Math is full of laws: group laws in abstract algebra, the law of sines in trigonometry, and De Morgan’s law in set theory, to name a few. And occasionally, the law is full of math. That was the certainly the case in recent patent dispute at the London Court of Appeals, as covered by The Independent.
Here’s the TLDR: two drug companies were arguing over a patent. Company A has a patent for a solution containing between 1 and 25 percent of a certain compound. Now company B has manufactured a very similar solution, containing .95 percent of the compound in question. But everybody knows that .95<1 so company B is obviously in the clear, right?
Wrong. The judge eventually decided that any number larger than .5 is actually the same as 1, since we can round .5 up to 1, and apparently this judge has no love for non-integers.
My immediate reaction as a mathematician is that this could all have been avoided if Company A had just used interval and set builder notation. A quick recap in case it's been awhile since you've seen interval notation. There are two types of intervals, closed and open. The closed ones have square brackets, like [1,25], and the open ones have round brackets, like (1,25). The first contains all numbers between 1 and 25 including 1 and 25, and the second contains all numbers between 1 and 25 excluding 1 and 25.
The whole point of interval notation (in my mind) is that it takes away any and all possibility for ambiguity. If I say that my solution contains m percent of some compound, where m is in the interval [1,25], I truly mean that the smallest possible value for m is 1 and the largest value is 25. For example, the number 0.9999…9. (that’s just some long string of nines), which by any convention of rounding would round to 1, is still, itself, smaller than 1 and therefore not part of the interval [1,25]. Because of course, significant digits aside, you can round and truncate wherever you please. So to say that anything larger than .5 is really the same as 1 is a bit arbitrary, why not say anything larger than .49 or even .499, you get the idea.
So I guess the upshot is this: when making large business deals, use the most rigorous language possible to describe numbers, because you can’t count on some guy in a powdered wig to do it for you.
Correction: I initially said that .9999… repeating nines forever was less than 1, but as several apt commenters pointed out, if it really goes on forever forever, that’s just 1 — a true but unsettling controversial fact the internet loves to argue about! So let’s say it’s .999…9 for some really long but finite amount of nines, then we’re ok.