The student's acceptance or rejection of 0.999 =1

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the acceptance or rejection of the equation 0.999... = 1, exploring the reasons behind students' confusion regarding this non-intuitive mathematical concept. Participants examine the implications of infinity in mathematics and how it affects understanding, with a focus on theoretical and conceptual aspects rather than practical applications.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants argue that students view 0.999... as a large number of 9's, leading to confusion about its equality with 1, which they perceive as more "realistic."
  • Others propose that the rejection stems from the invocation of infinity, which they consider an operationally useful but non-existent concept.
  • A participant suggests that if one rejects infinite sets, then the notation 0.999... lacks meaning, thus complicating the discussion of equality.
  • Some argue that while infinite sets may not represent material existence, they are still valid within mathematical frameworks that do not require real-world correspondence.
  • Concerns are raised about students' understanding of irrational numbers and infinite decimals, suggesting that they may grasp the concept of infinite decimals but struggle with the implications of infinity.
  • Participants discuss the role of convergence in understanding limits and how this concept is often not adequately explained to students, leading to confusion.
  • Some highlight that mathematical concepts do not need to correspond to physical reality, emphasizing the distinction between mathematics and physics.
  • There is mention of the challenges faced by students when learning about mathematical proofs and the timing of introducing complex concepts.

Areas of Agreement / Disagreement

Participants express a range of views on the nature of infinity and its implications for understanding the equation 0.999... = 1. There is no consensus on whether the confusion arises from the concept of infinity itself or from the teaching methods employed.

Contextual Notes

Limitations in understanding convergence and the implications of infinite sets are noted, as well as the potential disconnect between mathematical concepts and their physical representations.

  • #61
coolul007 said:
The point that I try to make is that a limit is NOT equality. A limit is a number that will not be reached and will not go beyond, because one can always add an iteration to the sequence. The fact that limits have been used so much to achieve success, doesn't mean that they are EQUAL. We also treat pi and e as if they were not limits, but numbers, nice habit, but not accurate.

Completely and utterly wrong. Each sentence is wrong. coolul007 you need to go back and study limits again, from the very beginning.
 
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  • #62
coolul007 said:
A limit is a number that will not be reached and will not go beyond, because one can always add an iteration to the sequence.
I think that you have a fundamental misunderstanding of what a limit actually means, based on what you are saying above, particularly the part about "that will not be reached".

There is a big difference between saying an = L and ## \lim_{n \to \infty} a_n = L##, and you seem to not be getting that difference.
 
  • #63
coolul007 said:
The point that I try to make is that a limit is NOT equality. A limit is a number that will not be reached and will not go beyond, because one can always add an iteration to the sequence. The fact that limits have been used so much to achieve success, doesn't mean that they are EQUAL. We also treat pi and e as if they were not limits, but numbers, nice habit, but not accurate.

There is little point in continuing this discussion. I do understand that you think that 0.999... is not equal to 1. I even understand why you think that. The problem is that you seem to be missing quite some basic knowledge about real numbers. Without this knowledge, I think it would be impossible to fully grasp the 1=0.9999... situation.

So, if you're truly interested in understanding the equality 1=0.9999..., then I can only suggest you to start studying limits and real numbers. Any good analysis book should cover these things very well. So please, do yourself a favor and try to study these things from the very beginning.

I'll keep this thread open to see if we can get further discussion. But if people keep commenting without taking the effort of studying limits and real numbers, then we will be forced to lock.
 

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