# What does 'infinitesimally small' mean?

If infinite is the opposite of infinitesimal, and small is the opposite of large, then:

• infinitely large ---------- Means "very large"
• infinitely small --------- Means "very small"
• infinitesimally large ---- Means "very small"
• infinitesimally small --- Means "very large" ???

However my interpretation of the last situation doesn't seem to match how people use the phrase, even though there is sort of a double negative present.

Can the 'negativeness' of the adverb cancel out the 'negativeness' of the adjective, or does the adverb make the adjective stronger?

• Infinitesimally large is a very awkward usage. It's like saying someone is short tall instead of short. – David M Feb 14 '14 at 15:16
• Infinitesimal derives from Leibniz and the birth of calculus: en.wikipedia.org/wiki/Infinitesimal 1/infinity is not just 'very' small, it is arbitrarily small. IOW, pick a size, and your infinitesimal will be smaller than that. – Wayfaring Stranger Feb 14 '14 at 16:15
• infinitesimal is the way mathematicians use the term, the idea is that if you `literally` go all the way to infinitely small, you get zero, whereas with infinitesimal you stop short by a so-called `epsilon`, which is a quantity so small that we cannot conceive it -- as opposed to zero which we can conceive. – PatrickT Feb 14 '14 at 19:25
• infinitesimal: 1/∞ – TecBrat Feb 15 '14 at 4:43
• If the premise is incorrect, then the answer is to explain that the premise is incorrect, and why. An incorrect premise doesn't make the question "off-topic." – phenry Feb 17 '14 at 0:44

## 8 Answers

"Infinitesimal" may be the opposite of "infinite", but it does not indicate any notion of logical negation. So infinitesimally small does not indicate a large object: it just emphasizes the smallness.

"Infinitesimally large" is not a very good phrase, and I would avoid it unless I wanted to play with irony (there is a conflict between the notions of 'infinitesimal' and 'large'). — Edited to add: as `Roger` points out in the comments, one can use "infinitesimally larger" to describe that one thing is larger than another, but only by an extremely small amount; here the fact that it's a comparison between two objects makes it a useable phrase (it's the difference in sizes which is small, not necessarily the 'largeness' of the objects themselves).

"Infinitely small", while perhaps not uncommon, is not quite as graceful as infinitesimally small. But not everyone knows the word 'infinitesimal'; as a computer scientist, I would only use it among physicists, computer scientists, mathematicians, and people whom I believe to have a large vocabulary. However, as with "infinitesimally larger", the phrase "infinitely smaller" is a very good phrase if you want to indicate that the sizes of two things are very different, and you wish to emphasize the smallness of one compared to the other.

• By contrast, you could say "infinitesimally larger" -- using the comparative -- to mean that something is larger than something else, but only by the tiniest bit. – Roger Feb 14 '14 at 14:42
• @Roger: excellent point! I hope you don't mind if I include that as an edit. – Niel de Beaudrap Feb 14 '14 at 14:46
• Don't mind at all. :) – Roger Feb 14 '14 at 14:51
• I don't like `infinitely smaller`, since it implies that there is an infinite difference between the absolute sizes of things (which would in turn imply that one was infinitely large). I think most people would recognize "infinitesimally small" as a way of saying that something is really tiny, even if they don't understand the mathematical details of exactly how tiny. – supercat Feb 14 '14 at 19:43
• @supercat: People have been known to occasionally exaggerate differences. But even so, I would be happy to say that the cardinality of the integers is "infinitely smaller" than the cardinality of the real numbers; and so there is at least one context (however technical) where this phrase is apropos. – Niel de Beaudrap Feb 14 '14 at 23:18

Actually, infinitesimal means "very, very small".

Hence,

infinitesimally small

means very, very smallishly small, which, though it may be a pleonasm, very understandable.

infinitesimally large

means very , very smallishly large.

And that, in no way conceivable to at least my poor brain, makes any sense.

Or indeed, it makes about as much sense as "largely small" or "heavily light".

Since neither "large" nor "small" are absolute positives or negatives, they do not "cancel" each other "out", just like a white and black would not when applied to the same object.

• “The flats in the compound were largely small, though a few bigger ones existed.” ;-) – Janus Bahs Jacquet Feb 15 '14 at 0:37

Niel de Beaudrap's answer has discussed the fact that there is no logical negation going on. A separate issue is that "infinitesimally small" can be applied not just to a number but instead to a thing to be measured, and small versus large size is only one type of measurement. For example, a banker's heart could be infinitesimally small, a socialite's infinitesimally heavy. The former says you'd need a microscope to see it. The latter says that she spent an entire 7 seconds today pondering the plight of Romanian orphans. Things can be infinitesimally important, infinitesimally valuable, and so on.

In a purely mathematical context, there is a longstanding tradition of describing things by their measure. For example, Euclid says that a certain triangle and a certain square are equal, meaning that their areas are equal. If we feel that this blurs distinctions, then we can use a word such as "small" that clarifies that we consider the two shapes equally small in measure, not equal in the sense of being congruent or constituting the same point-sets.

Since one answer has claimed that "infinitesimally small" is a pleonasm, here's an example that wasn't written by ignorant people or for ignorant readers:

Only for an infinitesimally small region of four-dimensional space, i.e., for one in which the potentials g_μν can be considered a constant -- is 'velocity' defined at all.

--Janssen et al., The Genesis of General Relativity, p. 403. This is a good example of the use of "small" to clarify the meaning. Suppose the authors had written simply:

Only for an infinitesimal region is 'velocity' defined at all.

That would have left us wondering what measurement they had in mind for the region that would make its measure infinitesimal.

