Why Division by Zero is a Bad Idea

[fresh_42]

A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:
What if it was allowed?
Then we would get a contradiction, and a contradiction is the greatest enemy of mathematical rigor. Many students tried to find a way to divide by zero once in their lifetime. To be honest: It is possible! We could allow it! However, that would come at a price. We had to give up other laws which we did not want to lose; most of all the fact that $0$ and $1$ are two different numbers. A division by zero often results in the equation $1=0$ which would make both of them pretty useless. But it also causes problems for other, often unexpected rules and laws that we do not want to give up.

Why division by zero is a bad idea as it implies …

Table of Contents...

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[berkeman]

Love the countdown from 10 to 1...

 

[symbolipoint]

I have not read the article, or the posting into the continued link.

Division by zero is meaningless. You could try this:

Start with a number, any non negative number greater than 0. Ten, for example. Now take a divisor, being 0. How many times can you subtract 0 from 10, revise the result, and then continue subtraction of 0, until your result is too small to subtract another 0? FOREVER! This will never end. Division By Zero is meaningless.

 

[fresh_42]

I have not read the article, or the posting into the continued link.

Division by zero is meaningless. You could try this:

Start with a number, any non negative number greater than 0. Ten, for example. Now take a divisor, being 0. How many times can you subtract 0 from 10, revise the result, and then continue subtraction of 0, until your result is too small to subtract another 0? FOREVER! This will never end. Division By Zero is meaningless.

Sorry, but your comment is meaningless. It contains a lot of hidden assumptions and "meaning" is not quantifiable, it is an opinion, not a fact. By what right do you make assumptions like $1\neq 0$? My article deals with that kind of assumption and is meant to answer the FAQ about "division by zero". Calling it "meaningless" is a patronizing standpoint towards all the kids who come here and ask about it.

 

[symbolipoint]

Sorry, but your comment is meaningless. It contains a lot of hidden assumptions and "meaning" is not quantifiable, it is an opinion, not a fact. By what right do you make assumptions like $1\neq 0$? My article deals with that kind of assumption and is meant to answer the FAQ about "division by zero". Calling it "meaningless" is a patronizing standpoint towards all the kids who come here and ask about it.

We (or as part of my instruction) were taught that "division by zero is meaningless". Your article explains this? My attempt to try dividing by zero confirmed, at least for me, what I was earlier taught. The simpler undergraduate and remedial level textbooks never treated details of "division by zero"; only doing not much more than ".... is not allowed.". (Those books being for Algebra 1 through Calculus 2/3).

 

[symbolipoint]

I just now looked at your points #10 and #9 in the article. We already know those. Those reasons are enough that division by zero is meaningless; or must we now just say, "a bad idea"?

 

[symbolipoint]

Sorry, but your comment is meaningless. It contains a lot of hidden assumptions and "meaning" is not quantifiable, it is an opinion, not a fact. By what right do you make assumptions like $1\neq 0$? My article deals with that kind of assumption and is meant to answer the FAQ about "division by zero". Calling it "meaningless" is a patronizing standpoint towards all the kids who come here and ask about it.

In pure truth I WAS in fact, along with other students in my classes, taught that division by zero is "meaningless". Not that it is a "bad idea". The choice of words used for our instruction was as I say.

 

[fresh_42]

It is obvious that I addressed a question that frequently comes up here. Those threads are usually several dozen posts long. Don't you think an answer in a single article is justified then? If you do not want to read it, or even attempt to understand that it wasn't written for you, why do you comment at all?

I have not read the article, or the posting into the continued link.

... but I have an opinion ...

Division by zero is meaningless.

... is a declaration of <censored>.

I tried a scientific approach, instead of calling out people who ask this question. If it does not please you, stop commenting. You already said that you stopped reading what you commented about here, so I do not need to say it.

I still believe that PF can be I place where kids and students learn something, despite some patronizing <censored> who always know things better and belittle those who have questions they consider "meaningless".

 

[fresh_42]

In pure truth I WAS in fact, along with other students in my classes, taught that division by zero is "meaningless". Not that it is a "bad idea". The choice of words used for our instruction was as I say.

