Most of the time, my goal as a tutor is to help students stop hating maths and, if I’m lucky, to grudgingly accept that there are some good bits to it. I’m not here to indoctrinate anyone into becoming a mathematician unwillingly.

And then sometimes, I get asked a question out of the blue that tells me I’ve done that by accident. Discussing the equation of a circle ($x^2 + y^2 = r^2$), my student asked: “What happens if you replace the 2s with 3s?”

It’s hard to think of a more mathematician-like question to ask – it’s a natural generalisation of the form, but in the better part of a decade in full-time tutoring, he was the first person to suggest it. Gold star!

The other curious thing is: my intuition about the answer was completely wrong. Like several of my Twitter friends, when I asked about it later, I said “I reckon it’s a bit like a circle with slightly squarer corners”. But then I went to Desmos and plotted it….

Not what I expected at all: an almost-straight line with a bump in it. For an even power, I was bang on, but for odd ones, the line-with-a-bump was the one.

After a bit of thought, I realised that that shouldn’t have been a surprise: rearranging to get $y = \sqrt{r^3 - x^3}$ gives something that’s pretty close to $y=-x$ as soon as $x$ is significantly bigger than $r$. When $r$ is more significant, you get something more like a quadrant of a circle with a sharper corner – if you change the power to 5, 7, 9 or higher, you’ll see it getting much sharper.

Desmos also gave me another surprise: it didn’t complain when I changed the power to 2.6. I’ve been trained by computers to think that you can’t take fractional powers of negative number, but that’s not quite true – you wouldn’t think twice about working out $(-8)^{\frac{1}{3}}$. In fact, you can take a fractional root of a negative number if and only if the bottom of the fraction is odd. Even denominators don’t work (as square roots of negative numbers are imaginary), and neither do irrational powers (because they can be expressed as the limit of fractions with either odd or even bottoms.)

For fractions with an odd bottom, odd tops give the same sort of behaviour as odd integer powers do (a line with a bump), while even tops give circle-esque behaviour, like even integers do.

There’s so many things to generalise, I’d prefer to leave it to you to play with. What about numbers between 0 and 1? Negative numbers? How big is the bump? Can you rotate it so the line is horizontal?

Have fun!

* Thanks to Aidan for the question