0.33333333

Jun 2014
650
54
USA
Ok let me straighten this out.
For me it’s about curiosity. The recurring decimal is the itch that I can never quite scratch. It frequents my dreams and invades my meditations. It’s the blot on my copy book, the fly in the ointment, the Ringo to my Paul McCartney.
...

I just cannot sate the feeling that the mathematics that underpins so many of the foundations on which we build our lives, so harmoniously accepts this troublesome and unsolvable conundrum. 1/3 isn’t 0.3333.
No mathematician thinks that 1/3 = 0.3333. That is your own misunderstanding. Even if I give you the benefit of implying $\frac{1}{3} = 0.\overline{3}$, you're still confusing things in that $0.\overline{3}$ is just a representation of $\frac{1}{3}$ in decimal form. There are uncountably many ways to represent $\frac{1}{3}$, where the decimal representation $0.\overline{3}$ is just one of them. This is nothing to lose sleep over.
 
Jun 2014
650
54
USA
Is this a figure of speech? I would like your reasoning on this.
Let “peach” = 1/3
Let [insert squiggly line here] = 1/3
... and so on.

I’m separating the representation of the number in whatever language we care to conjur from the number itself.
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
Also, because maths happens in the mind rather than the physical world, mathematicians are able to work with (some) infinite objects such as infinite decimals and infinite sequences.
 
Jun 2014
650
54
USA
Yes, so why is it uncountable?
Let each line segment between 0 and 1 inch in length represent 1/3. We can always come up with more arbitrary ways to represent 1/3...

Edit:

PS - Just in case you don't have a magnifying glass that powerful, for each real number x (of which there are uncountably many), let 1/3 = x/3x.
 
Last edited:
Aug 2012
2,496
781
Let each line segment between 0 and 1 inch in length represent 1/3. We can always come up with more arbitrary ways to represent 1/3...

Edit:

PS - Just in case you don't have a magnifying glass that powerful, for each real number x (of which there are uncountably many), let 1/3 = x/3x.
Wait, what? Aren't there at most countably many notations for each notatable number? (Excluding those that are not notatable at all, such as noncomputable numbers.)
 
Last edited by a moderator:
Jun 2014
650
54
USA
Wait, what? Aren't there at most countably many notations for each notatable number? (Excluding those that are not notatable at all, such as noncomputable numbers.)
If you are restricting your notations to formulas that can be produced using a finite formal language that takes into account only finite sentences, then sure, there are only countably many.

My only point is that a representation of a number ought not be confused as being the number itself. There certainly are uncountably many representations that we could conjure. The line segment 'thing' above demonstrates that, as there are uncountably many line segments that have lengths of between 0 and 1 inches long.
 
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Aug 2012
2,496
781
If you are restricting your notations to formulas that can be produced using a finite formal language that takes into account only finite sentences, then sure, there are only countably many.

My only point is that a representation of a number ought not be confused as being the number itself. There certainly are uncountably many representations that we could conjure.
Repeating a false claim doesn't count as evidence. Can you explain what you mean?


The line segment 'thing' above demonstrates that, as there are uncountably many line segments that have lengths of between 0 and 1 inches long.
That couldn't be more false, as it contradict known physics and profoundly misrepresents the nature of physical measurement.
 
Last edited by a moderator:
Oct 2009
942
367
Let each line segment between 0 and 1 inch in length represent 1/3. We can always come up with more arbitrary ways to represent 1/3...

Edit:

PS - Just in case you don't have a magnifying glass that powerful, for each real number x (of which there are uncountably many), let 1/3 = x/3x.
There are only finitely many atoms in the line segment, so finitely many ways to write a line segment, no?