It is possible, so it is not meaningless. It is a bad idea because the consequences are severe. People quite often think that
$$
\dfrac{1}{0}=\infty
$$
is a valid point of view. They deserve to know why it is not. And, and this is not trivial, that
$$
\dfrac{1}{\varepsilon }=\dfrac{\omega }{1}
$$
is something else.

 

[weirdoguy]

Um, but since when calling a question "meaningless" is patronizing? At my faculty we even had a lecture called "Meaningless questions in hydrodynamics". The word is not a problem, understanding its meaning is.

 

[fresh_42]

Um, but since when calling a question "meaningless" is patronizing?

It is a judgment. It also says, that people who ask this, have a meaningless question. This is patronizing.

At my faculty we even had a lecture called "Meaningless questions in hydrodynamics". The word is not a problem, understanding its meaning is.

"Meaningless" is a word that has no place in education. It belittles people. And it cannot be measured, it's merely an opinion by someone who does not want to be disturbed by someone else's question.

 

[symbolipoint]

It is obvious that I addressed a question that frequently comes up here. Those threads are usually several dozen posts long. Don't you think an answer in a single article is justified then? If you do not want to read it, or even attempt to understand that it wasn't written for you, why do you comment at all?


... but I have an opinion ...

... is a declaration of <censored>.

I tried a scientific approach, instead of calling out people who ask this question. If it does not please you, stop commenting. You already said that you stopped reading what you commented about here, so I do not need to say it.

I still believe that PF can be I place where kids and students learn something, despite some patronizing <censored> who always know things better and belittle those who have questions they consider "meaningless".

But I did read parts of the article. Later as you say division by zero is still meaningful - not yet convincing me.

 

[fresh_42]

Here is a set with four numbers $\{0,1,2,3\}$ that allows a division by zero:
$$
\begin{array}{|c|c|c|c|c|}
+ & 0 & 1& 2 & 3\\
\hline
0& 0 & 1&2&3\\
\hline
1& 1 & 2&3&0\\
\hline
2& 2 & 3&0&1\\
\hline
3& 3 & 0 &1&2\\
\hline
\end{array} \qquad
\begin{array}{|c|c|c|c|c|}
\cdot & 0 & 1& 2 & 3\\
\hline
0& 1 & 0 &2 &3\\
\hline
1& 0 & 1 & 3&2 \\
\hline
2& 2 &3 &1 &0\\
\hline
3& 3& 2& 0&1\\
\hline
\end{array} $$
This is an admissible structure. By which criteria do you call this set "meaningless"?

 

[symbolipoint]

It is a judgment. It also says, that people who ask this, have a meaningless question. This is patronizing.

"Meaningless" is a word that has no place in education. It belittles people. And it cannot be measured, it's merely an opinion by someone who does not want to be disturbed by someone else's question.

I am not so sure that we felt patronized when we were taught that Division by Zero is meaningless. We did not seem to feel disturbed nor uncomfortable about this. Our point of view might have been limited by very few of us going into detail not required of our at most intro. and intermediate level Calculus courses.

Possibly, just possibly, the more common wording used was "impossible" more than "meaningless" in regard to Division By Zero. Memory about this is not perfect.

 

[symbolipoint]

Here is a set with four numbers $\{0,1,2,3\}$ that allows a division by zero:
$$
\begin{array}{|c|c|c|c|c|}
+ & 0 & 1& 2 & 3\\
\hline
0& 0 & 1&2&3\\
\hline
1& 1 & 2&3&0\\
\hline
2& 2 & 3&0&1\\
\hline
3& 3 & 0 &1&2\\
\hline
\end{array} \qquad
\begin{array}{|c|c|c|c|c|}
\cdot & 0 & 1& 2 & 3\\
\hline
0& 1 & 0 &2 &3\\
\hline
1& 0 & 1 & 3&2 \\
\hline
2& 2 &3 &1 &0\\
\hline
3& 3& 2& 0&1\\
\hline
\end{array} $$
This is an admissible structure. By which criteria do you call this set "meaningless"?

Is that Arithmetic, as might be understood of a high school or early university student?

