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Square root of 2 
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#19
Feb2814, 04:59 PM

P: 25




#20
Feb2814, 05:45 PM

P: 354

You can look at this a different way. We know that 1.4 < √2 and 1.5 > √2. And we can keep going with every greater mathematical precision:
[tex]1.41421356^2 < 2 \ and \ 1.41421357^2 > 2[/tex] And so on. And, we can make these two numbers either side of √2 as close as we want. So, the question is simple: is there an "irrational" number in there that is equal to √2 or not? Suppose not, then the number line has gaps in it. And, it would get quite difficult to do maths if there are gaps where you'd like numbers to be. For example, if you draw the graph of [tex]y = x^22[/tex] This clearly crosses the xaxis twice (at √2 and +√2). But, if there are gaps where these numbers should be, then the graph crosses the xaxis between numbers, so there is no solution to the equation. The graph sort of ghosts thru the xaxis missing all the numbers. So, an axiom of mathematics is that there are no gaps. This is called the "completeness" axiom. It might be quite interesting to see how far you could go without the completeness axiom. But, intuitively, I don't like the idea of a curve crossing the xaxis where there is no number! Surely, wherever the curve crosses the xaxis, there must be a number there? 


#21
Mar214, 05:24 AM

P: 25

After the geometrical demonstrations I was shown, I was convinced [itex]\sqrt{2}[/itex] indeed exists, because it is possible to draw a line segment with a length of [itex]\sqrt{2}[/itex]. Then if you take that line segment and put it's begining at 0 in the real numbers' line, it's end would be the exact point of the line that has a distance equal to [itex]\sqrt{2}[/itex] from 0, in other words, that point is where [itex]\sqrt{2}[/itex] lies in the real number's line. Geometrically, it's easy to picture. Using just numbers, not so much, at least for me. I suppose my mind is limited to things more concrete like lines and points and circles, and is bad at conceiving purely abstract stuff. 


#22
Mar214, 05:32 AM

P: 25




#24
Mar214, 09:21 AM

P: 25




#25
Mar214, 09:27 AM

Mentor
P: 18,038

Anyway, a very good book is http://www.amazon.com/GeometryEucli.../dp/0387986502 In the second chapter he starts with constructing an axiom system using the undefined notions of "line", "point", "between" and "congruent". It might just be what you're looking for. The book is however a bit to the difficult side. 


#26
Mar214, 10:44 AM

P: 25




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