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This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 September 2022 and 9 December 2022. Further details are available
on the course page. Student editor(s):
Annie.nguyen0811 (
article contribs).
— Assignment last updated by Annie.nguyen0811 ( talk) 21:33, 27 October 2022 (UTC)
In Paragraph "Algebra" there ist written:
"It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0."
This is wrong, since the assumption b ≠ 0 is already violated in the next sentence.
Stephan Schief (
talk)
20:15, 8 May 2023 (UTC)
My edit cites two sources: a blog post from Kevin Buzzard and and StackOverflow answer from Arthur Azevedo de Amorim. I'm not concerned about the authors, given that they are a Lean maintainer and an author of Software Foundations respectively. But I'm open to alternatives from more established publishers. Lambda Fairy ( talk) 05:37, 2 December 2023 (UTC)
the article reads like an academic paper or a teaching aid. it speaks to the reader as if the reader is a student. sections such as Elementary arithmetic epitomise this. The article needs to be rewritten to avoid being a research paper or a teaching aid as opposed to be encyclopedic. It should not posit that the reader considers examples. The Fallacies section goes into details of mathematical examples, which treats the section as an academic exercise instead of an encyclopedic article. PicturePerfect666 ( talk) 17:56, 7 December 2023 (UTC)
To clear things up Wikipedia states as follows on what Wikipedia is not " [articles should not ] presented on the assumption that the reader is well-versed in the topic's field." the flede begtins as follows"
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming ); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").
- directly from a maths teaching textbook or aid.
This is confusing and dense and challenging for the reader with limited initial knowledge of the topic to understand the subject of the article, hence it reads like something from academia. PicturePerfect666 ( talk) 19:32, 7 December 2023 (UTC)
If you believe I am being confrontational that is not my intention here and not the tone being aimed for...but remember to assume good faith. If you think that I am not acting in good faith then please consider why you believe this and re-evaluate, as I can assure you my actions are exclusively in good faith. I am simply asking for you to state what your concerns with the article are. You have said you think the article has problems, yet won't state what these are, so no one else can attempt to improve the issue you see the article as having. I may have used direct language, but this is because you won't tell anyone what you see the issues with the article are. Please share what you think the issues with the article are. No one here can read your mind. — Preceding unsigned comment added by PicturePerfect666 ( talk • contribs)
The end of the article contains a simple list of "Sources" which are not referenced in the article. It is therefor challenging to know what these refer to specifically or how they can be scrutinised as verifiable or simply padding. This needs to be updated and improved. PicturePerfect666 ( talk) 18:03, 7 December 2023 (UTC)
Wikipedia [articles should] not rely on users going to complex and dense academic papers or specialist resources to understand [them].This particular article doesn't require familiarity with "dense academic papers" to follow, though some sections assume familiarity with introductory textbook material about various topics, some of which are relatively advanced. But even if it did, Wikipedia articles should cover their subject reasonably comprehensively, including advanced material, while staying as accessible as practical in each section. Sometimes arcane and difficult prerequisites are necessary; we can't include the multiple years' worth of technical coursework in the middle of every article which would be necessary to make every part completely accessible to a lay audience. What we can do is cover the lay-accessible parts of the subject as clearly and completely as we can, organize them toward the top of the article if possible, and then strive to make more advanced parts accessible to the broadest audience we can without going excessively off topic to do so. – jacobolus (t) 23:43, 7 December 2023 (UTC)
I'm trying to rewrite the lead section for clarity, but I don't think I've done an amazing job so far. Does anyone want to take another crack at it, or maybe polish up some of my sentences? For example, can someone improve this vague sentence I wrote without making it too long or technical? "When a real function involves division by a quantity that can become zero for some values, that is a type of singularity." @ XOR'easter? – jacobolus (t) 19:47, 7 December 2023 (UTC)
I think this section would be better titled something like "other number systems", since the focus is the extended real numbers and the projectively extended real numbers, and the material about Wheel theory could be moved there.
