Naked Science Forum
General Science => General Science => Topic started by: latebind on 02/04/2010 15:38:13
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Supposed you have a right angled triangle with each of the smaller sides being 1 and the hypotonuse therefore being √2 (square root of 2).
the √2 is an irrational and forever repeating number, bound to go on for billions of centuries. There is not enough computers or paper on earth to actually store just this one number, yet a humble line of pythagorus is able to effortlessly depict it.
So how is it that a finite line can depict an irrational number? does that seem rational?
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Not sure to have understood your problem, maybe you are looking for the concept of "series":
http://en.wikipedia.org/wiki/Series_%28mathematics%29
you can sum infinite numbers and finish with a finite number.
Example: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2.
To show such results you have to study the series.
In the specific case you are asking, the irrational number 1.414213562.... is the sum of infinite numbers, even more little, which gives at the end a finite number, that is √2.
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Irrational numbers are numbers that cannot be expressed as the ratio of two integers. An interesting fact, for which there is a proof, is that despite there being (obviously) an infinite number of both types of number, there are more irrational numbers than rational numbers.
I probably did not express this well enough for a rigorous mathematician but I hope good enough for scientists :-)
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I'd just like to make my question a little clearer.....
Ok, if you look at a line, you will notice it has a finite beginning and a finite end, now suppose this line has a length of √2 (pythag).
We know already that √2 has no finite end, but yet the line stops in just the right place doesn't it?
I'm not saying the line should be infinite or something, what I am actually saying is that the line is absolutely the perfect length, and this suggests that √2 actually has a finite length or how else could the line be able to display it perfectly??
IE: if √2 has no finite end, then how does the line manage to have a finite end? I could understand if the line was off by 0.000001% , but in pyhtag the line is perfect and that is scary!
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Just wanted to add a little more food for thought...
Imagine you wanted to draw this line with a ruler.
Suppose you just happen to have a ruler where the measurement-bars are a millionth of an atom-width apart, this ruler would be far from accurate enough to depict √2 perfectly.
I actually don't know what my point is other than the fact that I am baffled by how perfectlt this irrational length line is shown in a pythag triangle.
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[???]
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Is this a consequence of decimal counting? Is the hypotenuse always irrational for any counting base (binary, etc, etc)?
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It is not a consequence of the counting base. An irrational number is an irrational number whatever the base - see the definition if irrational number I gave earlier.
Latebind, I have trouble understanding why you think that any number should be able to be defined in numerical terms, even if the terms can be infinite and non-recursive. Isn't infinity big enough to give you the precision you need? It is a fact that there exists more irrational numbers than rational numbers. It is an interesting and profound fact but nothing beyond comprehension given a glass of wine and and a few hours reflection :-)
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Is this a consequence of decimal counting? Is the hypotenuse always irrational for any counting base (binary, etc, etc)?
I suppose you could count things in square roots of 2. The hypotenuse would be 1 unit while the remaining sides would be 1/Sqrt[2] units. :p
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Is this a consequence of decimal counting? Is the hypotenuse always irrational for any counting base (binary, etc, etc)?
I suppose you could count things in square roots of 2. The hypotenuse would be 1 unit while the remaining sides would be 1/Sqrt[2] units. :p
Does that have anything to do with the invention of the radian?
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Is this a consequence of decimal counting? Is the hypotenuse always irrational for any counting base (binary, etc, etc)?
I suppose you could count things in square roots of 2. The hypotenuse would be 1 unit while the remaining sides would be 1/Sqrt[2] units. :p
Does that have anything to do with the invention of the radian?
Yes, but I think we are confused enough.
Imagine I draw a line on a piece of paper, and I measure it.
First I use a ruler calibrated in cm, the line between 4 and 5 cm. Then I get a better ruler, marked in mm. Now I know that the line is between 4.2 and 4.3 cm.
OK, so I get a vernier gauge and measure it again. The line is between 4.25 and 4.26 cm.
I can carry on doing this for ever, using finer and finer scales but, I never get to the point where the line stops exactly on one of the marks of the ruler. If it's too close to say if it's more or less than 4.255 then I just need a finer ruler, one that tells me that it's between 4.25500000000000003 and 4.25000...4.
Almost any line I draw has this property of being irrational.
The weird lines are ones that are exactly 1.603 units long or whatever.
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Just wanted to add a little more food for thought...
Imagine you wanted to draw this line with a ruler.
Suppose you just happen to have a ruler where the measurement-bars are a millionth of an atom-width apart, this ruler would be far from accurate enough to depict √2 perfectly.
I actually don't know what my point is other than the fact that I am baffled by how perfectlt this irrational length line is shown in a pythag triangle.
Numbers are mathematical abstract concept; the reality is another thing (it's physics, not mathematics).
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Hi thank you for your post. But one thing is not clear that what the author of this post wanted to mean ? And I have seen some people's comment also and I saw they are also not very clear about this.
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Hi thank you for your post. But one thing is not clear that what the author of this post wanted to mean ? And I have seen some people's comment also and I saw they are also not very clear about this.
Well, to be honest, this is a very hard concept to get across.
It is a very abstract problem, so I would advise you to put on your lateral thinking cap before reading the next part....
It is a question that is intended to probe the very deep relationship between the real-world(physics) and our man-made abstraction to represent it (mathematics).
I was just trying to explore something that I personally found very unusual - the fact that a perfectly normal line has an imperfect and irrational represenation in our maths system.-√2
I mean, we see the line as rational because it has a clear beginning and end point, yet mathematics does not see it the same way -- Where has the process gone wrong? Is our maths system in need of a few patches and upgrades?
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I would not take the words "rational" or "irrational" to be in any way related to the normal english meaning. "Irrational" simply means that the number cannot be expressed as the ratio of two integers. It does not mean that it is somehow not subject to reason and logic.
Maths and physics is full of ordinary words being used for an entirely different concept. "Imaginary" numbers have nothing to do with them being fictitious any more than are "Real" numbers the only ones that actually exist. In physics we call Quarks with daft names like Up, Down, Charm, Strange, Top and Bottom. The words don't really reflect position or a beguiling or weird attitude :-)
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In physics we call Quarks with daft names like Up, Down, Charm, Strange, Top and Bottom. The words don't really reflect position or a beguiling or weird attitude :-)
Dang! I was sure that quark was giving me the old "come hither".