Is 1/3 Really Equal to 0.333...? Find Out the Easier Way with This Trick!

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The discussion centers on the mathematical equivalence of 1/3 and 0.333..., leading to the conclusion that 1 equals 0.999... when multiplied by three. Participants debate the simplicity and clarity of using 1/3 = 0.333... as a starting point versus directly proving 1 = 0.999... through algebraic manipulation. Some argue that the former approach is more intuitive for learners, while others emphasize the need for rigorous proof from first principles. The conversation touches on the representation of rational numbers and the nature of decimal expansions, highlighting that both forms are equally valid yet can be perceived differently. Ultimately, the discussion illustrates the complexities and nuances in understanding infinite decimals and their equivalences.
  • #91
Selak3 said:
I guess the nature of reality does have something to do with this.
Is there a part of the universe that is indefinitely divisible? Ie: you keeping cutting
a part of reality in half but it never complete disappears? (ie, you get infinitely close
to 0 but never quite reaching it).
Would the definition of 1/x as x approaches to infinity need to be modified?
Would one need to modify mathematics in these cases?
Would one want mathematics to reflect reality?o:)

You have only a rough Idea what mathematics is. Mathematical theories are consistent- that is all we can ask of them. If there is some new fact of "reality" (I'm not sure what you are talking about here- a sort of physics perhaps?) that makes the mathematical model being used not correct, that means you have to change your model. That happens all the time. Mathematics itself stays the same. The way mathematics is applied changes.

(Arildo typed a shorter response and got it in before me! But we are really saying the same thing.)
 
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  • #92
Selak3 said:
Would one need to modify mathematics in these cases?
Physics, perhaps. Mathematics is only modified through addition of more concepts, or the finding of a logical flaw in existing concepts. Reference to models of reality is irrelevant.
Selak3 said:
Would one want mathematics to reflect reality?o:)
No, but that's the exact job description of physics. :biggrin: Mathematics is more explorative in terms of abstract objects and relationships than the stagnation that would result if one had to wait for empirical models of reality.
 
  • #93
As a follow-up to my own, and hallsofIvy's comments (and hypermorphism's):
To be sure, "reality" provides a spur to develop new mathematics and "old" mathematics is used in order to create a "reality model".

This, however, does not impinge upon whether or not a given set of axioms defines a consistent or inconsistent mathematics.
 
  • #94
Bleh, this archived thread keeps popping up. I think it's time to put an end to this necromancy.
 

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