It should IMO be preceded by a section (perhaps titled "Calculus") discussing the treatment in calculus and real analysis of the limits of functions with real numbers as the domain/codomain, where infinity is often treated as a limit but not a number per se. This section would discuss infinite singularities, and possibly contrast them with other kinds of singularities, and would then include material about indeterminate forms and L'Hospital's rule.
It might be worth adding a section about complex analysis and the idea of zeros and poles and meromorphic functions. – jacobolus (t) 23:20, 7 December 2023 (UTC)
In regards to Jacobulus's edits, latest here: This article is about division by zero. Therefore acknowledgment that there is sometimes such a thing as division by zero should not be deferred to the fourth paragraph. I am also not sure why Jacobulus preferred to speak of the projectively extended real line rather than the more-used Riemann sphere (when the real line is extended it is more common to add a +∞ and a −∞, which doesn't allow division by zero because it isn't clear which one to pick). -- Trovatore ( talk) 00:22, 8 December 2023 (UTC)
don't think the one-point compactification of the reals is nearly as usedit is used ubiquitously throughout most areas of mathematics and applications to science and engineering, but as a quirk of history is unfortunately rarely explained in a clear or systematic way. – jacobolus (t) 00:55, 8 December 2023 (UTC)
shaped like a sphere– yes, exactly, it's shaped like a sphere. For example, the natural setting for "Möbius transformations" (which, contra Wikipedia's article, should fundamentally be defined geometrically, as general transformations generated by reflections and circle inversions) is the "inversive plane" a.k.a. Möbius plane, which can be naturally modeled either by the Euclidean plane with a single extra point at infinity or by the 2-sphere; as it happens the inversive geometry of these, because all of the relevant relationships are preserved by the stereographic projection, is the same. If we want to represent Möbius transformations numerically, one natural representation is using complex numbers.
I'd need to see examples for your claimFor example, any time you see the trigonometric tangent function (or in general the slope of a geometric line) you are properly dealing with the projectively extended real numbers, which is the natural codomain there. – jacobolus (t) 04:05, 8 December 2023 (UTC)
In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as , where is the dividend (numerator). In ordinary arithmetic, for example on the real numbers, the result of this operation is undefined. However, there are other mathematical contexts in which division by zero is allowed.
Let imagine a graph (1/x) is divided into 2 halves where the negative half is x<0 and the positive half is x>0. The primary proof against 1/0 being a defined value is that the two halves directly contradict one another (negative half shows -∞ but positive half shows ∞) so it must be an undefined value right? No, so lets change our way of thinking. We will be adding a new rule which states: "Any division that might return a negative result shall be written in a way in which a negative number will never be dividing another or divided against.". Since this only applies to HOW we write down or calculate the equations, this does not change anything ((wrong) 2÷-5 = -0.4 --> (correct) -(2÷5) = -0.4). By applying the rule, we would never actually divide by the negatives which completely removes the negative half. This will leave only the positive half with nothing contradicting it, which could only mean x/0 = ∞.
I don't study math like a mathematician but this is easy. I need yall to say im wrong rn holy moly. Also the rules are made ambiguous, think of the best case scenario of my explanation as the primary one. SussusMongus ( talk) 04:52, 16 December 2023 (UTC)
Note: SussusMongus copied this discussion over to Wikipedia:Reference_desk/Mathematics#Overlooked proof. Seems fine to let the conversation live over there.– jacobolus (t) 07:26, 19 December 2023 (UTC)
if a/b=c then cb=a, right? then, 0/0=0 as 0*0=0. anyone find any sources for this? 49.37.202.122 ( talk) 16:07, 5 January 2024 (UTC)
I split this section up and rewrote the first part (the second half, now titled 'Inverse of multiplication') still needs a rewrite). Does it make sense to folks? This kind of informal discussion about concrete interpretations of division and the way zero might fit in seems valuable to me to include up front, but I'm just one person here. Hopefully it doesn't seem like belaboring the point or getting to far into the weeds. I'll try to find and insert a few relevant reliable sources if I can. – jacobolus (t) 01:41, 7 January 2024 (UTC)
Multiplication of two real numbers and is a function . Does it have the inverse? No, because it's not a bijection.
