Is writing $m=\text{undefined}$ formally correct?

Question

I often find some of my students write $m=\text{undefined}$ in their answer sheets. Is it formally correct? Should I suggest them to write $m$ is undefined instead?

For example the question is

Determine the slope $m$ for a line passing through $(1,2)$ and $(1,1)$. Use the previously explained equation $m=\frac{y_2-y_1}{x_2-x_1}$.

\begin{align} m &= \frac{2-1}{1-1}\\ &= \frac{1}{0}\\ &= \text{undefined} \end{align}

Should I suggest them to write as follows? \begin{align} m &= \frac{2-1}{1-1}\\ &= \frac{1}{0} \end{align} $m$ is undefined.

Answer 1

I would write in words, something like:

The specified line is vertical, and the usual equation for slope does not apply since it would involve dividing by zero. So the slope of this line is undefined; it does not have a slope.

I would avoid writing $\frac{1}{0}$ at any time, and would avoid saying that $m$ is equal to anything at all.

Strictly speaking the question is not well posed, because you are asking them to "determine" something which does not exist. Better wording would be something like

Determine the slope $m$ of the line, if it exists. If the slope is not defined, explain why not.

Answer 2

Actually, the gradient formula applies only when $x_1\neq x_2,$ so, technically, the first equality is incorrect; i.e., $$m\neq\frac{2-1}{1-1}=\frac10.$$ m is (uhm, by definition) undefined.