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

r2 = p

By the definition of a rational number

r = a/b

where a and b are mutually prime integers. Then

pa2 = b2

implies that b2 is divisible by p. Therefore (by Euclid), b is divisible by p. That is,

b = ps

Where s is some integer. Substituting, we have

pa2 = p2s2

Dividing by p we have

a2 = ps2

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 nth root of any prime is irrational:

Suppose that p1/n, where n is an integer greater than 1, is rational. Then

p = an/bn

where a/b is an irreducible fraction and b > 1. Therefore an/bn 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 = rn/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:

n2 = 2m2

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

4k2 = 2m2

We can simplify this to:

m2 = 2k2

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.