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A proof using M-Properties

1. Introduction

I found a proof, that the square root of 2 is not a rational number, in Roger Penrose's book 'The Road to Reality'. I realized that I could use M-Properties to construct a similar proof to determine the rationality of any number having a given M-Property. This item is loosely based on Roger Penrose's proof.

Roger Penrose uses the fact that the square root of an even number is also an even number, whereas I use the multiplication table for the m2 set to achieve the same result. The advantage of using the concept of M-Properties is that a proof with the same format can be used to investigate the square roots of other numbers. See Section 3.

For completeness, I have included this brief definition of M-Properties.

Definition
Let a number, which is a multiple of 2, have the property m2. So an even number has the property ‘m2’, and an odd number has the property m2 + 1.
Let a number, which is a multiple of 3, have the property m3. Numbers which are not a multiple of 3 have the property m3 + 1 or m3 + 2.
In general, let a number which is a multiple of n have the property ‘mn’.
Let these properties be called M Properties.
Define a set of numbers: mn, mn + 1, ... mn + (n - 1) as the mn set.

2. To show that the square root of 2 is not a rational number

This is the Multiplication Table for the m2 set.
The results are shaded. For example, m2 x (m2 + 1) = m2
times m2 m2 + 1
m2 + 1 m2 m2 + 1
m2 m2 m2
Assumption: The square root of 2 is a rational number.

Let the ratio for √2 be a/b, where 'a' and 'b' are positive integers.

a/b = √2

(a/b)² = 2

a² = 2b² ................................... (1)

As 2b² is m2, a² must be m2 ................... From (1)
So, a must be m2. * .................................. Using the Multiplication Table
([m2]² is the only square which is m2)
So, a² must also be m4.
So, b² must be m2 .................................... From (1)
So, b must be m2. ** ................................ Using the Multiplication Table

Let a = 2c, and b = 2d .............................. a and b are both m2 (from * and **)

4c² = 2(4d²)

c² = 2d² ............................................ (2)

As (2) has the same format as (1), the process can be repeated endlessly.
So, a and b can be divided by 2 endlessly and yet the results have to be finite positive integers.
This is not possible. So, ‘a’ and ‘b’ etc. cannot be finite positive integers.
So, √2 is not a rational number.

3. To show that the square root of 3 is not a rational number

This proof has the same format as on Section 2, but uses the Multiplication Table for the m3 set.
times m3 m3 + 1 m3 + 2
m3 + 2 m3 m3 + 2 m3 + 1
m3 + 1 m3 m3 + 1 m3 + 2
m3 m3 m3 m3
Assumption: The square root of 3 is a rational number.

Let the ratio for √3 be a/b, where 'a' and 'b' are positive integers.

a/b = √3

(a/b)² = 3

a² = 3b² ................................... (1)

As 3b² is m3, a² must be m3 ................... From (1)
So, a must be m3. * .................................. Using the Multiplication Table
([m3]² is the only square which is m3)
So, a² must also be m9.
So, b² must be m3 .................................... From (1)
So, b must be m3. ** ................................ Using the Multiplication Table

Let a = 3c, and b = 3d .............................. a and b are both m3 (from * and **)

9c² = 3(9d²)

c² = 3d² ............................................ (2)

As (2) has the same format as (1), the process can be repeated endlessly.
So, a and b can be divided by 3 endlessly and yet the results have to be finite positive integers.
This is not possible. So, ‘a’ and ‘b’ etc. cannot be finite positive integers.
So, √3 is not a rational number.

4. The square roots of other numbers

It seems that this process is valid for any √n where n is a prime number; or any multiple of primes as long as any given prime is not a factor an even number of times.

For a graphical proof of this, see Shaded Patterns (section 6) in the M-Properties topic. There is only one instance of (mn)² on the principal diagonal of the pattern of a prime number set, or of a set which is a multiple of two different prime numbers.

Mike Holden - Sep 2005
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