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I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old.

After several more mundane questions he asked his daughter what 1/0 evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually ∞.

I thought the Dad's statement was a little irresponsible. Does that seem like reasonable attitude? I suppose this question is partly about morality.

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$1/0$ is meaningless (and not infinity) in any context they might have been discussing. –  Isaac Solomon May 13 '12 at 9:41
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Whatever the way you think it, $\infty$ is not a number. So both answer are not mutually exclusive. –  Lierre May 13 '12 at 10:07
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The IEEE standard specifies exception handling for division-by-zero, which includes 1/0 among other cases, but the default result of signed infinity (which is a NaN) is not the whole specification. At a minimum a compliant floating point implementation must also set a status flag that reflects the exception. –  hardmath May 13 '12 at 11:31
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How about migrating to parenting.stackexchange.com ? –  Michael Greinecker May 13 '12 at 17:07
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I am with @MichaelGreinecker on this. Morality is off topic at Maths.StackExchange. And The mathematical part of the question is certainly a duplicate of previously asked questions. –  Willie Wong May 14 '12 at 7:47
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12 Answers

up vote 67 down vote accepted

The usual meaning of $a/b=c$ is that $a=b\cdot c$. Since for $b=0$ we have $0\cdot x=0$ for any $x$, there simply isn't any $c$ such that $1=0\cdot c$, unless we throw the properties of arithmetic to the garbage (i.e. adding new elements which do not respect laws like $a(x+y)=ax+ay$).

So "undefined" or "not a number" is the most correct answer possible.

However:

It is sometimes useful to break the laws of arithmetic by adding new elements such as "$\infty$" and even defining $1/0=\infty$. It is very context-dependent and assumes everyone understands what's going on. This is certainly not something to be stated to kids as some general law of Mathematics.

Also:

I believe that the common misconception of "$1/0=\infty$" comes from elementary Calculus, where the following equality holds: $\lim_{x\to 0^+}\frac{1}{x} = \infty$. This cannot be simplified to a statement like $\frac{1}{0}=\infty$ because of two problems:

  1. $\lim_{x\to 0^-}\frac{1}{x} = -\infty$, so the "direction" of the limit matters; moreover, because of this, $\lim_{x\to 0}\frac{1}{x}$ is undefined.
  2. By writing $\lim f(x)=\infty$ we don't really mean that something gets the value "$\infty$" - in Calculus $\infty$ is what we call "potential infinity" - it describes a property of a function (namely, that for every $N>0$ we can find $x_N$ such that $f(x_N)>N$ and $x_N$ is in some specific neighborhood).
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Was there a reason for the downvote? Maybe I can fix whatever is wrong with my answer. –  Gadi A May 13 '12 at 11:43
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I did not downvote, but I can speculate. Given the topic, and especially the question, I don't think there is room for sloppiness and omissions here. e.g. the "usual meaning" you cite says that 0/0=1 and 0/0=2. "$1/0=\infty$" is not 'breaking the laws of arithmetic'. Instead, it is either flat out wrong, or it is a true statement in a different arithmetic system (e.g. projective numbers). Also, before just now, I have never seen anyone use the phrase "potential infinity" in this context. (but I've never seen a coherent description of what the phrase means, so I can't judge if its appropriate) –  Hurkyl May 13 '12 at 12:31
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"By writing $\lim f(x)=\infty$ we don't really mean that something gets the value "∞"" - this is wrong. Somebody who uses this notation means the expended real number system en.wikipedia.org/wiki/Extended_real_number_line. In this system the limit gets the value ∞, otherwise this notation would be plainly wrong. –  Anixx May 13 '12 at 14:21
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This does not have any connection with potential infinity versus actual infinity, this is from a different part of mathematics. The "number" ∞ and its properties can be defined in mathematical systems regardless whether they allow actual infinity. –  Anixx May 13 '12 at 14:29
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I suspect part of the confusion may be because of what IEEE 754 floating point numbers do when you divide by zero: 1/0 is postive infinity, (-1)/0 is negative infinity, and 0/0 is NaN. –  Ken Bloom May 13 '12 at 15:48
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The teacher is right: $1{\color{red}/}0$ is undefined. (if he said 'doesn't have an answer', then he is being somewhat sloppy)

However, the father is right: $1{\color{green}/}0 = \infty$.