That being said, we can define a function, say, , which is a bijective function and thus it has the inverse , but the function is not a multiplication (of two numbers), it's a multiplication by a specific number.
I hope this explains why we can't say "Division is the inverse of multiplication", because this sentence doesn't specify the specific number. For example, we could say something like this: "Division by 2 is the inverse of multiplication by 2", but I haven't seen such usage (it's much simpler just to write the function in such cases). — Preceding unsigned comment added by Robertas.Vilkas ( talk • contribs) 21:35, 24 February 2024 (UTC)
I am hoping someone can fix the following issues in the intro paragraph starting with "Calculus":
It's not correct that a positive ratio of functions whose denominator tends to 0 tends to infinity, because the numerator could be going to 0 faster.
There is no reason to require a "positive fraction". (It would even be OK for it to be complex; one just needs the denominator to be nonzero in a punctured neighborhood. Can someone figure out a nice way to say this without getting bogged down in details?)
We should avoid conflating "becoming arbitrarily large" and "tending to infinity". The function sin(1/x)/x as x approaches 0 becomes arbitrarily large but does not tend to infinity.
It is 0/0 that is the indeterminate form, not the quotient of functions. So the last sentence of the paragraph needs to be rewritten. Ebony Jackson ( talk) 21:31, 9 March 2024 (UTC)
In the "Elementary arithmetic" section; "The meaning of division" - there is, and I quote: "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server " http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1:0}" - I suppose a mathematical function was put there, but has failed for reasons beyond my comprehension. I hope someone may be able to fix it? Or was it only on my browser? I used two browsers and both showed this error.
Althuios ( talk) 14:58, 29 April 2024 (UTC)
This is the
talk page for discussing improvements to the
Division by zero article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Archives:
Index,
1,
2Auto-archiving period: 730 days
![]() |
![]() | This ![]() It is of interest to multiple WikiProjects. | ||||||||||||||||||||
|
This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 September 2022 and 9 December 2022. Further details are available
on the course page. Student editor(s):
Annie.nguyen0811 (
article contribs).
— Assignment last updated by Annie.nguyen0811 ( talk) 21:33, 27 October 2022 (UTC)
In Paragraph "Algebra" there ist written:
"It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0."
This is wrong, since the assumption b ≠ 0 is already violated in the next sentence.
Stephan Schief (
talk)
20:15, 8 May 2023 (UTC)
My edit cites two sources: a blog post from Kevin Buzzard and and StackOverflow answer from Arthur Azevedo de Amorim. I'm not concerned about the authors, given that they are a Lean maintainer and an author of Software Foundations respectively. But I'm open to alternatives from more established publishers. Lambda Fairy ( talk) 05:37, 2 December 2023 (UTC)
the article reads like an academic paper or a teaching aid. it speaks to the reader as if the reader is a student. sections such as Elementary arithmetic epitomise this. The article needs to be rewritten to avoid being a research paper or a teaching aid as opposed to be encyclopedic. It should not posit that the reader considers examples. The Fallacies section goes into details of mathematical examples, which treats the section as an academic exercise instead of an encyclopedic article. PicturePerfect666 ( talk) 17:56, 7 December 2023 (UTC)
To clear things up Wikipedia states as follows on what Wikipedia is not " [articles should not ] presented on the assumption that the reader is well-versed in the topic's field." the flede begtins as follows"
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming ); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").
- directly from a maths teaching textbook or aid.
