answersLogoWhite

0

Let's assume that p is prime and the square root of p is rational. This means there are (positive) integers a, b such that sqrt(p) = a/b. Therefore: p = (a/b)*(a/b) = (a^2)/(b^2) This shows that a/b already has to be a (positive) integer, so that (a^2)/(b^2) is also one. (If a/b is not an integer, multiplying it by itself wouldn't create one, since no elements would come in that you could cancel the numerator and the denominator with.) So we have shown that (a^2)/(b^2) = (a/b)^2 = p. But this means that p isn't prime, because it has a/b (an integer) as a divisor, so we have a contradiction of the given fact that p is prime. This makes our assumption that sqrt(p) is rational false, and therefore proves that if p is prime, sqrt(p) is irrational.

User Avatar

Wiki User

10y ago

What else can I help you with?

Related Questions

Is the square root of 19 a rational number?

No. 19 is a prime number, and all prime numbers have irrational roots.


Why are the square roots of prime numbers greater than one always rational?

They aren't. The square roots of prime numbers are always irrational.


Why is square root 31 irrational?

In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals.The square root of 31 is one such.


Is the square root of 68 irrational or rational?

The square roots are irrational.


Is the square root of 15 rational or irrational?

IRRATIONAL sqrt(15) = sqrt(3 x 5) =sqrt(3) X sqrt(5) Since the 'square roots' of prime numbers , '3' & '5' in this case, are irrational , then the square root od '15' is irrational .


Is the square root of 163 rational or irrational?

The square roots of 163 are irrational.


The square root of 84 is it rational or irrational?

The square roots of 84 are irrational.


The square root of 50 is it rational or irrational?

The square roots of 50 are irrational.


Is the square root of 29 a rational number?

NO!!! '29' is a Prime Number. The square roots of prime numbers are irrational sqrt(29) = 5.385164807.... NB Irrational numbers are those where the decimals go to infinity AND there is no regular order in the decimal digits.


Can square roots be irrational?

Yes


Are all square roots irrational?

No.


Is The square root of 0.151 rational or irrational?

The square roots are irrational.