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.
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:


$\lim_{x\to 0^-}\frac{1}{x} = -\infty$, so the "direction" of the limit matters; moreover, because of this, $\lim_{x\to 0}\frac{1}{x}$ is undefined.
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 $N&gt;0$ we can find $x_N$ such that $f(x_N)&gt;N$ and $x_N$ is in some specific neighborhood).
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;


"anything multiplied by zero is zero" which contradicts...
"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.