Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$?

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.

Solution 1:

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).

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.

Solution 5:

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.