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Is −9 a smaller number than −8?

And is −9 a lower number than −8?

What is the difference between lower and smaller here?

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I remain of the opinion that – swanky graphics notwithstanding – the opinion of an octagenarian maths teacher should not be the deciding factor in establishing what if any difference there is between smaller and lower in this very precise context. –  FumbleFingers Jun 23 '11 at 2:34
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@Fumble I agree. The NGram diagrams are telling a different picture here. I will unaccept the answer and when I have more time I will see what answer I finally accept. –  Johannes Schaub - litb Jun 23 '11 at 9:52
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4 Answers

5 is smaller than 3

My math teacher used to say that in this inequality 5 is smaller than 3, but 3 is less than 5.

Joking aside, only less than and greater than are uniformly understood as < and > relation respectively; the other words (e.g. smaller, lower) are often used colloquially to mean less than, whose absolute values are less than, written smaller, etc however their usage are more ambiguous and so should be avoided when writing mathematics.

EDIT:

One could argue just about anything with Google NGram:

enter image description here

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@Lie Ryan: Definitely use greater/less when writing in a maths context, as you say. And computing, where it's common to write and even say LT/GT (even some computer languages themselves support these forms). But outside these specialist areas it's not just 'colloquial' to use smaller - it's overwhelmingly more 'standard'. –  FumbleFingers May 22 '11 at 20:01
    
@Lie Ryan: re your recently-added NGram; OP's question isn't about the relative prevalence of less than, smaller than, and lower than. It asks whether there's any difference between smaller number and lower number, for which one of the biggest differences is in fact their relative prevalence. –  FumbleFingers May 26 '11 at 13:20
    
@FubmleFingers: at least in a mathematician's POV, "lower" and "smaller" has no well-defined meaning; only less and greater has well defined meaning when comparing numbers. I'd personally suggest the OP to ask whoever wrote the sentence what they actually mean when they wrote that. –  Lie Ryan May 26 '11 at 15:06
    
@Lie Ryan: That's just not true - "smaller" is used by mathematicians all the time. And "lower" is too, especially in the phrase lower bound. –  psmears Jun 21 '11 at 8:25
    
Having looked again, I think an NGram for less than is totally irrelevant to OP's question, in that relatively few of the 'hits' will have anything to do with numbers. It remains the case that (for numbers, at least) smaller remains the comparator of choice, though semantically lower is equivalent. –  FumbleFingers Jun 22 '11 at 0:57
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I would generally say that ‘bigger’ refers to magnitude (distance from zero) whereas ‘higher’ refers to value (distance from negative infinity), but in context anything is possible. We don't tend to use ‘big’ or ‘small’ in mathematical contexts very much. If you want to be correct, I would recommend ‘greater’ and ‘less’. In fact, the signs < and > are read ‘is less than’ and ‘is greater than’ respectively.

In http://oi56.tinypic.com/2lieama.jpg 5 is smaller and lower than 3, but still greater.

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I would prefer greater and lesser, but otherwise this is a good answer. –  Bradd Szonye Sep 8 '13 at 20:02
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You can't argue with the whole English-speaking world, and the fact of the matter is they overwhelmingly prefer smaller to lesser or lower.

NGram usage

I think if pressed most people would agree that -1 is a 'bigger' number than -2. But negative numbers don't exactly have real-world correlates, so we all tend to be a bit vague on that one.

Even more vague - for those who know what imaginary numbers are - is the question of whether i is 'smaller' than 2i

LATER: I have no opinion on whether either of OP's examples is more 'correct' than the other. They're both fine to me. I'm simply making the point that of the two specific usages being asked about, on average people prefer to use the former. Here's a somewhat more specific NGram for those who still want to dispute that point... ngram

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thanks for your answer. what does the y-axis represent? –  Johannes Schaub - litb May 22 '11 at 16:26
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@Johannes Schaub - litb: Sorry, I should have said it came from NGrams. The y-axis is "frequency of occurence" as a percentage of the total number of words in the corpus (the millions of books indexed by NGram). So the more leading zeros you see, the rarer the search term is. But NGram always scales the y-axis for you. –  FumbleFingers May 22 '11 at 16:45
    
@FumbleFingers: NGram is deceiving; lesser is non-grammatical or at least very uncommon in mathematics or everyday life as "less" already implies a comparison. See my edit. –  Lie Ryan May 26 '11 at 4:51
    
@Lie Ryan: I see nothing deceptive there. I agreed in my comment against your answer that specialist contexts such as maths and computing normally use greater/less. It's hardly deceptive to point out that outside those specialist contexts people use smaller far more often than lesser. –  FumbleFingers May 26 '11 at 13:25
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@Lie Ryan: "Lesser number" is definitely used. It is not redundant at all - lesser is the comparative form of "little". –  psmears Jun 21 '11 at 8:53
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My gut instinct as a PhD mathematician is that −9 is lower than −8, but not smaller than −8.

There is not a technically correct term without more context being given. If you are talking about the integer numbers (including positive and negative numbers) then I wouuld prefer to say that -9 is lower than -8. If you are talking about the magnitude of the numbers, rather than their place on the integer number line, then -8 is smaller and lower than -9 (although I would prefer to say smaller). As there is not a technically correct answer, these phrases are often used interchangeably in practice. Mathematical notation, such as -9<-8 and |-8|<|-9|, would need to be used to avoid misinterpretation.

"Less than" is also a good alternative phrase. It is the words intended by the < sign.

In summary, I would say that a small number is close to 0 and, in a context where negative numbers make sense, a low number is close to minus infinity. It is best to give more context of what kind of numbers you are talking about.

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This answer can be improved by adding citations: reliable facts and references which show that the answer is correct. "As a PhD mathematician" is a good start. –  MετάEd Sep 19 '12 at 17:38
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@ΜετάEd Chogg speaks from an explicitly mathematical perspective; his degree, provides (for me at least) sufficient authority for his "technical" observations. This must be assumed to be pretty primitive stuff in his field. +1 A clear and subtle answer. –  StoneyB Sep 22 '12 at 4:03
    
Good point. I would never the less like to know a text which discusses this point. –  Chogg Sep 8 '13 at 17:18
    
Good point. I would never the less like to know a text which discusses this point. A very early stage book on number theory might do it, but I don't know which. A reference to this ambiguity is given (again without reference) on en.wikipedia.org/wiki/Small_number. –  Chogg Sep 8 '13 at 17:40
    
I just updated my answer to be clearer. –  Chogg Sep 8 '13 at 17:50
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