This is confusing and dense and challenging for the reader with limited initial knowledge of the topic to understand the subject of the article, hence it reads like something from academia. PicturePerfect666 ( talk) 19:32, 7 December 2023 (UTC)
If you believe I am being confrontational that is not my intention here and not the tone being aimed for...but remember to assume good faith. If you think that I am not acting in good faith then please consider why you believe this and re-evaluate, as I can assure you my actions are exclusively in good faith. I am simply asking for you to state what your concerns with the article are. You have said you think the article has problems, yet won't state what these are, so no one else can attempt to improve the issue you see the article as having. I may have used direct language, but this is because you won't tell anyone what you see the issues with the article are. Please share what you think the issues with the article are. No one here can read your mind. — Preceding unsigned comment added by PicturePerfect666 ( talk • contribs)
The end of the article contains a simple list of "Sources" which are not referenced in the article. It is therefor challenging to know what these refer to specifically or how they can be scrutinised as verifiable or simply padding. This needs to be updated and improved. PicturePerfect666 ( talk) 18:03, 7 December 2023 (UTC)
Wikipedia [articles should] not rely on users going to complex and dense academic papers or specialist resources to understand [them].This particular article doesn't require familiarity with "dense academic papers" to follow, though some sections assume familiarity with introductory textbook material about various topics, some of which are relatively advanced. But even if it did, Wikipedia articles should cover their subject reasonably comprehensively, including advanced material, while staying as accessible as practical in each section. Sometimes arcane and difficult prerequisites are necessary; we can't include the multiple years' worth of technical coursework in the middle of every article which would be necessary to make every part completely accessible to a lay audience. What we can do is cover the lay-accessible parts of the subject as clearly and completely as we can, organize them toward the top of the article if possible, and then strive to make more advanced parts accessible to the broadest audience we can without going excessively off topic to do so. – jacobolus (t) 23:43, 7 December 2023 (UTC)
I'm trying to rewrite the lead section for clarity, but I don't think I've done an amazing job so far. Does anyone want to take another crack at it, or maybe polish up some of my sentences? For example, can someone improve this vague sentence I wrote without making it too long or technical? "When a real function involves division by a quantity that can become zero for some values, that is a type of singularity." @ XOR'easter? – jacobolus (t) 19:47, 7 December 2023 (UTC)
I think this section would be better titled something like "other number systems", since the focus is the extended real numbers and the projectively extended real numbers, and the material about Wheel theory could be moved there.
It should IMO be preceded by a section (perhaps titled "Calculus") discussing the treatment in calculus and real analysis of the limits of functions with real numbers as the domain/codomain, where infinity is often treated as a limit but not a number per se. This section would discuss infinite singularities, and possibly contrast them with other kinds of singularities, and would then include material about indeterminate forms and L'Hospital's rule.
It might be worth adding a section about complex analysis and the idea of zeros and poles and meromorphic functions. – jacobolus (t) 23:20, 7 December 2023 (UTC)
In regards to Jacobulus's edits, latest here: This article is about division by zero. Therefore acknowledgment that there is sometimes such a thing as division by zero should not be deferred to the fourth paragraph. I am also not sure why Jacobulus preferred to speak of the projectively extended real line rather than the more-used Riemann sphere (when the real line is extended it is more common to add a +∞ and a −∞, which doesn't allow division by zero because it isn't clear which one to pick). -- Trovatore ( talk) 00:22, 8 December 2023 (UTC)
don't think the one-point compactification of the reals is nearly as usedit is used ubiquitously throughout most areas of mathematics and applications to science and engineering, but as a quirk of history is unfortunately rarely explained in a clear or systematic way. – jacobolus (t) 00:55, 8 December 2023 (UTC)
shaped like a sphere– yes, exactly, it's shaped like a sphere. For example, the natural setting for "Möbius transformations" (which, contra Wikipedia's article, should fundamentally be defined geometrically, as general transformations generated by reflections and circle inversions) is the "inversive plane" a.k.a. Möbius plane, which can be naturally modeled either by the Euclidean plane with a single extra point at infinity or by the 2-sphere; as it happens the inversive geometry of these, because all of the relevant relationships are preserved by the stereographic projection, is the same. If we want to represent Möbius transformations numerically, one natural representation is using complex numbers.