But also, the title is right: $1{\color{blue}/}0 = \mathrm{NaN}$.

The problem is that the topic fits into what I believe to be a significant gap in mathematics education: people aren't taught syntax and mathematical grammar at all, so they don't have the ability to make precise statements about what they mean -- or even to know that it's an issue!

(I've added color to emphasize that I mean three different things in those three statements!)

The first version of division is what is taught in elementary school; the teacher is right on that point. $1{\color{red}/}0$ is a syntax error: $(1,0)$ isn't in the domain of ${\color{red}/}$, and so it is illegal to write the expression evaluating ${\color{red}/}$ at $(1,0)$.

The second version, however, is the division of projective numbers. The projective numbers are very useful for algebraic purposes, and even for some analytic purposes: e.g. it can be convenient to have $\tan$ be projective-valued, so that $\tan(\pi/2) = \infty$. I was being a little forgiving when I said the father was right, though -- I find it more likely he was thinking about the extended real numbers (but not knowing that by name!), and simply made a common mistake.

The third version is back to ordinary division, but in a syntax/semantics based on something like partial functions or composition of relations. A rough description is that in so far as functions $\{ * \} \to \mathbb{R}$ correspond to elements of $\mathbb{R}$, the partial function $\{ * \} \to \mathbb{R}$ with empty domain corresponds to $\mathrm{NaN}$.

On this last point, note that to some extent we force students to actually think in terms of this family of concepts with notation like $1 \pm \sqrt{2}$ and $x^3/3 + C$, and questions like "What is the domain of $1/(1-x)$?". But IMO, these notions are somewhat incongruous with what students are actually taught about functions.

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You just had to use colors... :D –  The Chaz 2.0 May 13 '12 at 13:14
    
For non-coloured, I suppose you could use $1 \div 0$ for the red slash, $\frac {1} {0}$ for the blue slash, and $1/0$ for the green slash. –  Joe Z. Feb 13 '13 at 16:45
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The real numbers line can be extended in multiple ways. One way is to add one element, unsigned infinity $\infty$ (this is called projective extended real numbers), another way is to add two elements, negative and positive infinities $-\infty$ and $+\infty$ (this is called affinely extended real numbers). When moving to complex numbers, the things become even more complicated, one can add complex infinity and infinities corresponding to any direction on the complex plane.

Thus there is no universally-accepted way to extend the real line and complex plane with infinities. For example if one only adds positive and negative infinities $-\infty$ and $+\infty$, the expression 1/0 still has no answer.

Claiming that something is equal to $\infty$ may be ambiguous depending on whether the real line is extended with signed or unsigned infinity, so when using $\infty$ without a sign one should specify whether unsigned infinity or positive infinity is meant.

It should be noted that in calculus "$\infty$" is usually used to designate positive infinity, so teaching the kids that this means unsigned infinity may complicate their future experience with calculus.

It should be noted that finding value for 1/0 can be viewed as solving the equation $\frac 1x=0$. In that case both positive and negative infinities fit. So we face a similar problem as with expression $\sqrt{2}$. While there are two real numbers which squared give 2, by convention only the positive one is considered the value of the expression $\sqrt{2}$. But for dividing there is no such convention.

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I think @KMX and @Glougloubarbaki gave good answers (and two others, whose names I forgot). I have more to add, and strong feelings about this, so am answering despite how many others have piled up already!

First, some assumptions
Children were 11-12 years old. In the U.S. public school system, that is equivalent to last year of elementary school (6th grade) to perhaps 7th or even 8th grade, which are middle school/ junior high. Most 6th, 7th and 8th grader's are not learning first year algebra yet. Some basic number theory (primes, integers), as well as fractions, exponents, roots, probably reinforcing how to do long division. Also, very basic geometry, like polygon names and area of quadrilaterals and circles. And units of measurement, Celsius to English conversion, scientific notation.

Given that context, with no knowledge of algebra or calculus, only simple number theory, it seems reasonable that the teacher described 1/0 as "undefined". "Not a number" is equally acceptable.

