View Full Version : Is zero a finite number?

Mariemarie

08-29-2005, 12:10 AM

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?

David Simmons

08-29-2005, 01:37 AM

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.

Orbifold

08-29-2005, 01:39 AM

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. 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.

BrainGlutton

08-29-2005, 02:19 AM

From 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.

Wendell Wagner

08-29-2005, 09:52 AM

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.

Left Hand of Dorkness

08-29-2005, 10:00 AM

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

ultrafilter

08-29-2005, 02:26 PM

In common usage, zero may or may not be finite, but in technical usage, it most definitely is.

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.

From http://en.wikipedia.org/wiki/Infinitesimal:

No such thing in the standard real numbers.

Left Hand of Dorkness

08-29-2005, 05:17 PM

An interval of length 0 is still length 0.

In what sense is it an interval, then?

Daniel

Left Hand of Dorkness

08-29-2005, 05:20 PM

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

ultrafilter

08-29-2005, 05:24 PM

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.

Hypnagogic Jerk

08-29-2005, 05:50 PM

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?

ultrafilter

08-29-2005, 06:00 PM

Yes.

Left Hand of Dorkness

08-29-2005, 08:51 PM

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

Mariemarie

08-31-2005, 10:55 AM

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.

Left Hand of Dorkness

08-31-2005, 04:04 PM

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

Pleonast

08-31-2005, 05:06 PM

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.

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