Thommo wrote:Is that right? Isn't the algebraic completion of the reals unique up to isomorphism?
Sorry Thommo, who was that at?
Edited to include the quote when I realised I was at the top of a new page.
Paradoxical Numbers
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Thommo wrote:Is that right? Isn't the algebraic completion of the reals unique up to isomorphism?
Gareth wrote:
Firstly, he was prepared to admit that he was wrong. That's huge. Show me a man who never admits when he's wrong and I'll show you a fool.
Secondly, he had the humility to ask for help, even if others might think it was a stupid question. That is a fast track to wisdom. I recommend it.
logical bob wrote:Sorry Thommo, who was that at?
I think there might have been some mix-ups in the story there. There's this story about Hippasus being killed for revealing the secret for constructing the Dodecahedron, though.scott1328 wrote:The Pythagoreans killed the man who first demonstrated that √2 is irrational. It overthrew their mathematics until Eudoxus theory of proportion salvaged the mess.
Not sure what the point of confusion is (though it obviously isn't yours).Thommo wrote:logical bob wrote:Sorry Thommo, who was that at?
Ughaibu. He said it was a matter of definition rather than algebra, but the algebra follows from the definition - the answer would be the same whatever we assigned to be the square root of -1, up to isomorphism.
He obviously realised at about the time I posted though, since he edited saying he got it just moments after I hit submit, so it became a bit redundant as no explanation was actually needed.
Gareth wrote:DavidMcC.
Thanks for that. I don't get it though. If x to the fourth is 1, what else could x be but 1?
I didn't crib it by the way. I thought of squaring both sides all be myself, so as not to get stuck with impossible square roots.
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VazScep wrote:I think there might have been some mix-ups in the story there. There's this story about Hippasus being killed for revealing the secret for constructing the Dodecahedron, though.scott1328 wrote:The Pythagoreans killed the man who first demonstrated that √2 is irrational. It overthrew their mathematics until Eudoxus theory of proportion salvaged the mess.
Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”[8] Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.
The Fundamental Theorem of Algebra? It's mostly analysis.Thommo wrote:Yes, it's just a special case of the number of solutions of a polynomial of degree n being n, if you count repeated roots separately.
I forget how the proof goes though. It's all algebraic either way.
b). then x squared is minus 1
Take A-level maths, and you should be introduced to complex numbers, a universe in which there are numbers that square to any negative number you like.LjSpike wrote:I haven't read all the way through, but I spot a problem:b). then x squared is minus 1
You can't have a square being a negative number (or any even power for that fact) as an NEGATIVE x NEGATIVE = POSITIVE. You also can't fix this statement by saying it means to be the root of x as the root of x can be 2 values, negative or position
So from point B I've been put off, because secondary school education tells me its not possible.
LjSpike wrote:I haven't read all the way through, but I spot a problem:b). then x squared is minus 1
You can't have a square being a negative number (or any even power for that fact) as an NEGATIVE x NEGATIVE = POSITIVE. You also can't fix this statement by saying it means to be the root of x as the root of x can be 2 values, negative or position
So from point B I've been put off, because secondary school education tells me its not possible.
Or at least, its not real.
So if anything your number is a dream.
Scarlett and Ironclad wrote:Campermon,...a middle aged, middle class, Guardian reading, dad of four, knackered hippy, woolly jumper wearing wino and science teacher.
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