Big picture
School teachers have degrees in education, with specialty training by subject matter. Instruction is not done in isolation, but as part of a coherent curriculum. If the teacher defined it in this particular way, it is likely part of a longer term mathematics instruction plan which will redefine the meaning of 1/0 in more appropriate ways when the time is right, i.e. when the accompanying content is at a sufficiently advanced level so that it will make intuitive sense.

Unintended consequences
The father needs to be very careful here, because he might undermine the teacher's credibility to his children. Worse yet, his children will potentially give incorrect answers on tests, based on the father's comments. This puts the children in an awkward situation which they should not be subject to at this point in their lives. If the father has concerns about the quality of instruction, he should take it up with the teacher directly, not in the way he did. At least, not with children who are only 11 or 12 years of age.

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@J.M. mentioned that children should know that adults can sometimes be wrong, too. I agree with J.M. but it is very context-dependent, and usually with different sorts of subject matter than math, or much more blatant "wrong-ness* in a math answer. –  Feral Oink May 13 '12 at 16:22
    
Celcius to English? What English temperature scales are there? Kelvin would be the closest (although he was Irish by birth and arguably Scottish by adopted), I suppose, but I don't think they teach absolute temperature at that age. –  Peter Taylor May 14 '12 at 9:13
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If this were Europe, many of the assumptions would be untrue and we wouldn't be arguing now... –  hexafraction May 14 '12 at 20:18
    
@PeterTaylor That's Celsius (not Celcius) to English... as in degrees Fahrenheit. Here, have a look at this, Why Fahrenheit is better than Celsius isomorphismes.tumblr.com/post/3767526267/… Do note that I agree with you, about the superiority of Kelvin in general ;o) though! –  Feral Oink May 14 '12 at 23:29
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I don't get it. What's "English" about a temperature scale invented by a German who was born in Poland and lived in the Netherlands? –  Peter Taylor May 15 '12 at 7:58
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This has been treated extensively both in this question and in others. But I think the clearest formulation possible is this : for me the most correct answer is "NaN", Not a Number.

It is not wrong to say that $1/0 = \infty$ from a topological point of view (be it an Alexandrov compactification, or a projective space), and it's actually very useful and natural in some contexts.

However, as pointed out by everyone, it cannot be made coherent with arithmetic laws, addition and multiplication. So it's "not a number".

That said, for pedagogical purposes I think it's best to not go into these things and just illustrate the arithmetic problems arising when defining division by 0 (and maybe say something like "one might define 1/0 in some contexts, but we will not because of these problems"). Indeed arithmetic notions are much more intuitive than abstract, topological constructions at this stage.

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It can be made coherent with arithmetic laws -- it's just that those laws do not have identical form to the laws for real numbers. –  Hurkyl May 13 '12 at 14:22
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Well I think people are taking your quest as a math rather than the demand you have made. I see you actually have asked two questions together:

Q#1) What is the actual answer for 1/0 to tell a 11/12 year oldie?

Q#2) Was father's attitude towards answering his daughter wrong?

So I would rather choose to answer something like this:

A#1:

Since the question is very simple so is the answer (Gadi's way???) . But the main thing involved is that how to tell such answers to your little ones who are trying to understand things which become important in their upcoming lives. I dont know what was the actual answer from the "teacher" but whatever teacher answered I consider it correct because s/he answered the kid since s/he knew that it was better to explain them that it has nothing. The girl must have skipped all explanation the the teacher would have made at the time when the question was answered but she told her dad what she understood from that.

I guess, teachers and parents should answer their students and kids by keeping the age in mind.

A#2:

I consider the "teacher taught it wrong" attitude from father was NOT good. Instead he could have explained it politely with a better way so that the daughter would not get a feeling that the teacher dont misguide but there are better ways to explains such things. And parents are the next best choice to get best answers.

Being parent, I am going through this phase. Its tough, its difficult how to teach your kids everything when there are many things having different answers on different ages.

We know Newton's laws are not 100% perfect, but they are taught. Why dont we explain theory of relativity from the beginning instead? Because they are more complex than understanding Newton's laws.

Anyway, I think instead of answering (Gadi's) mathematically (which a 11/12 age group would not understand) we should explain it in words with as many examples as one could reach a better understanding level depending on their age group so that in coming years they understand it more as they advance.