I'd need to see examples for your claimFor example, any time you see the trigonometric tangent function (or in general the slope of a geometric line) you are properly dealing with the projectively extended real numbers, which is the natural codomain there. – jacobolus (t) 04:05, 8 December 2023 (UTC)
In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as , where is the dividend (numerator). In ordinary arithmetic, for example on the real numbers, the result of this operation is undefined. However, there are other mathematical contexts in which division by zero is allowed.
Let imagine a graph (1/x) is divided into 2 halves where the negative half is x<0 and the positive half is x>0. The primary proof against 1/0 being a defined value is that the two halves directly contradict one another (negative half shows -∞ but positive half shows ∞) so it must be an undefined value right? No, so lets change our way of thinking. We will be adding a new rule which states: "Any division that might return a negative result shall be written in a way in which a negative number will never be dividing another or divided against.". Since this only applies to HOW we write down or calculate the equations, this does not change anything ((wrong) 2÷-5 = -0.4 --> (correct) -(2÷5) = -0.4). By applying the rule, we would never actually divide by the negatives which completely removes the negative half. This will leave only the positive half with nothing contradicting it, which could only mean x/0 = ∞.
I don't study math like a mathematician but this is easy. I need yall to say im wrong rn holy moly. Also the rules are made ambiguous, think of the best case scenario of my explanation as the primary one. SussusMongus ( talk) 04:52, 16 December 2023 (UTC)
Note: SussusMongus copied this discussion over to Wikipedia:Reference_desk/Mathematics#Overlooked proof. Seems fine to let the conversation live over there.– jacobolus (t) 07:26, 19 December 2023 (UTC)
if a/b=c then cb=a, right? then, 0/0=0 as 0*0=0. anyone find any sources for this? 49.37.202.122 ( talk) 16:07, 5 January 2024 (UTC)
I split this section up and rewrote the first part (the second half, now titled 'Inverse of multiplication') still needs a rewrite). Does it make sense to folks? This kind of informal discussion about concrete interpretations of division and the way zero might fit in seems valuable to me to include up front, but I'm just one person here. Hopefully it doesn't seem like belaboring the point or getting to far into the weeds. I'll try to find and insert a few relevant reliable sources if I can. – jacobolus (t) 01:41, 7 January 2024 (UTC)
Multiplication of two real numbers and is a function . Does it have the inverse? No, because it's not a bijection.
That being said, we can define a function, say, , which is a bijective function and thus it has the inverse , but the function is not a multiplication (of two numbers), it's a multiplication by a specific number.
I hope this explains why we can't say "Division is the inverse of multiplication", because this sentence doesn't specify the specific number. For example, we could say something like this: "Division by 2 is the inverse of multiplication by 2", but I haven't seen such usage (it's much simpler just to write the function in such cases). — Preceding unsigned comment added by Robertas.Vilkas ( talk • contribs) 21:35, 24 February 2024 (UTC)
I am hoping someone can fix the following issues in the intro paragraph starting with "Calculus":
It's not correct that a positive ratio of functions whose denominator tends to 0 tends to infinity, because the numerator could be going to 0 faster.
There is no reason to require a "positive fraction". (It would even be OK for it to be complex; one just needs the denominator to be nonzero in a punctured neighborhood. Can someone figure out a nice way to say this without getting bogged down in details?)
We should avoid conflating "becoming arbitrarily large" and "tending to infinity". The function sin(1/x)/x as x approaches 0 becomes arbitrarily large but does not tend to infinity.
It is 0/0 that is the indeterminate form, not the quotient of functions. So the last sentence of the paragraph needs to be rewritten. Ebony Jackson ( talk) 21:31, 9 March 2024 (UTC)
In the "Elementary arithmetic" section; "The meaning of division" - there is, and I quote: "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server " http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1:0}" - I suppose a mathematical function was put there, but has failed for reasons beyond my comprehension. I hope someone may be able to fix it? Or was it only on my browser? I used two browsers and both showed this error.
Althuios ( talk) 14:58, 29 April 2024 (UTC)