# Is zero a finite number? **URL:** \ **Category:** Great Debates\ **Created:** [August 29, 2005, 4:10am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063 "2005-08-29T04:10:46Z")\ **Posts on this page:** 16\ **Page:** 1
**Author:** ![Mariemarie](https://avatars.discourse-cdn.com/v4/letter/m/58f4c7/32.png) [@Mariemarie](https://boards.straightdope.com/u/Mariemarie)\ **Post date:** [August 29, 2005, 4:10am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/1 "2005-08-29T04:10:46Z")
So, my friend was talking about how an electronic (musical) keyboard works, saying “the processor scans the switch states, so even if you press F and G at exactly the same time, there is a finite interval between the moment F plays and the moment G starts playing, but it’s so small that your human ear and brain do not perceive the difference.” I corrected him: “you mean a _non-zero_ finite interval”. He said that would be redundant, that zero is not a finite number. I disagree. Is to say that zero is not finite the same as saying zero is infinite, i.e. zero equals infinity, or maybe that zero is an infinitesimal? Does the answer change depending on whether you’re talking mathematics or physics? Is zero a valid finite number for apples but not for time? What say ye, O Teeming Millions? Is zero finite? ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [August 29, 2005, 5:37am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/2 "2005-08-29T05:37:19Z")
I’m not a number theorist but zero certainly behaves like a finite number. It is a definite place on the real number line and on the complex number plane. It is also subject to the ordinary mathematical operations of addition and subtraction since the definition of zero is that there is a number, b, such that a + b = a. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [August 29, 2005, 5:39am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/3 "2005-08-29T05:39:18Z")
I was about to chime in and say that as a mathematician I’d never heard of zero not being finite, but then out of curiousity I went to [dictionary.com](http://dictionary.com). Lo and behold: > [@](#): > > fi·nite > adj. > […] > 2 Mathematics. > > a. Being neither infinite nor infinitesimal. > b. _Having a positive or negative numerical value; not zero._ > c. Possible to reach or exceed by counting. Used of a number. > d. Having a limited number of elements. Used of a set. So it appears your friend may have the dictionary on his side. I guess 2(a) is the key thing here. Personally, I wouldn’t have thought “finite” and “infinitesimal” were mutually exclusive, but it’s not something I’ve ever really thought about. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [August 29, 2005, 6:19am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/4 "2005-08-29T06:19:04Z")
From [http://en.wikipedia.org/wiki/Infinitesimal:](http://en.wikipedia.org/wiki/Infinitesimal:) > [@](#): > > In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. A number x ≠ 0 is an infinitesimal iff every sum |x| + … + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number. Infinitesimals, obviously, are not real numbers, so “operations” on them aren’t familiar. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [August 29, 2005, 1:52pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/5 "2005-08-29T13:52:02Z")
Under the standard definition of “finite,” zero certainly is a finite number. However, expressions like “a finite amount of time,” meaning “a nonzero amount of time” are semi-common. It’s used to say that, while the amount of something might be small, it’s more than zero. The use of the expression comes from the infinitesimal notion in calculus. It would take a fair amount of explanation, but when calculus was first discovered, the proofs for it used a mathematical notion called “infinitesimals.” The mathematics for working with infinitesimals was pretty vague, so in the nineteenth century a completely different way of doing proofs in calculus was invented using deltas and epsilons, and that’s how calculus is presently taught. Actually, it is possible to make the notion of infinitesimals mathematically precise, and this was done by Abraham Robinson in the mid-twentieth century. In any case, you should just take the phrase “a finite amount of X” as being an idiomatic phrase in which the word “finite” means something different from its usual meaning. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 29, 2005, 2:00pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/6 "2005-08-29T14:00:55Z")
I would argue that nonzero finite interval is redundant not because nonzero equals finite, but because nonzero equals interval. It’s an amount of time between two set points; if the amount of time is zero, there is no time between these points. On the other hand, the “finite” part was redundant, too, inasmuch as you can’t have an infinite intervale between two notes in a piece of music that has already been played. Daniel ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 29, 2005, 6:26pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/7 "2005-08-29T18:26:22Z")
