# Irrationals Proof

This is my proof that the square roots of all prime numbers are irrational:
The proof is by contradiction. First assume that there does exist some rational number *r*
that when squared is a prime number *p*. Then

*r*^{2} = *p*

By the definition of a rational number

*r* = *a*/*b*

where *a* and *b* are mutually prime integers. Then

*pa*^{2} = *b*^{2}

implies that *b*^{2} is divisible by *p*. Therefore
(by Euclid), *b* is divisible by *p*. That is,

*b* = *ps*

Where *s* is some integer. Substituting, we have

*pa*^{2} =
*p*^{2}*s*^{2}

Dividing by *p* we have

*a*^{2} = *p**s*^{2}

Thus *a* is divisible by *p*, a contradiction (because *a*/*b* is irreducible).
Therefore, our assumption that some irrational number squared is prime is false.

## Extension to All Integer Roots

The *n*th root of any prime is irrational:
Suppose that *p*^{1/n}, where *n* is an
integer greater than 1, is rational. Then

*p* =
*a*^{n}/*b*^{n}

where *a*/*b* is an irreducible fraction and *b* > 1. Therefore
*a*^{n}/*b*^{n}
is not an integer yet is equal to a prime, a contradiction.

## A Rational Number Divided by an Irrational Number is Irrational

Let *r* be a rational number and let *s* be an irrational number. Then suppose that
*r*/*s* is rational. Then *r*/*s* = *m*/*n* where *m* and
*n* are integers. Then *s* = *r**n*/*m*, so *s* is rational,
a contradiction.

## Historical Note

At one time the existence of the irrational numbers was kept secret because that knowledge was
thought to have shattered the foundation of the Pythagorean religion. Later, however,
Euclid published a proof that the square root of two is irrational in his *Elements*,
volume III. Here is Euclid's proof:
Consider this figure of a unit square and its diagonal, AD:

The proof is by contradiction. Assume that the ratio of a side of the square to its diagonal
is rational and is equal to *n*/*m*, where *n* and *m* are integers.
By the Pythagorean theorem:

*n*^{2} = 2*m*^{2}

This implies that *n* is even and *m* is odd. Let *n* = 2*k*. Then

*4k*^{2} = 2*m*^{2}

We can simplify this to:

*m*^{2} = 2*k*^{2}

which implies *m* is even, a contradiction. Therefore, length of the diagonal of a unit quare, equal
to the square root of two, is irrational.

Email: Richard dot J dot Wagner at gmail dot com
Irrational.html; This hand crafted HTML file created December 7, 2001, by
Rick Wagner

last updated October 21, 2010, by Rick Wagner.

Copyright © 2001-2010 by Rick Wagner, all rights reserved.