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December 21st, 2018, 09:35 PM   #1
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how to draw a non-terminating decimal on a number line

Hi,
it is possible to draw 3.456 on a number line and graph it, but how is it possible to draw the following non-terminating decimals, i.e 2.3460274501837330986734345672095867, pi, 4.265686866868686868686868, on a number line and graph?
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Last edited by skipjack; December 22nd, 2018 at 01:19 AM.
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December 22nd, 2018, 09:28 AM   #2
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Like all drawing of points on lines, its an approximation. You make that approximation as accurate as you deem necessary for your purpose. There isn't a graphics package out there that will accurately plot the terminating decimals 2.3460274501837330986734345672095867, or 4.265686866868686868686868 on a number line - or at least the screen/printer won't display is accurately - they will all approximate. Doing it by hand is even less realistic.
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Last edited by v8archie; December 22nd, 2018 at 09:34 AM.
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December 22nd, 2018, 11:17 AM   #3
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Can you cut a pie in thirds? Isn't each third .333...? How did you do that?
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December 22nd, 2018, 11:42 AM   #4
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Not exactly, no. The chances of me doing so are vanishingly small.

Last edited by v8archie; December 22nd, 2018 at 11:44 AM.
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December 22nd, 2018, 11:55 AM   #5
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Quote:
Originally Posted by Maschke View Post
Can you cut a pie in thirds? Isn't each third .333...? How did you do that?
Actually I can't. But apparently I am quite capable of cutting a pie into 3 pieces which have varying sizes. What can I say? I'm a klutz at hand-eye coordination!

-Dan
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December 22nd, 2018, 12:41 PM   #6
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You can draw 0,333... or 1/3 by triangle congruence
Lets take another example about $\displaystyle \sqrt{2}$
Take a square with Area=1 , the diagonal is $\displaystyle \sqrt{2}$
Use some equipments to copy the diagonal to axis
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Last edited by idontknow; December 22nd, 2018 at 12:43 PM.
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December 22nd, 2018, 01:34 PM   #7
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In general you can draw neither a square nor a triangle. Even if you could, in general you can't draw a straight line between the two. Far less can you do these things on a given line. Every step in the geometric construction you use is subject to errors.

Of course, in mathematics we usually pretend that we can do these things without error, but then we also pretend that we can mark any number, rational or not, on a number line.

PS: to mark $\pi$ on a number line, you can roll a wheel of diameter 1 a whole turn along it (from zero).
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December 22nd, 2018, 01:53 PM   #8
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Quote:
Originally Posted by v8archie View Post

PS: to mark $\pi$ on a number line, you can roll a wheel of diameter 1 a whole turn along it (from zero).
You can roll a wheel but not cut a pie?

OP, the point is that in the real world, we can't measure anything exactly at all. Archie is right, we can't cut a real pie in exact thirds. Even if we accidentally did, we couldn't measure it to be sure.

There is another aspect of this, which is that many of the familiar transcendental numbers are in fact computable. We can product a line segment of length pi by writing a program to successively compute the decimal digits of pi. We take each digit as an instruction to move that many tenths forward from where we are. The sequence converges to pi, and the program itself is finite. So we can in fact measure pi, if we allow computer programs that are themselves finite but are allowed to run as long as we like.

On the other hand most real numbers are not computable. Their digits are essentially random, and no algorithm can generate them. Do those numbers exist? We just assume they're there wherever "there" is. They're a mathematical abstraction.

Why do we take the mathematical real numbers as the correct model of the ancient idea of a continuous line? It's only an assumption. It's not a fact of the world.
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Last edited by Maschke; December 22nd, 2018 at 02:37 PM.
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December 22nd, 2018, 02:29 PM   #9
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You can roll a wheel but not cut a pie?
Use the wheel to cut the pie!

Done.
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December 23rd, 2018, 12:56 AM   #10
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Quote:
Originally Posted by idontknow View Post
You can draw 0,333... or 1/3 by triangle congruence
Lets take another example about $\displaystyle \sqrt{2}$
Take a square with Area=1 , the diagonal is $\displaystyle \sqrt{2}$
Use some equipments to copy the diagonal to axis
pi=22/7
and 22/7=3.1428571428571428571428571428571...........
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