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 p^1/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