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"I consider the 'teacher taught it wrong' attitude from father was NOT good." - I don't know about you, but somehow I think children should know that adults can sometimes be wrong, too... –  J. M. May 13 '12 at 15:29
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Adults could be wrong. But I am talking about the "attitude". The point here is that, if teach was wrong, then Parents must contact the teacher to know why they told a wrong thing instead telling them on the spot that teacher was wrong. Teacher is the person whom children meet every day for few hours a day. If they know that this person is NOT truthful then they wont cooperate with them and result in severe consequences that are NOT considered at the time of taking that. So my point is still valid regarding the attitude. there must be some way to work on it and then tell the right answer. –  KMX May 13 '12 at 19:06
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I'm assuming "NaN" means "not a number". Certainly $\infty$ is not a number. There are indeed various different sorts of infinite numbers (cardinalities, non-standard reals, surreals, ordinals, the "infinities" of generalized functions like Dirac's delta and its derivatives, . . .), but $\infty$ itself is not a number.

In some contexts $1/0=\infty$ makes sense. In thinking about projective geometry, trigonometric functions, or rational functions, it makes more sense to have just one $\infty$ at both ends of the real line that $+\infty$ and $-\infty$ as separate entities. But in other contexts, one should distinguish between the latter two things.

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... and any comments about IEEE standards should be in another place. (That Dad made no mention of them.) –  GEdgar May 13 '12 at 23:27
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You ask:

Is it wrong to tell children that 1/0 = NaN is incorrect, and should be $\infty$.

(My) answer:

Absolutely. When we talk about multiplication and division with numbers, I think that it important to get things right. If people would only grow up learning that $\infty$ is not a (real) number, then I think that it would relieve a lot of confusion about how math treats infinity. And it is definitely not right to say that the statement "$1 / 0$ is not defined" is incorrect. That is what is taught in all basic calculus classes. Why would this be incorrect?

Now, if the question is whether one can ever come in a situation where it is appropriate to write $\frac{1}{0} = \infty$, then the answer is definitely yes. This is illustrated in some of the other answers to your questions. But to say that $1$ divided by $0$ is infinity and then just stop, then I would agree that this is wrong (I might even go so far as to call it irresponsible).

When people ask me why we can't divide by $0$ and if they don't want to "leave it", then I would simply start talking about groups and rings and fields as sets with the operations. When you do that carefully you clearly see why it doesn't make sense.

The wonderful thing about math is that we never have to answer why something isn't defined, the burden of proof is in the hands of the person who claims that something is true.

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I'm sorry but I think there is a little misconception here: 1/0 is not infinity, never was, never will be. This would imply that $$0\cdot\infty = 1$$ which is absolutely wrong.

The correct mathematical formula is $$\lim_{x \to 0+}(1/x) = +\infty$$ The teacher is therefor right, and it's actually the parent that was wrong. 1/0 is undefined, and so is $$0*\infty$$ even in term of limits.

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Actually $\lim 1/0$ is completely undefined. –  Asaf Karagila May 13 '12 at 23:41
    
It's actually $$\displaystyle\lim_{x\to 0^+}(1/x)=+\infty$$ since dividing a positive number by another positive number returns also a positive number :) –  Samy Arous May 13 '12 at 23:42
    
@Asaf Karagila, yes I made a mistake there, it's the right limit of 1/0 which is +infinity, where the left limit is -infinity :) This is kinda of old, haven't done that in years –  Samy Arous May 13 '12 at 23:44
    
No, $\lim 1/0$ has an undefined term within the limit, when we write $\lim_{x\to0^+}1/x$ we mean that $x$ is smaller and smaller and approaches $0$, but it is never really $0$ because the terms inside the limit need to be well-defined. As you argue, in the context of real analysis $1/0$ is not really defined, but we can treat it as a limit. This means that you cannot really write it inside a limit, it makes no sense. –  Asaf Karagila May 13 '12 at 23:47
    
Oh, you mean the shortcut I took? well that's what was confusing everyone else, so I put it. I know it's mathematically wrong. removed it. It's a language shortcut that tends to become a rule overtime :p –  Samy Arous May 13 '12 at 23:50
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Claiming $\frac 10=\infty$ is wrong (if $\infty$ means just positive infinity, as usual). Look yourself, $\frac10$ should be equal to $-\frac10$ because $0=-0$, but $\infty\ne-\infty$.