In common usage, zero may or may not be finite, but in technical usage, it most definitely is. > [@Left Hand of Dorkness](#): > > I would argue that nonzero finite interval is redundant not because nonzero equals finite, but because nonzero equals interval. An interval of length 0 is still length 0. > [@BrainGlutton](#): > > From [http://en.wikipedia.org/wiki/Infinitesimal:](http://en.wikipedia.org/wiki/Infinitesimal:) No such thing in the standard real numbers. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 29, 2005, 9:17pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/8 "2005-08-29T21:17:07Z")
> [@ultrafilter](#): > > An interval of length 0 is still length 0. In what sense is it an interval, then? Daniel ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 29, 2005, 9:20pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/9 "2005-08-29T21:20:32Z")
If I may clarify, the friend said that he’d not used the word “nonzero” because that would have been redundant. However, he said: > [@](#): > > even if you press F and G at exactly the same time, there is a finite interval between the moment F plays and the moment G starts playing, but it’s so small that your human ear and brain do not perceive the difference. Given that context, were the interval equal to zero, the sentence would be absolutely bizarre; similarly, were the interval infinite, the sentence would be absolutely bizarre. Therefore, the word “finite” is redundant in the sentence. Normally, this isn’t a big deal (language redundant: help understand!) but if he’s gonna bring up the issue, he oughtta be consistent, is all. Daniel ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 29, 2005, 9:24pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/10 "2005-08-29T21:24:48Z")
> [@Left Hand of Dorkness](#): > > In what sense is it an interval, then? Of course, this in the technical sense, and not common usage. An open interval [a, b] is the set of all points x with a \< x \< b. If a = b, the interval is a single point. ---
**Author:** ![Hypnagogic\_Jerk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hypnagogic_jerk/32/4252_2.png) [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)\ **Post date:** [August 29, 2005, 9:50pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/11 "2005-08-29T21:50:48Z")
> [@ultrafilter](#): > > Of course, this in the technical sense, and not common usage. An **open** interval [a, b] is the set of all points x with a \< x \< b. If a = b, the interval is a single point. Do you mean a **closed** interval? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 29, 2005, 10:00pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/12 "2005-08-29T22:00:38Z")
Yes. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 30, 2005, 12:51am UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/13 "2005-08-30T00:51:54Z")
> [@ultrafilter](#): > > Of course, this in the technical sense, and not common usage. An open interval [a, b] is the set of all points x with a \< x \< b. If a = b, the interval is a single point. Makes sense. This technical definition doesn’t make sense within the quote, though :). Daniel ---
**Author:** ![Mariemarie](https://avatars.discourse-cdn.com/v4/letter/m/58f4c7/32.png) [@Mariemarie](https://boards.straightdope.com/u/Mariemarie)\ **Post date:** [August 31, 2005, 2:55pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/14 "2005-08-31T14:55:52Z")
Thank you for your replies. I hadn’t considered that the word finite might have different meanings in common usage and in technical usage. I googled the phrase “nonzero finite”, and it returned some IEEE documents discussing floating point computation, among others, so those people consider zero to be a finite number. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [August 31, 2005, 8:04pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/15 "2005-08-31T20:04:28Z")
To summarize: 1. You shouldn’t correct a friend on something like that. 2. If you’re gonna correct him, the word he should’ve used instead of “finite” is “itty-bitty.” Daniel ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [August 31, 2005, 9:06pm UTC](https://boards.straightdope.com/t/is-zero-a-finite-number/319063/16 "2005-08-31T21:06:21Z")
In physics, zero is not considered finite (but neither is it infinite), as a matter of convenience. My understanding is that it is because you can’t actually measure a zero interval, even in principle. The best you can do is say something is less than a given value (an upper bound). This is similar to measuring an “infinite” value: you can only state a lower bound. So a physicist talking about a “finite” value is only indicating that it could be measured in principle. (And to answer the inevitable question: I’m talking about finite intervals here. You could label a point on a measurement scale as “0” and measure something at the “0”. But that’s only a reference point, not an interval.) You can also think in terms of a logarithmic scale. Both zero and “infinity” are off-scale. Any real measurement is going to fall somewhere on that log scale. Zero and infinite are not possibilities.