*Zero is a strange number - learners sometimes even doubt if it is a number... *

In my articles in *Mathematics in School*, I often address 'Questions pupils ask' (I even have a book of that name, Foster, 2017). In this blogpost, I'm going to address a question that came from a learner and that relates to zero and units:

*“Is there any point writing ‘metres’ after zero, as it will just be equal to zero?”*

In other words, is ‘zero metres’ just exactly the same as the number ‘zero’, *without* any units? (“Zero m is just zero!”)

Teachers often stress the idea that ‘$1$ metre’ is *not at all* the same as the number $1$. We know that $1$ is a dimensionless number, whereas $1$ m is a length. So, we would never dream of writing:

$1$ m $= 1$,

as this would be a dimensional catastrophe – as bad as saying something like ‘$1$ m $= 1$ kg’.

But is zero a different matter? If pupils think of ‘$1$ m’ as ‘$1$, multiplied by a metre’, as indeed it looks symbolically, then ‘$0$ m’ is ‘$0$, multiplied by a metre’, which is surely just zero, since zero multiplied by *anything* is just zero. Sometimes, when students are simplifying algebraic expressions (e.g., collecting like terms), they might simplify something like $$8a - 3m + a + m - 2a + 2m$$ by writing $7a + 0m$, but the teacher would probably say that there is no need to write the $+$ $0m$. Is it any more relevant to mention that "we haven't got any $m$'s" as it is to mention the absence of other quantities ($7a + 0m + 0r + 0mr + 0m^3$, etc.). In a particular context it might be worth being explicit about the zero $m$s (I sometimes find this useful when solving simultaneous equations by elimination, for instance, so as to keep everything nicely lined up in columns), but in general we wouldn't regard it as worth mentioning, so we would simplify to just $7a$. So, this feeling that $0m$ is just 'nothing' perhaps suggests that ‘zero metres’ should also be simplified to ‘zero’; i.e., nothing at all. Having no metres just means that you have nothing at all. Indeed, perhaps, on all the different dimensional scales of different quantities, the zeros coincide:

$0$ m $= 0$ kg $= 0$ °C $= 0$.

Clearly, it it not necessary to know any conversion factors to know that zero in *any* length unit, say, will be zero in any other length unit:

$0$ cm $= 0$ m $= 0$ inches $= 0$ furlongs $= ... $

But can it make any sense to write two zero measures on *different* dimensional scales as though they are equal, like $0$ m $= 0$ kg? This *really* looks wrong.

If it is right, does this mean that if a question says, “Give the units in your answer” that that is a subtle, unintended clue that the answer *won’t be zero*? Should a student *not* be penalised in an exam for omitting the units for a question where the answer is zero, such as this one?

*The temperature on Tuesday is 2°C.On Wednesday, it is 3°C warmer than it was on Tuesday. On Thursday, it is 8°C colder than it was on Wednesday. On Friday, it is 3°C warmer than it was on Thursday. What temperature is it on Friday? Give the units in your answer. *

Surely not, as here the units are *very* necessary, since an answer of ‘$0$’ for temperature could be $0$ °F or $0$ K, since the zeroes of temperature certainly *don’t* coincide with one another, let alone with the zero for kilograms. Maybe ‘all zeroes are equal’ only applies to quantities with a ‘true zero’, as opposed to those with ‘arbitrary zeroes’, like temperature.

But can this be right? Zero oranges and zero apples do not necessarily represent the same state of affairs: just because I’ve run out of oranges, it doesn’t necessarily follow that I must have run out of apples too.

One possible response is to say that, with measurement, 'zero never really means zero'. So, a statement like '$0$ m' really means 'zero metres, to some degree of accuracy', so this represents not *a point* but *an* *interval* on the metres number line, such as $-0.5 \leq $ length $ \lt 0.5$. (Although can length be negative? Possibly, say if it's a difference between two other lengths.) And now clearly $-0.5 \leq $ length $ \lt 0.5$ is different from, say, $-0.5 \leq $ mass $ \lt 0.5$, so the problem goes away. But in pure mathematics we *can* have an exact zero that is not rounded to any degree of accuracy.

Questions like this can make zero seem like ‘not a number’, or at least not like any other number, in ways that students may find disturbing. I remember a Year 10 (age 14-15) student remaining behind after a mathematics lesson to ask me a question she didn't want others to hear. I assumed it would be something personal, but it turned out she was embarrassed to ask ("This is probably a really silly question", etc.) the question: "I have always wondered, but is zero actually a number?"

It might be tempting to dismiss such questions. Of course zero is a number - it's on the number line. What else would we put half way between $1$ and $-1$? Would we want the number line to have a tiny, infinitesimally small gap 'at zero'? And if zero isn't a number, what else would it be? But there are instances where zero does indeed seem to be in a class of itself. One example is that it's neither positive nor negative. There are three kinds of real number: positive ones, negative ones, and then a class all of its own for the single number zero. I sometimes catch myself saying that the topic of 'directed numbers' refers to positive and negative numbers, or I might even call the topic 'positive and negative numbers', forgetting to say 'and zero', which is 'central' to the whole thing. It is interesting to contrast the question "Is zero a number?" with the (possibly related) question "Is infinity a number?", which I suspect different mathematics teachers would answer in different ways.

A question like the one I've discussed in this blogpost may feel very abstract, and why should we worry about such an unusual question? I am not claiming that lots of students are asking this particular question all the time. But uncertainty over things like this perhaps contributes to students' feelings that mathematics doesn't make any sense. Sometimes, I suspect, students who are confused or stuck in mathematics, and who we might regard as 'having difficulties', are in that position because they have thought *further* (rather than less) than their peers.

### Questions to reflect on

1. Do you think that $0$ cm is 'the same thing' as $0$ kg? Would you dare to put an equals sign between them? Why/ why not?

2. Would you mark a zero answer wrong for not having the units?

3. What other issues do you see students having with zero?

### References

Foster, C. (2017). *Questions Pupils Ask*. Mathematical Association.

"Sometimes, I suspect, students who are confused or stuck in mathematics, and who we might regard as 'having difficulties', are in that position because they have thought further (rather than less) than their peers." Further than their teachers, in many cases. For one, distances aren't mathematical entities; 1m is outside of mathematics, so this conversation doesn't really belong in a mathematics class. The mathematical part of this problem might be to determine how many multiples (a number) of some standard quantity (in units such as metres, litres, etc.) measure a given quantity (length, volume, mass, etc.). That so-called 'mathematics teachers' can't make this distinction reflects on the levels of mathematics understanding within the profession. The argument that 0m is 0 therefore even more meaningless than comparing apples and oranges. The other issue is confounding measure (cardinality) and order. Positions (order) can be relative in ways that sizes can't. If 0 is used to represent/label some point (your origin), you're using numbers to refer to positions relative to the origin with respect to some other point (which defines your unit). Negative numbers make perfect sense in the context of positions, but negative lengths don't make sense. Temperature conversions involve both two distinct origins and two distinct units (centigrade and fahrenheit), that's why they shouldn't both be labelled 0; they may be labelled 0C or 0F to indicate their respective origins. The label 2C represents a point two multiples of the C-unit in a certain direction, just as -2F represents the point for which the origin, 0F, is two F-units to the right of that point. All too often we blame students for the failings of their teachers.

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