# What is the upper bound on “several”?

In this answer on Stack Overflow, the term "several" is used as an indeterminate number, the actual value of which is literally in the quintillions:

Zero is one of several values that can be represented exactly.

To my ear, this is an exceedingly strange use of "several", which led me to believe that the writer was confused.

I realize that trying to truly pin down "several" is probably a hopeless task, but I'm curious if anyone else would use it for such an immensely vast quantity, and if there's regional variation in the usage.

So: How many is "several"? Would anyone else use "several" for "quintillions"?

This question addresses similar issues, but doesn't seem to have the answer I'm interested in (much of the discussion mentions lower bounds for "several", but not upper bounds).

• @Stephen: I changed the title of this question to try to focus on the unique aspect of what you were asking. Hopefully this will help keep this away from being too close to a General Reference question. If you disagree with my edit, please revert it. – MrHen Aug 2 '11 at 1:22
• Compared to all possible numbers, only a small proportion of them are represented exactly on that computer. Zero is one of several that can be represented exactly. But most (indeed, almost all) numbers cannot be represented exactly. – GEdgar Aug 2 '11 at 2:51
• @GEdgar: to my mind "several" is not a relative quantifier; it is absolute. I would not describe the population of Boston as "several people", despite the fact that most (indeed, almost all) of the people on Earth do not live in Boston. Do you actually use "several" in such a relative sense? – Stephen Canon Aug 2 '11 at 3:10
• The only possible answer is: 42. – F'x Aug 2 '11 at 16:14

As I posted in a comment, one of the definitions of several is:

more than two but fewer than many

A further search for a definition of many yields:

Being one of a large indefinite number; numerous

In my own experience in American English, I wouldn't use "several" to mean "quintillions". Many would fit the context better.

• I think you're on sticky ground with "several" even when you get to double figures. By the time you get to 20-30+ it's "quite a few", or at least "dozens". And I really can't imagine a context where "several" could mean "several hundred", because you'd always say the latter. – FumbleFingers Aug 2 '11 at 0:47
• @FumbleFingers: that certainly matches with my expectations; I would hesitate to use several for much more than 12-15. – Stephen Canon Aug 2 '11 at 1:13
• This is my intuition as well. Few -> several -> many. If you'd feel comfortable classifying something as many, you shouldn't feel comfortable classifying it as several. – Jeremy Aug 2 '11 at 5:27
• I'm with you, @Jeremy. I hope my answer didn't imply otherwise. – simchona Aug 2 '11 at 5:41

I consider several to be workable as a comparison between sizes of numbers. If I say, "Several of the planets in the solar system are larger than Earth" it would make sense that somewhere around 3 or 4 planets would qualify.

If I were to say, "Several of the planets in our galaxy are larger than Earth" I would think it fair to consider the number referenced by this several to be larger than the one comparing planets in only our solar system.

This isn't a hard and fast rule that can be applied by a strict measurement — which is exactly why the word several is being used in the first place. The point is to convey the idea of an amount that is akin to "a few" or "some but not most."

So to directly answer your question: No, there is no upper-bound on the amount conveyed by several. More specifically, the upper-bound on several grows as the container one level up grows.

• Fascinating. I don't think I've ever heard "several" used as a relative measure before today. – Stephen Canon Aug 2 '11 at 6:33

Thank you for providing a link the original Stackoverflow question. I think that there is a mathematical aspect to the answer that should be examined. There is also a numerical methods aspects that needs to be considered (that is, how computers represent numbers).

If the question were "how many numbers may be represented exactly using a floating point representation," then the answer would be similar to your (2^64 - 2^53), or something like 1.843 x 10^19. Yes, Big number. I agree, too big for "several."

However, the context of the original question is using cosine. While the range of cosine can be [-infinity, infinity], the domain can only be [-1.0, 1.0]. As you probably know, cosine is only different from [0, 2*pi] and then starts to repeat itself.