A more correct answer to your question would be that $\frac10=\tilde{\infty}$, where $\tilde{\infty}$ is complex infinity. It can be expressed in terms of positive infinity though:

$$\frac10=\tilde{\infty}=\infty(-1)^\infty=\infty e^{i\infty}$$

This answer will be the correct one given we have extended complex numbers with $\infty$.

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Computers are so ubiquitous and important, I'm not sure that one shouldn't use the IEEE floating point notation as the "default" to teach in middle school (where I assume you'd be talking about some representation of real numbers):

scala> val x = 1.0/0.0
x: Double = Infinity

The mathematician in me cringes, but if you're teaching things in order for them to be maximally useful, I think there's a pretty good argument for this. However, saying that

scala> val y = 1/0
java.lang.ArithmeticException: / by zero

is the wrong answer is itself wronger than giving this as an answer; as this question and associated answers demonstrate, the actual situation is more involved than any simple declaration of fact.

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In IEEE floating point, $a+(b+c)$ and $(a+b)+c$ are not necessarily equal (try e.g. 1.0, 1.0e20 and -1.0e20). Are you sure we should teach this in middle school? –  Robert Israel May 13 '12 at 20:39
    
@RobertIsrael - No, but when there are multiple conventions, I would tend to choose the one that aligns with IEEE floating point over the more common mathematical formulations if I was concerned about general utility that most efficiently teaching mathematics. –  Rex Kerr May 13 '12 at 22:36
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No! IEEE FP is a representation of underlying mathematical concepts. Students must have at least some idea of these concepts before the idea of IEEE Infinity makes sense. And then, if you're teaching things in order for them to be maximally useful you would certainly steer clear of IEEE floating point standards! –  Bennett McElwee May 13 '12 at 23:38
    
@Rex: I don't understand your choice. To efficiently teach mathematics, one should teach things in a way that is easy to understand, and unlikely to produce difficult-to-correct misconceptions. IEEE arithmetic would fail on both counts. e.g. while $\infty$ is often a useful approximation to very large numbers, your plan would reinforce the misconception that that is all $\infty$ is -- shorthand for "a large, unspecified real number". –  Hurkyl May 14 '12 at 1:20
    
@Hurkyl - I wouldn't teach that e.g. 10^4000 is equal to infinity--that would be absurd. Rather, I'd use the IEEE convention for 1.0/0.0. Any time IEEE FP doesn't align with any reasonable mathematical formulation, I would avoid it. –  Rex Kerr May 14 '12 at 5:11
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As @hardmath indicated there is indeed a floating point representation for infinity.

added:

To address the non-math side of this question. IMHO, it was bad form for the father to contradict authority to his child and sets a bad example without setting a better attitude such as "there may be another answer". I think... it's infinity, what do you think? Here he shows his child he can be vulnerable have a different opinion, even if it technically wrong, and that can be OK if you consider the possibility that someone could be wrong and find a way to verify supposition of facts and confirm or deny them in a rational non-confrontational way. That also demonstrates tolerance and rational objectivity to determine truth by asking if she knows the reason in a non-confrontational way and learn to discuss differences of opinions without fear of criticism as overheard in this example.

" If 'a' is zero and the payload is zero, then it represents infinity." Is the technical reason why the teacher was correct.

However for teaching purposes such as the concept of the distance of the universe, the Big Bang and the weight of an atom, it may be useful to consider teaching in "metaphors" so that approaching a zero numerator or denominator, which may become a Singularity when you approach infinity with a real number.

It is also useful to define the order of rules when the paradox of;

  1. "anything multiplied by zero is zero" which contradicts...
  2. "anything divided by zero is infinity"

    • ...Actually (2) really bad boo boo ;) (2.) is not true, as it is only a conditionally true if the numerator is also zero, otherwise, it is NaN.

So explain to the child that 0/0 is a special case with religion. ;) or something of

greater than human intelligence.

Fun Fact: Thanks to John Wallis who introduced the infinity symbol to mathematical literature with his works on the collisions of heavenly bodies at university in the women's dorm. Consider how fortunate we are that we dont have to call it a Wallis instead of ∞ infinity. However John eventually got his immortality with an asteroid in 2000 and his name assigned to it.

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