Always simplify your answer

Einstein is supposed to have said that “Everything should be made as simple as possible, but no simpler”. Mathematics questions often say 'Simplify your answer', or, if not explicitly stated, then this is often assumed, but is it a 'simple' matter to say what 'Simplify' actually means?

A student was calculating the radius of a circle with unit area. They wrote

$$\pi r^2=1$$

$$r^2=\frac{1}{\pi}$$

$$r=\frac{1}{\sqrt{\pi}}$$

$$r=\frac{\sqrt{\pi}}{\pi}$$

When challenged about the final step, they said that they were 'rationalising the denominator'. The teacher said, "You mean 'irrationalising' the denominator?", since $\pi$ is irrational. But the attempt at humour was not really right, because the denominator was irrational before and after this step. However, I have some sympathy with what the student was doing, presumably by analogy with things like

$$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.$$

Dividing by a rational number, like $3$, is much 'nicer' than dividing by an irrational number, like $\sqrt{3}$, and so rationalising denominators feels like a good thing to do, and comes under the heading of 'simplifying your answer'. But with $\sqrt{\pi}$, of course, that is different, because $3$ is rational, whereas $\pi$ is not. But $\pi$ is typographically almost a numeral, and we may sometimes think of it in that way, and, in cases like this, the fact that $\pi$ happens to be irrational feels separate from the square-rooting issue. Somehow, $\frac{\sqrt{\pi}}{\pi}$ does kind of look nicer than $\frac{1}{\sqrt{\pi}}$; perhaps $\sqrt{\pi}$ seems even more irrational than $\pi$? In fact, although it is irrational, we tend to think of $\pi$ as a beautiful, elegant number, whereas a decimal approximation, like $3.14$, although rational, does not seem anywhere near so nice. And I suppose $\sqrt{\pi}$ seems uglier than $\pi$, although $\sqrt{\pi}$ does turn up in some interesting places (e.g., the Gaussian distribution).

This got me thinking about how confusing it can be for students to appreciate what counts as 'simplified', and there is some mathematical aesthetics here along with some perhaps rather arbitrary inconsistencies. Students may first meet the idea that there are multiple ways of representing the same thing when they encounter equivalent fractions. There, writing a fraction in its 'simplest form', or 'lowest terms', means reducing it to the smallest possible integers. There is something intuitive about 'simple' and 'small integers' being the same thing.

But things soon become more complicated (see Foster, 2021). Everyone would agree that $2x$ is simpler than $3x-8x+7x$, say, but is $2(x+1)$ simpler than $2x+2$? Simplifying algebra sometimes seems to mean writing in the most condensed form, "using the least possible amount of ink", but of course $\frac{\sqrt{3}}{3}$ uses more ink, and more/larger numbers, than $\frac{1}{\sqrt{3}}$, since $3>1$. We would probably prefer to write $-1+x$ as $x-1$, and this uses slightly less ink (we save a '$+$' sign, Note 1), but we would not always do this. If we were writing complex numbers in 'real-part, imaginary-part' form, we might prefer $-1+i$ to $i-1$, especially if we are combining (adding, say) several complex numbers, and don't want to mix up the real and imaginary parts.

Similarly, if solving a set of three simultaneous equations in three unknowns, we might prefer to write something like $-x+0y+2z$, so as to keep the unknowns aligned and in order, rather than 'simplifying' this to $2z-x$. Is $x^{-1}+y^{-1}+z^{-1}$ simpler or less simple than its equivalent form, $\frac{x+y+z}{xyz}$? I think it depends on the context. There are lots of situations in which we seem to prefer using more ink. And we would certainly rather write an exact number like $e^{\pi}-\pi$, rather than a very good approximation to this, $20$, which is unarguably 'simpler' and certainly uses less ink (Note 2).

Conversion of units provides another possible example. If you were calculating $1 \text{ cm} + 1.54 \text{ cm}$, to obtain $2.54 \text{ cm}$, would you regard it as 'simplifying' to convert this to $1 \text{ inch}$? What if you happened to end up with an answer of $7.62 \text{ cm}$ or $3.81 \text{ cm}$? Would you spot that they were 'simple' multiples of an inch, and, if so, would you convert to inches? I suppose it would depend on the context, but I don't think I would do this unless there was a good reason.

Ambiguity over 'simplification' continues as the mathematics becomes more complicated. Differentiating $\sin^2 x$ to obtain $2\sin x \cos x$, should the student 'simplify' this to $\sin 2x$? What if they were instead given $\frac{1}{2}\sin^2 x$ to differentiate, and so obtained $\sin x \cos x$, this time without the factor of $2$. Any pressure to go to $\frac{1}{2}\sin 2x$ feels less here. If, instead of using the chain rule on $\sin^2 x$, they had used trigonometric identities to convert to $\frac{1}{2}(1-\cos(2x))$, then they would 'instantly' obtain $\sin 2x$ as the derivative. But, otherwise, I would not expect students to switch $2\sin x \cos x$ into $\sin 2x$. But am I being inconsistent over identities? If they obtained an answer of $\sin^2x+\cos^2x$, then I certainly would expect them to simplify this to $1$!

I think it's pretty difficult to explain what exactly we mean by 'Simplify', and to specify what counts as simplified and what doesn't. When I devise trigonometric identity questions, with the instruction 'Simplify', I try to ensure that there is an equivalent form to the expression that I provide that is uncontroversially by far the shortest and 'simplest'; otherwise, it is hard to say that the question has a right answer. But how do I judge the student who arrives at an equivalent expression to that, if all the statements, including the starting one, are equivalent. Agonising over things like this reminds me of the method Paul Halmos (1985) recounted being taught by one of his students for how to answer any trigonometric identities question:

If you're told to prove that some expression A is equal to a different-looking B, you put A at the top left corner of the page, B at the bottom right, and, using correct but trivial substitutions, keep changing them, working from both ends to the middle. When they meet, stop. If the identity you were given is a true one (it always is), everything on the page is true. To be sure, somewhere near the middle of the page there is a gigantic step, probably as big as the original problem, but very few paper graders will ever find it, or, if they find it, dare to mark you down for it - it is, after all, true! (Halmos, 1985, p. 25).

I think that the idea that every expression has a unique, 'most simplified' form is not really right - and finding this magic form (and knowing when you've got it) is certainly a hard thing to communicate to students. Perhaps we need to be open about the fact that the simplest, most elegant way to leave an answer is to some extent a matter of judgment.

Questions to reflect on

1. How do you explain to your students what is required for 'simplified' answers?

2. Can you think of other examples of ambiguous or confusing situations involving simplification?

Notes

1. I suppose if we wanted to be super-picky about this, we could argue about whether the '$-$' in '$-1$' might be written as a smaller line, like '$\text{-}$', than the '$-$' in '$1-x$'.

2. No one knows 'why' $e^{\pi}$ (Gelfond's constant) is so close to $\pi + 20$. Maybe it doesn't really make sense to ask for 'explanations' of things like this (see https://en.wikipedia.org/wiki/Mathematical_coincidence).

References

Foster, C. (2021). Questions pupils ask: What are 'like terms'? Mathematics in School, 50(4), 20–21. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20What%20are%20'like%20terms'.pdf

Halmos, P. R. (1985). I want to be a mathematician: An automathography. Springer Science & Business Media.