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

  • Context: High School 
  • Thread starter Thread starter phoenixthoth
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Discussion Overview

The discussion centers around the relationship between the fractions and their decimal representations, specifically examining whether 1/3 is equal to 0.333... and the implications of this equality for understanding that 1 is equal to 0.999.... The conversation explores various methods of demonstrating these relationships, the intuitiveness of different starting points, and the conceptual challenges faced by learners.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that starting with 1/3 = 0.333... and multiplying by three to show 1 = 0.999... is an easier method, as it avoids more complex algebra.
  • Others argue that this method assumes the validity of 1/3 = 0.333... without providing a rigorous proof, which they believe is necessary for a solid understanding.
  • There is a suggestion that different starting points can lead to varying levels of understanding, with some finding 1/3 = 0.333... less mystifying than 1 = 0.999....
  • Some participants mention that many people accept 1/3 = 0.333... but struggle with the idea that 1 = 0.999..., indicating a potential intellectual crisis for learners.
  • A later reply introduces the concept of defining 0.999... as the least upper bound of its finite approximations, which leads to the conclusion that 0.999... = 1.
  • Another participant presents a limit-based argument to show that 0.999... equals 1, although they acknowledge potential inaccuracies in their reasoning.

Areas of Agreement / Disagreement

Participants express differing views on the ease and validity of various methods for demonstrating the relationship between these decimal representations. There is no consensus on which method is superior or more intuitive, and the discussion remains unresolved regarding the best approach to explain these concepts.

Contextual Notes

Some participants highlight the need for rigorous proofs from first principles, while others suggest that intuitive methods may suffice for teaching purposes. The discussion reflects a range of mathematical backgrounds and pedagogical approaches, indicating that the topic is complex and multifaceted.

  • #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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