The big argument: m/n or n/m?

17 November 2025

Must be $m/n$, argues Ashleigh Ratcliffe

How is this even a question?

$m/n$ makes so much more sense. It’s natural to say $m$ over $n$, it’s alphabetical order! Don’t ask me ‘why $m/n$?’ Ask the Phoenicians why they decided on the order … m n …!

Just look at it:
$$
\frac{m}{n}
$$
It is aesthetically pleasing. You have two bumps over one bump. When we are young, we are taught that division is splitting into equal parts. For example $6/3$ is dividing $6$ into $3$ equal parts. Now introduce $m$ and $n$: `Split $m$ into $n$ equal parts.’ We can split the letter $m$ into $n$ equal parts: we can make $m$ by putting two $n$s together and removing the gap, and we can split $m$ into $n$s just as easily! But, how many $m$s can you get out of an $n$? Not as simple. I can’t split an $n$ into $m$s.

Furthermore, when picking a number, $n$ is the obvious choice. It is the first letter in the word! So, again, what is division? How many $m$-ultiples of a $n$-umber, ie $m/n$.

Even in practical applications, we would be likely to use $n$ to represent a count. And, what do we normally divide by? A count! Introducing… $n$, the denominator, the $n$-atural choice.

$n/m$, argues Ellen Jolley

Everyone knows the letters you choose for your symbols is not simply an arbitrary decision but in fact loaded with meaning: see the Tom Lehrer song There’s a delta for every epsilon, never the other way around.

Similarly, for fractions we have a clear algorithm by which we assign our roles to our letters, which is as follows:

$n$ is the standard algebraic symbol for a \emph{n}atural number, and so is assigned to the first natural number in our expression: this is the numerator.
$m$ lacks the cultural significance of $n$, and is a mere stand-in for when $n$ has already been occupied. It is therefore assigned to the second natural number in the expression: the denominator.

It is also worth noting that should the fraction consist not of natural numbers but of decimals or, worse, algebraic expressions, it would be wholly inappropriate to use either $m$ or $n$ at any stage. For polynomials, I recommend $p/q$, assigned according to the above algorithm.

We must not allow ourselves to be swayed by trivial alphabetical considerations.

Ashleigh Ratcliffe

Ashleigh is a PhD student and graduate teaching assistant at the University of Leicester. Her main mathematical interests are in number theory. She is passionate about outreach and inclusion in mathematics, volunteers as a STEM ambassador and is a representative for the Piscopia Initiative.