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The square root of 2

One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the contrary and showing that doing so leads to a contradiction (hence the proposition must be true).

Assume that √2 is a rational number. This assumption implies that there exist integers m and n with n ≠ 0 such that m/n = √2.

Then √2 can also be written as an irreducible fraction m/n (the fraction is shortened as much as possible). This means that m and n are coprime integers, i.e., they have no common factor greater than 1.

From m/n = √2 it follows that m = n√2, and so m2 = (n√2)2 = 2n2.

So m2 is an even number, because it is equal to 2n2, which is even.

It follows that m itself is even (since only even numbers have even squares).

Because m is even, there exists an integer k satisfying m = 2k.

We may therefore substitute 2k for m in the last equation of (3), thereby obtaining the equation (2k)2 = 2n2, which is equivalent to 4k2 = 2n2 and may be simplified to 2k2 = n2.
Because 2k2 is even, it now follows that n2 is also even, which means that n is even (recall that only even numbers have even squares).

Then, by (5) and (8), m and n are both even, which contradicts the property stated in (2) that m/n is irreducible.

Since we have found a contradiction, the initial assumption (1) that √2 is a rational number is false; that is to say, √2 is irrational.

This proof can be generalized to show that any root of any natural number is either a natural number or irrational.


See http://en.wikipedia.org/wiki/Irrational_number

Irrational Numbers

I am reading an awesome book, "God Created the Integers", compiled and commented by Stephen Hawking. I admit I am not particularly well-versed in mathematics, but I have the (uninformed) opinion that irrational numbers are a side-effect of a fixed-base mathematical system, which can only approximate certain (i.e irrational) values, and not express them definately.

Now, I may be completely wrong, but I am working on a proof that (X * X) = 2, has a rational VBM solution. VBM is a mathematical system that I have developed in which irrational numbers do not exist because irrational numbers are, by their property of having an infinite decimal expansion, undefined (or unspecified) values.

This is a very complicated exercise and I might find out that I am completely mistaken about irrational numbers. But at the end I will hopefully have learnt something.

Read this

http://freethought.mbdojo.com/nobeliefs.html

Phyllotaxis

Pick up a pinecone and count the spiral rows of scales. You may find eight spirals winding up to the left and 13 spirals winding up to the right, or 13 left and 21 right spirals, or other pairs of numbers. The striking fact is that these pairs of numbers are adjacent numbers in the famous Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21... Here, each term is the sum of the previous two terms. The phenomenon is well known and called phyllotaxis. Many are the efforts of biologists to understand why pinecones, sunflowers, and many other plants exhibit this remarkable pattern. Organisms do the strangest things, but all these odd things need not reflect selection or historical accident. Some of the best efforts to understand phyllotaxis appeal to a form of self-organization. Paul Green, at Stanford, has argued persuasively that the Fibonacci series is just what one would expects as the simplest self-repeating pattern that can be generated by the particular growth processes in the growing tips of the tissues that form sunflowers, pinecones, and so forth. Like a snowflake and its sixfold symmetry, the pinecone and its phyllotaxis may be part of order for free

Stuart Kauffman At Home in the Universe, Oxford University Press, 1995, p 151. (1) Available from Amazon.com

Daniel Krugel

It is only very recently, 6 days ago, through James Randi's newsletter, that I heard of the man named Daniel Krugel. This man, who lives in South Africa, claims that he has developed a device that can accurately locate matter - biological or non-biological - if he inputs a part of that matter, or something closely related to that matter, into the machine. For instance, he claims to be able to find missing persons if he has a piece of the missing person's hair.

Mr Krugel and his device got quite alot of attention when two parents from England contacted him, claiming that their 3-year-old daughter, Madelaine McCann, is missing. The story goes that the McCanns were staying in Portugal, in Praia da Luz, when their daughter disappeared from their hotel room while they were having dinner with friends while Madeleine stayed in the hotel room alone. The parents contacted Mr Krugel and asked him if he could locate their daughter. According to Mr Krugel, the parents provided him with something that he could use to locate Madelaine, and after 'using' his device, he said he 'thinks' that Madelaine is still in Praia da Luz.

Mr Krugel got quite alot of publicity, and was featured in two of South Africas leading investigative journalism TV shows, Carte Blanche and 3rd Degree. I used to always have respect for 3rd degree in particular. But both shows supported MR Krugel's claims, so obviously, 'investigative' does not come into the picture anymore.

This claim is obviously false. It is not possible to locate matter in this way. So I have contacted Mr Krugel and asked him about his device. I am still corresponding with Mr Krugel, and he has provided me with details that he asked me to handle confidentially, so I am not going to post our correspondences here for now. Suffice to say, Mr Krugel's recent statement to me was that his device shows that Madeleine is still in Praia da Luz.

It is perhaps quite surprising that Mr Krugel has offered to demonstrate his device to me in private. If it comes to that, I will definately take him up on his offer. But first I want to find out some more. I will keep you informed.