square root of 2

Originally posted by @uzless
still stuck on our current number system. poor duchess. It's just a tool to explain what we see around us....what happens when that tool can't explain what we see?

A better number theory would start with the premise that there is only 1 number that exists and that number is 1. All other numbers just tell you how many more than 1 you have. Start with that premise as your basis for a new number theory and imagine the possibilities......

This is known as Robinson Arithmetic, provided that the smallest number is zero and it's successor is known as 1 and 1's successor is known as 2 and so on. Sorry, but the mathematicians got there first.

Originally posted by @deepthought
I've got my head around this, your proof relies only on the fundamental theorem, since it states that the prime factorization is unique. Consequently the left and right hand side cannot have different numbers of factors of any of the primes. Even so, the proof of the fundamental theorem of arithmetic involves the Chinese Remainder Theorem - which isn't the most straightforward and why Blood's teachers gave him the other proof.

Euclid's lemma rather than the Chinese Remainder Theorem is used in the proof. I really should look these things up before posting...

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That was remiss of you. In the meantime I had a look at the Wikipedia page on root two which gives no fewer than six proofs. The most terse is simply that it is a special case of the Rational Roots Theorem (for polynomials with integer coefficients) which has the consequence that the roots of x^r = m, where r and m are natural numbers, must either be integer or irrational (neatly covering all generalisations). Both the proofs we presented are there, although the exposition is horrible it took me a while to realise they were the same proofs, as well as a couple of geometric proofs, the fifth proof considered limits of sequences and show that a quantity that is integer must be strictly between zero and one and established a contradiction that way.

The last one was interesting, being acceptable to intuitionists, as it avoids proof by contradiction. Irrationals have the property that the difference between them and any rational is non-zero. It then demonstrates a lower bound on the difference between root two and a/b.

https://en.m.wikipedia.org/wiki/Square_root_of_2

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Sadly they seem to never have regarded me as a reliable go-between.

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Maybe Trump can have Kim Jong-un intercede with Beijing on this issue.....

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Did you really think I was serious?

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Wow, that is really stupid. I WISH I was paid for that🙂

BTW, I am trying to solve this stupid little algebra problem, could you help? I tried several tactics but never seemed to get it to simplify, factorization and such didn't seem to do much:

Solve for x: x/(x-1)=1/(x+1)
I tried (x-1)*x/(x-1) = x-1/x+1, where it seems that leaves x=(x-1)/(x+1) but that doesn't seem to me to actually solve for x. What am I doing wrong?

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Forgive my stupidity but how did you get to the very first step? Also I saw one where both x's were x-1, what would that solution be? X^2+2X?

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Is that called a commutative rule? Using that rule, the next one X/(X-1)=1/(x-1) comes out the same! X=i.