How Can We Understand Division by Zero Through Logical Reasoning?

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Discussion Overview

The discussion revolves around the concept of division by zero, exploring logical reasoning rather than mathematical definitions. Participants examine the implications of attempting to divide a number by zero, addressing both intuitive and formal perspectives on the issue.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that dividing a number by zero means attempting to distribute that number among no individuals, leading to contradictions about the existence of parts.
  • Others challenge the initial definitions of division, stating that the logic presented is flawed and that division by zero is inherently undefined.
  • One participant suggests that the problem arises from misunderstanding the nature of division, emphasizing that dividing by zero does not yield a meaningful quantity.
  • Another viewpoint posits that division can be understood as the inverse of multiplication, and thus dividing by zero does not make sense within this framework.
  • Some participants mention that while division by zero is typically considered undefined, there are contexts in advanced mathematics (like projective reals) where it might be treated differently.
  • A later reply questions the applicability of the initial logical reasoning to other numbers, suggesting that the logic may not hold for non-integer values.

Areas of Agreement / Disagreement

Participants express a range of views, with no consensus on the nature of division by zero. Some agree that it is undefined, while others propose alternative interpretations or challenge the definitions used in the discussion.

Contextual Notes

Limitations include varying definitions of division, the reliance on intuitive versus formal reasoning, and the potential for misunderstanding the implications of dividing by zero in different mathematical contexts.

luther_paul
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Lets try arguing in not mathematical terms but logical reasoning. if a number x is divided by any number n, it means giving every individual n a single part from x... if our number is zero how are we going to divide x to zero individuals...? we have a x objects to be divided among no one... either the number stays as it is or a the number is gone. but the number cannot be gone because it was distributed to no one.. how is this?
 
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Lets try arguing in not mathematical terms but logical reasoning. if a number x is divided by any number n, it means giving every individual n a single part from x... if our number is zero how are we going to divide x to zero individuals...? we have a x objects to be divided among no one... either the number stays as it is or a the number is gone. but the number cannot be gone because it was distributed to no one.. how is this?

(1) It is not a good idea to use mathematical notation for something that is not given in "mathematical terms".
(2) Your logic is wrong.
(3) You definition of division is wrong.
 
"..., with all parts of equal magnitude..." needs to be added to your specification.

Zero parts means no parts at all. But they must finally add up to 1. If the total magnitude of parts is less than 1, then the parts must be too small.

For example, if you guess that 2 is 20/5, then five 2s comprise only 10. That's too small, so 20/5 must be larger than 2. Likewise, five 3s is smaller than 20, so 20/5 must be larger than 3. But five 5s is larger than 20, so 5 is too big for 20/5. It turns out that five 4s is just right, total magnitude 20. So 4 is 20/5.

Back to 1/0:
If you guess 1,000,000, then NO 1,000,000s is 0, and less than 1, so 1,000,000 is too small for the magnitude of a part. You can easily see that any whole number x is just too small to make NO xs equal to 1. One might be tempted to try a surpassing infinite magnitude. But "NONE of those" still adds up to 0, not 1. So you are left with no viable answer.

In brief: for any magnitude x, x*0 = 0, so 1/0 has no viable solution.
 
going back to what you have stated that

...if a number x is divided by any number n, it means giving every individual n a single part from x...

This simply means that each person from the n people would be receiving x/n parts of the x (that is if the number x is to be divided equally among the n people). But did you give any part of the x to any one? the answer is no. so the answer is still 0. got my point?
 
(1) It is not a good idea to use mathematical notation for something that is not given in "mathematical terms".
(2) Your logic is wrong.
(3) You definition of division is wrong.


Mathematics says this wrong because it defined something that makes it wrong. mathematics defined everything, but logic can never be defined. its our mind that makes logic!
 
Mathematics says this wrong because it defined something that makes it wrong. mathematics defined everything, but logic can never be defined. its our mind that makes logic!

I would recommend that you actually study logic before you make such sweeping statements.
 
better try to understand the logic behind the problem itself. however, it is still MATHEMATICAL in nature ( the way you would be presenting the solution to the problem is mathematical in nature)
the other explanation (still using mathematical reasoning) is by {cross multiplication} however this does not make any sense at all.

[zz)]
 
thanks for the advice... id better study more!
 
The problem is that x/0 doesn't equal anything it is undefined.
 
  • #10
The problem is that x/0 doesn't equal anything it is undefined.

you better try to explain more what you have written. it is very confusing.

or do you mean "The problem is that x/0 doesn't equal anything.[/size] it is undefined. ?
 
  • #11
Originally posted by oen_maclaude
The problem is that x/0 doesn't equal anything it is undefined.

you better try to explain more what you have written. it is very confusing.

or do you mean "The problem is that x/0 doesn't equal anything.[/size] it is undefined. ?

What do you mean with "equal anything"?
 
  • #12
you better ask ruud about it. just an empahasis on the sentence he/she had written specially on the period (.)
 
  • #13
Originally posted by luther_paul
Lets try arguing in not mathematical terms but logical reasoning. if a number x is divided by any number n, it means giving every individual n a single part from x... if our number is zero how are we going to divide x to zero individuals...? we have a x objects to be divided among no one... either the number stays as it is or a the number is gone. but the number cannot be gone because it was distributed to no one.. how is this?
I think one of the main problems with this statement is that it does not make sense to divide something among zero people while looking for the amount each person received. If you have a cake and nobody wants it, then yes the cake is still there, but this is not the same as dividing the cake among zero people then finding how much cake each person has because there are no people to receive the cake. You should not be looking for the remainder of the cake but the amount of cake each person received. And as I have hopefully shown this causes the problem to fall apart.
 
  • #14
i agree with the second comment on this thread, the problem is in the definition of division. devision is simply the inverse operation of multiplication. if we ask "how many will we have, if we have n, m number of times?" we have n*m=nm. if we do the inverse of this and as "how many will times can nm be divided into groups of n?" which will translate nm/n=m.

all this to say:

if we have x zero times, we have nothing, which we write x*0=0. if we ask how many times we can divide x into groups of zero (x/0), it just doesn't make since. This is why it is "undefined".


math commandment IV. Thou shalt not divide by zero!
 
  • #15
gymnast_nerd said:
math commandment IV. Thou shalt not divide by zero!

math Ecclesiastes III. Sometime you can divide by 0. :-p

In the projective reals, or the trivial field, for example.
 
  • #16
CRGreathouse said:
math Ecclesiastes III. Sometime you can divide by 0. :-p

In the projective reals, or the trivial field, for example.

Yes. Things always look different when you see things in terms of circles rather than (extended) lines.
 
Last edited:
  • #17
luther_paul, your logical example only seems to apply for all natural numbers. This is the example given to children in primary school so they can understand division.

Division can be extended upon, and it's no surprised your example breaks down. Unless you believe otherwise, please explain to me using the same logic how to divide by 1.4, or even -2.
 

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