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.

I thought 0.9999… = 1

https://en.wikipedia.org/wiki/0.999…

Actually, 0.99999… (nines forever) is equal to one, not less than one.

https://www.math.hmc.edu/funfacts/ffiles/10012.5.shtml

Yes! Indeed it is, I guess we want nines for a really really long time, but just short of forever. That’s one of those weird things in math.

(I may be mistaken here, but:)

If we take the hyperreals of non-standard analysis, introduced by Abraham Robinson in the 1960s, then 0.999… is the “halo” of 1, and 1 the “shadow” of 0.999… ; This is a small, but important distinction, since it means (under the hyperreals) 0.999…< 1, but we are still afforded the rigor and power of standard analysis.

This means the judge’s (or blog author's) intuition that 0.999… < 1 can be bolstered by some clever court attorney citing non-standard analysis. To be still yet more rigorous, we may need what number system we are concerned with, and whether or not the halo of the interval should be excluded if pertinent.

[KARIN USADI KATZ AND MIKHAIL G. KATZ, A STRICT NON-STANDARD INEQUALITY .999 . . . < 1] http://arxiv.org/pdf/0811.0164.pdf

Maybe the judge isn’t so dumb. Maybe the production process might have an average of 0.95% in solution but a standard deviation that implies a significant proportion of production will exceed 1%.

Unfortunately I doubt that interval notation would have helped in this situation by itself. The root of the problem as you have presented it is ignorance on the part of the judge wrt approximation. Interval notation, while precise, would probably not be sufficient to keep this judge from making the same mistake. Company A also would have needed to add information about the accuracy of determining the percentage of substances in a solution as well as the clinical or projected efficacy of solutions at various levels including levels below the minimum 1%.

As an attorney, I can tell you that judges often interpret language differently than the authors intended. Sometimes they come up with a meaning different than anyone who previously read it.

As one of those ‘students’ who has objected

to the equality, and as one who has considered the question for decades, I have found something with sits well in my mind: .9999…. is _asymptotic_ to 1.

The asymptote in this case would be a Cauchy convergent sequence in this complete metric space of real numbers. By definition all such sequences converge to something in R. Since the metric space is dense (ie sequences converge to a real number) this number is unique and the same value =1. In non standard reals the same sequence could converge to some value infinitesimally away from 1 (since non standard reals include infinitesimals but we cannot say that for standard reals which do not include infinitesimals).

Infinite nines is by definition as 0.999… which in the reals actually means the concept of limits/convergence and not literally infinite nines since standard reals do no include infinity as a value for n nines (only limit of finite n nines).