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

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SUMMARY

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 argue about the simplicity of using 1/3=0.333... as a starting point for demonstrating 1=0.999..., suggesting that this approach may be more intuitive for learners. The conversation highlights the representation of rational numbers as infinite series and the implications of decimal expansions in mathematics.

PREREQUISITES
  • Understanding of rational numbers and their decimal representations
  • Familiarity with infinite series and limits
  • Basic algebraic manipulation of equations
  • Knowledge of the properties of real numbers and fields
NEXT STEPS
  • Study the concept of infinite series in mathematics
  • Learn about the properties of real numbers and their decimal expansions
  • Explore proofs of the equivalence between 0.999... and 1
  • Investigate the Archimedean principle and its implications in real analysis
USEFUL FOR

Mathematicians, educators, students of mathematics, and anyone interested in the foundations of number theory and decimal representation.

  • #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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