Googling for examples of "infinitesimally small," I also noticed that the phrase can come up in translation.

Cette difficulté peut être levée en supposant la différence de température entre le corps A et le corps B infiniment petite[...]

--Carnot, Reflexions sur la puissance motrice du feu, p. 27. Some translators render "infiniment petite" as "infinitesimally small," when "infinitely small" would have been more literal, and "infinitesimal" more idiomatic.

An argument against "infinitely small" is that "infinitely" doesn't literally mean "very, very," it means "enendingly." If a road is infinitely long, it means that you can walk on it for as long as you like, but you won't reach the end. That doesn't quite logically make sense when you apply it to something very, very small -- a small thing isn't "unendingly small" (which is what infiniment petite means).

• I dunno, I think "infinitesimal region" is fairly clear, though perhaps only in a sense that technically-minded people can appreciate. (BTW: I think you misspelled "unendingly" in the beginning of the last paragraph?) – David Z Feb 15 '14 at 0:11
• Infinitely or unendingly small means, for example, the 'last' item in the series `1/10` (i.e. `0.1`), `1/100` (i.e. `0.01`), `1/1000`, `1/10000`, etc. – ChrisW Feb 18 '14 at 19:05

'Infinitesimally small', to me, is an unequivocal redundancy, since 'infinitesimal' means 'immeasurably or incalculably small', as in 'an infinitesimal difference'. Source citation: 'Merriam-Webster's Collegiate Dictionary', 11th Edition.

In the sense that flammable and inflammable both mean liable to catch fire, we often see a redundant phrase added so infinitesimal means too small to be significant, as does infinitesimally small. Infinitessimally large would be an oxymoron (i.e. a contradiction in terms). Another example would be 'secret ballot' = 'secret secret vote' or 'black panther' = 'black black leopard'. Most examples are generated by the news media, not out of ignorance but in the belief their readers will not get the point otherwise

• "not our of ignorance, but out of a firm belief in the ignorance of their readers". A very good description of the press :D – oerkelens Feb 14 '14 at 14:48
• In my answer I've given an example that was not written by ignorant people or for ignorant readers. This usage isn't a pleonasm. – Ben Crowell Feb 14 '14 at 17:55

I was taught that infinite described numerical bounds while infinitesimal described a quantity. Where infinite is a significant scope and infinitesimal is an insignificant quantity.

For example, if I manufactured an infinite amount of steel cups on a steel die there would be an infinitesimally small difference between any two adjacent cups (a few atoms). However, as the action of stamping the die causes a small amount of material to rub off both the die and press, comparing two cups thousands or millions of units apart will begin to yield a significant change as the die and press begin to wear out.

Therefore:

• infinitely large means "very large" or describes the upper bound
• infinitely small means "very small" or describes the lower bound
• infinitesimally small means so small as to be insignificant
• infinitesimally large has no logical meaning

Additionally, saying infinitesimally small is in effect, saying a small amount of insignificance. Infinitesimal should be regarded as an absolute Boolean.

This usage varies between different physical sciences as well as math and physics depending on the context used. This is because Infinity as a concept can be both paradoxical and rational. When taken out of the realm of scientific writing or engineering into common usage, the definitions tend to be a composite of the solved equation or problem in its own discipline.

I would say something is infinitesimally small if its size is negligible compared to the scale we're discussing.

A function g(x) is negligible with respect to x if: g(x) <= 1/f(x) where f is a polynomial function

Note that this is defined in relative terms (in respect to x). A polynomial, such as x squared, x cubed, etc, will always dominate, so x is always orders of magnitude larger than g(x).

Adapted (by me) from this pdf on cryptography from Stanford's Dan Boneh

http://theory.stanford.edu/~trevisan/books/crypto.pdf

pages 16-17

"The motivation for the asymptotic denition, is that we take polynomial time to be an upper bound to the amount of steps that any efficient computation can take, and to the "number of events" that can take place. This is why we bound Eve's running time by a polynomial. The motivation for the denition of negligible functions is that if an event happens with negligible probability, then the expected number of experiments that it takes for the event to happen is superpolynomial, so it will 'never' happen."

• The question isn't asking for a definition of "infinitesimal," it's asking whether "infinitesimally small" is a pleonasm. The definition you're proposing for an infinitesimal also isn't consistent with the way the word is used in the mathematical literature, where it was traditionally understood as meaning the kind of "dx" you see in calculus, and was later reinterpreted more rigorously in formalisms like non-standard analysis and smooth infinitesimal analysis. – Ben Crowell Feb 16 '14 at 5:02

infinitely large ---------- Means "very large"

Technically, "infinitely large" means "larger than any finite number"; for example:

• Bigger than 10?
• Yes.
• Bigger than a zillion?
• Yes.
• How about bigger than a googolplex?
• Yes: bigger than any finite number you can name.

Similarly, "infinitessimal" means "smaller than any finite (non-zero) number":

• Smaller than 0.000 ... a hundred billion zillion zeroes ... 0001?
• Yes: smaller than that.

The question isn't asking what infinitesimal means. It's asking whether "infinitesimally small" is a pleonasm.

"Infinitesimally large" doesn't make much sense, and I've never seen or heard it used: it would mean 'not large'.

"Infinitesimally larger" makes some sense: an infinitesimal number is "infinitesimally larger than zero"; but again I've never seen it used.

"Infinitesimally small" is a synonym of "infinitesimal": meaning "very, very small", "extremely small", "vanishingly small", "smaller than anything".

• The question isn't asking what infinitesimal means. It's asking whether "infinitesimally small" is a pleonasm. – Ben Crowell Feb 16 '14 at 4:58