Many authors will tell you how to write mathematics clearly and correctly.
But few will tell you how to write it with style and panache, so as to attract the oohs, aahs, and swiveling heads of passers-by.
In that spirit, allow me to channel the men’s fashion guy on Twitter. (Note to the men’s fashion guy on Twitter: please never look at me or my clothes.) Here are a few side-by-side case studies in how to make your mathematics look good:
Sure, some polynomials look fabulous when factored. Also, some athletic 23-year-olds look good in midriff-baring tops. This doesn’t mean we should all try it.
Better to leave something to the imagination; it’s a sign of maturity.
They’re called radicals for a reason, folks. Don’t conform to algebraic conventions. Give the people something to talk about.
I’m not against f-1 notation in general. That would be like opposing casual wear at the office; no point shaking one’s fist at a ship that long ago sailed. (And anyway, why should x-1 be reserved for reciprocals? Isn’t that just a clever and illuminating convention? Any use of negative exponents is already a high-fashion abstraction.)
Anyway, in this particular case, it’s madness to use the dainty superscript when there’s a robust and appealing alternative.
Okay, yes, if you’re actually calculating anything from the limit definition of a derivative, you should go with the more familiar h going to 0 definition.
But be honest. Are you working with the definition of a derivative? Is this the 19th century? Are you a yeoman farmer and/or a Cauchy-era analyst?
No?
Well, then, you’re not bringing this definition to work. You’re using it to make a point: namely, that the derivative is what happens to a slope as the two points draw closer together. And that point is best made with this stylish latter version.
I hesitate to wade into the long-simmering /phi vs. /varphi debates.
But c’mon, folks.
If we can’t agree on such an obvious matter of aesthetics, then I fear we may be approaching the end of our existence as a coherent civilization. Perhaps, in a few decades, the /phi advocates can be resettled on the surface of the moon, where they can build their own sorry little society, beyond the intimidating shadow of our superior fashion sense.
Ah, variance, you minxy concept.
I almost went the other way on this one. After all, is this not the opposite of my advice on the definition of a derivative? Here, am I not promoting easy manipulation over conceptual illumination?
Indeed I am. And that’s because we’re perpetually manipulating variance. The only thing you want to do with that first definition is change speedily into the second one.
I know I’ll ruffle some feathers with this one. Good. Those feathers look silly. They need ruffling.
Now, have I disrupted the beauty of an equation that “unites the five fundamental constants of mathematics”?
Or, have I just revealed that “-1 + 1 = 0” is not as profound a sentiment as some folks think?
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