So what if we framed the question as these two questions:

``````How many floating point numbers may be represented between 0 and 2*pi?
And how many cosines of those floating point numbers may be represented exactly by a floating point number?
``````

A detailed analysis of this question would probably need to be migrated back to Stackoverflow. I am going to hazard a guess that there are only 5. (Can you think of any cosine values other than 0, .5, 1.0, -.5, and -1.0 that may be exactly represented in floating point? Those correspond to pi/2, pi/3, 0, 2*pi/3, and pi. I may be wrong about only 5 numbers for cosine that may be exactly represented, so the question is not rhetorical.)

The remaining cosines are probably irrational and do not have an exact FP representation.

If the original trigonometry answer meant "zero is one of 5 values that can be represented exactly," then I think the answerer used several correctly. If the actual number is 42, then several is less right. But we are not talking about a quintillion of numbers. The range and the domain are too restricted.

• +1 for what I originally thought was just creative lateral thinking. In fact it now seems logical to me that when talking about how many cosine values can be exactly represented, this should be in the context of cosines of exactly representable starting values. I'm no mathematician, so I can't say whether 5 is a good "guess", but it feels at least plausible to me. Certainly not unreasonable to say there are only several, rather than many. – FumbleFingers Aug 2 '11 at 17:38
• You run into an interesting issue if you adopt with this viewpoint: there is only a single floating-point value which is exactly the cosine of a floating-point value, and it isn't zero -- as I proved in my answer to the question, there is no floating-point number `x` such that `cos(x) == 0`. – Stephen Canon Aug 2 '11 at 18:59
• @Stephen, your link in this question pointed to the accepted answer, which used "Several." I consider your answer (which I didn't see until you referred to it above) to be superior to the accepted answer, in that was thorough and better analyzed. But let's get back to the question you posed here. You have the wherewithal to answer: How many FP numbers are there between 0 and 2*pi? And how many of those FP numbers generate rational cosines? (I searched for this on math.stackexchange.com and may pose the Q's there.) I suspect that the latter is in the "several" range, or possibly 0. – rajah9 Aug 3 '11 at 14:06
• There are approximately 4.6 quintillion double-precsion numbers in the interval [0,2π). Of those, only one has a cosine that is an exact floating-point value: cos(0) = 1. – Stephen Canon Aug 3 '11 at 17:18
• I asked this question on math.stackexchange.com/q/55422/5220 . There was a reference to Lindemann's theorem that agrees completely with your single floating-point value of cos(0) = 1. I think that the respondent to SO question that prompted this question may have used "several" to mean "a handful of numbers (where I don't know how big a 'handful' is)." If he had known that there was exactly 1, he would not have used "several." Consider: how many sequences of "2222" are there in the first billion digits of pi? I might respond with "several," even though the true count is 0 or 42. – rajah9 Aug 9 '11 at 13:34

This is purely subjective, but in terms of differentiating many and several, I would restrict myself to using 'several' when it represents a countable number, whereas many just means a large number.

• Just to be clear, do you mean "countable" in the mathematical sense of the word, or in the "small enough that I could count it if I bothered to" sense of the word? – Stephen Canon Aug 2 '11 at 6:35
• Wouldn't trillions be technically countable? – simchona Aug 2 '11 at 6:38
• @Stephen Canon: "Could I be bothered" sense of the word. – Jordaan Mylonas Aug 3 '11 at 11:48

"Several" implies 4 to 12. I think the word has maintained this definition because it comes in pretty handy. It loses value when it's any more ambiguous than that.

• Do you have a source to back this up? – simchona Aug 2 '11 at 7:01
• How did you come up with 4 to 12?! – Matthew Aug 2 '11 at 7:08
• I thought this page full of deliberation should have "several" more concrete answers. I see the question as an opinion poll :) – Matt Montag Aug 2 '11 at 7:37
• Not an opinion poll, exactly; more of a usage poll. I'm curious how common this relative sense of "several" is, and if there's any regional variation – Stephen Canon Aug 2 '11 at 16:06