Is Dividing 1 by 3 an Invalid Problem?

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

The discussion revolves around the mathematical operation of dividing 1 by 3, specifically questioning whether this operation can be considered valid given that the result is a repeating decimal (0.33333...). Participants explore the implications of this representation and the nature of division itself.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that dividing 1 by 3 results in an infinite repeating decimal, which raises the question of whether the operation can ever be considered to end.
  • Others argue that the notation 0.333... represents a completed process and is equivalent to 1/3, thus implying that the division is valid and finished.
  • A participant proposes examining the division in a base 3 number system, where 1/3 equals 0.1, suggesting that different number systems can represent rational numbers differently.
  • There is a discussion about the nature of division itself, with some participants clarifying that division is an inverse operation to multiplication and questioning what it means for division to "end."
  • Some participants highlight the distinction between the representation of numbers and the actual numbers themselves, emphasizing that 0.333... is a representation of 1/3 rather than a process that continues indefinitely.
  • Concerns are raised about division by zero, with participants noting that it is fundamentally different from dividing by a non-zero number and cannot be defined meaningfully in standard arithmetic.

Areas of Agreement / Disagreement

Participants express differing views on whether the division operation can be considered valid or if it is inherently flawed due to the nature of the repeating decimal. There is no consensus on the implications of this discussion, as multiple competing perspectives remain.

Contextual Notes

The discussion touches on the limitations of representing certain rational numbers in decimal form and the implications of infinite series and limits in mathematics. Participants also note that the understanding of division and number representation can vary based on the mathematical framework being used.

  • #61
HallsofIvy said:
No, it wasn't "chosen". 0.9999... means .9+ .09+ .009+ .0009+ .00009+ ...= .9(1+ .01+ .001+ .0001+ .00001+ ...)

That last is a "geometric series" which is taught in any good "precalculus" or "algebra II" class.
a(1+ r+ r^2+ r^3+ r^4+ ...) has sum a/(1- r) as long as |r|< 1.

Here, a= 0.9 and r= .1. a/(1- r)= .9/(1- .1)= .9/.9= 1.

The point is that it was chosen to mean exactly that, not that we chose to do our calculations correctly...
 

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