# Is it wrong to tell children that 1/0=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.

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).