt(2) The square root of '2' is irrational, so the square root of '32' is irrational.
How is root 2 irrational?
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Is the square root of 200 rational or irrational?
The square root of 200 is irrational.
Is the square root of any composite numbers irrational?
The square root of a composite number is not always an irrational number. e.g. 4 is a composite number and its square root is 41/2 = 2, which is rational number.
How do you find an irrational square root?
An irrational number is a number that never ends. An example of an irrational square root would be the square root of 11.
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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.
Is the square root a 2 an irrational number?
Yes, the square root of 2 is an irrational number.
Prove that square root of 2 is an irrational number?
The square root of 2 is 1.141..... is an irrational number
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irrational
Is the square root of 32 rational?
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Is the square root of 2 plus the square root of 8 a rational or irrational number?
irrational
Is root 2 a rational number?
The square root of 2 is irrational. In general, the square root of a positive integer is either an integer (if you take the square root of a perfect square), or it is irrational.
Is the square root of 2 irrational or rational?
The square root of (any number that isn't a perfect square) is irrational.
Is the product of two irrational numbers irrational?
Sometimes it is and sometimes it isn't. The square root of 2 and the square root of 3 are both irrational, as is their product, the square root of 6. The square root of 2 and the square root of 8 are both irrational, but their product, the square root of 16, is rational (in fact, it equals 4).
Is the square root of two fifths rational or irrational?
the square root of 2 fifths of irrational is 0.565685425