 

[fresh_42]

Possibly, just possibly, the more common wording used was "impossible" more than "meaningless" in regard to Division By Zero. Memory about this is not perfect.

Impossible is wrong, too. It is possible, see post #13.

Many students tried to find a way to divide by zero once in their lifetime. To be honest: It is possible! We could allow it! However, that would come at a price. We had to give up other laws which we did not want to lose ...

That is the point. It simply leads to constructions that we do not want to have. Judgments like "impossible" or "meaningless" are based on hidden assumptions, mainly on what people would call "normal arithmetic". Hiding assumptions and base a judgment on those is not my understanding of science.

 

[fresh_42]

Is that Arithmetic, as might be understood of a high school or early university student?

Depends on the laziness of the teacher.

 

[symbolipoint]

Depends on the laziness of the teacher.

That almost looks like a table of computation results but your second construction grid seems too strange. Your article referred from the post #1 makes far better progress that the post #13.

 

[gentzen]

Not sure whether you heard of inverse semigroups or inverse rings (or von Neumann regular ring) before. Those are perfectly legitimate algebraic structures. It is easy to turn the multiplicative semigroup of a field into an inverse semigroup: simply define $0^{-1}=0$. Inverse rings have a sufficiently nice equational theory, so no need to reject this definition on algebraic grounds. It is not continuous (and it is easy to construct examples where continuity "requests" a different result than this definition), which could be a valid reason to be a bit unhappy about it.

I once wrote a summary of known simple algebraic properties of inverse semigroups and related structures:
https://gentzen.wordpress.com/2014/...nverse-semigroups-and-strongly-regular-rings/

 

[DrClaude]

Isn't point 6 a circular argument? It appears to me to be a simple replacement of $n \rightarrow 0$ with $1/n \rightarrow \infty$.

 

[Mark44]

Um, but since when calling a question "meaningless" is patronizing?

It is a judgment. It also says, that people who ask this, have a meaningless question. This is patronizing.

It is a judgment about the question, but not necessarily about the person asking the question.

"Meaningless" is a word that has no place in education.

Baloney. A person who is in the process of learning may well have ideas that are in conflict with reality, and so could ask a question that is "meaningless."

It belittles people.

Correcting misguided students can be done in a way that does not belittle them, so the blanket statement above is incorrect.

Here is a set with four numbers {0,1,2,3} that allows a division by zero:

The division performed in the second table would not pass judgment in any court of law as a definition of either multiplication or division. For example, one of the table entries has $1 \cdot 2 = 3$. To say the least, this is a very unusual result.

 

[fresh_42]

It is a judgment about the question, but not necessarily about the person asking the question.

No, not in this world. The decoupling of them is artificial and far from reality. Belittling someone's curiosity is ad hominem. Full stop.

Baloney. A person who is in the process of learning may well have ideas that are in conflict with reality, and so could ask a question that is "meaningless."

This only shows a teacher's didactic incompetence in my mind. Yes, old school is based on punishment, I know this. However, I do not find this concept convincing.

 

[fresh_42]

Isn't point 6 a circular argument? It appears to me to be a simple replacement of $n \rightarrow 0$ with $1/n \rightarrow \infty$.

The second equation, yes, but it isn't a conclusion, just a different way to write something. The problem lies in the first "equation".

 

[Mark44]

No, not in this world. The decoupling of them is artificial and far from reality. Belittling someone's curiosity is ad hominem. Full stop.

Maybe belittling is the way you do things, as for example your reply to symbolipoint in post 4:

Sorry, but your comment is meaningless.

However, it is possible to set someone straight in this world without belittling them.

 

[fresh_42]

Maybe belittling is the way you do things, as for example your reply to symbolipoint in post 4:

The art of the argument. Pure defence.

However, it is possible to set someone straight in this world without belittling them.

Not by using judgments as arguments. Patronizing is a declaration of didactical bankruptcy.

 

[PeterDonis]

By what right do you make assumptions like $1\neq 0$? My article deals with that kind of assumption

By making it by fiat, as far as I can see. Your #10 in the article claims that $1 = 0$ is a contradiction. No argument is given, so you're assuming that $1 \neq 0$. So what's the problem with @symbolipoint doing the same?

 

[fresh_42]

By making it by fiat, as far as I can see. Your #10 in the article claims that $1 = 0$ is a contradiction. No argument is given, so you're assuming that $1 \neq 0$. So what's the problem with @symbolipoint doing the same?

That is wrong. I explicitly said

To be honest: It is possible! We could allow it! However, that would come at a price. We had to give up other laws which we did not want to lose; most of all the fact that 0 and 1 are two different numbers.

That was the difference. Number 10 came right behind that text.

 

[PeterDonis]

Here is a set with four numbers $\{0,1,2,3\}$ that allows a division by zero

Ok, then this set of numbers contradicts #10 in your article. Is your article wrong?

 

[Greg Bernhardt]

This reminder to keep discussions productive and constructive while focused on the material. The term "meaningless" is rooted in semantics which is rooted in logic which is rooted in mathematics. Discussing the meaning of something is perfectly fine and not a judgment of the author. If there is disagreement on a claim, it's within the purpose of this community to debate.

 

[PeterDonis]

That is wrong.

No, it isn't. I'm not arguing that you were wrong to say that division by zero could be allowed. I am arguing that you are unfairly criticizing @symbolipoint for making an assumption that you yourself make in #10 of the article. Your #10 assumes without argument that $1 \neq 0$ (if you want to quibble, you could say that it assumes that $1 = 0$ would be "useless", not that it would be an outright contradiction, but I don't see how that changes anything material). So on what grounds are you criticizing @symbolipoint for making the same assumption?

If you're going to call out other people for lack of logical rigor, you need to first make sure your own logic is impeccable.

 

[fresh_42]

The division performed in the second table would not pass judgment in any court of law as a definition of either multiplication or division. For example, one of the table entries has $1 \cdot 2 = 3$. To say the least, this is a very unusual result.

To say the least, you haven't understood that $0$ is the neutral element of multiplication in that example and that it was a potential counterexample. Your rhetoric said "meaningless" and that is not warranted. It cannot be proven. Even "unusual" is borderline since counterexamples are usually unusual.

I'm so tired of your attempts to bully pull me into personal arguments in a language you know better than me.

 

[PeterDonis]

Number 10 came right behind that text.

The text that explicitly says "0 and 1 are two different numbers". In other words, $1 \neq 0$. So, again, on what grounds are you criticizing @symbolipoint for saying the same thing?

 

[fresh_42]

No, it isn't. I'm not arguing that you were wrong to say that division by zero could be allowed. I am arguing that you are unfairly criticizing @symbolipoint for making an assumption that you yourself make in #10 of the article. Your #10 assumes without argument that $1 \neq 0$ (if you want to quibble, you could say that it assumes that $1 = 0$ would be "useless", not that it would be an outright contradiction, but I don't see how that changes anything material). So on what grounds are you criticizing @symbolipoint for making the same assumption?

If you're going to call out other people for lack of logical rigor, you need to first make sure your own logic is impeccable.

The difference is that @symbolipoint said "meaningless" based on hidden assumptions. I criticized a) that "meaningless" is nothing an argument can be based on, and b) that his assumptions are not stated; mine were.

 

[martinbn]

You could use rings with zero divisors, where division cannot be defined also for non-zero elements. It may be insightful.

 

[PeterDonis]

The difference is that @symbolipoint said "meaningless" based on hidden assumptions.

I don't think his assumption that $1 \neq 0$ is any more "hidden" than yours. Both of you are simply relying on the obvious fact that in order to have a useful number system at all, different numerals, such as $0$ and $1$, should refer to different (i.e., unequal) numbers.

In fact @symbolipoint doesn't even mention the number $1$ in his post; the number he picked is $10$. So his assumption is actually that $10 \neq 0$, not that $1 \neq 0$. Where in your article do you explicitly state that you are assuming that $10 \neq 0$? Wouldn't be better to just acknowledge that both of you are assuming that numbers that aren't $0$, um, aren't $0$? And that that assumption is fine because you have to make it to have a useful number system at all?