# What does finite mean in mathematical terms?

## Main Question or Discussion Point

What does "finite" mean in mathematical terms?

I was reading a discussion on here about whether or not Pi is infinite. It seems intuitive to me that it's not, but I'm having trouble grasping the idea that it (or any other number) is finite.

From what I understand, if something is finite, it means it is limited or bounded. By that definition, how can a number be finite? What is it "bounded" by? Itself?

Unless my understanding is wrong, and finite just means "not infinite". Is this the correct interpretation, or is there another one?

First, finite/infinite only applies to sets. A set S is said to be infinite if it possesses a proper subset that is one-to-one to S, a set is finite otherwise. Now a positive integer can be defined from set theory; one way is to start with 0 as the empty set, 1 as {0}, 2 as {1} and so on. Given those, or similar, definitions, a positive integer could be said to be a finite set. A number on the real line could be defined to be the set of all points preceding it from the left. It could be defined as all the points lying exactly to the right and to left of it. This would mean all real numbers are infinite sets. Otherwise, a statement like "pi is infinite" makes little sense.

DaveC426913
Gold Member

Pi is not infinite. Pi is irrational (cannot be expressed as a fraction of integers) and it is transcendental (no finite sequence of algebraic operations on integers could ever produce it).

What I think you may have read is that Pi, expressed as a decimal number, has an infinite number of digits.

This is the thread I'm referring to:

I understand that Pi is not infinite, but what I'm not quite sure of is how a number itself is finite.

A number on the real line could be defined to be the set of all points preceding it from the left. It could be defined as all the points lying exactly to the right and to left of it. This would mean all real numbers are infinite sets.
I understand the concept of finite and infinite sets, but what I get from this definition is that a statement like "5 is finite" makes no real sense. If that is true (and I'm not sure about that), then similarly a statement like "Pi is finite" shouldn't make sense either, unless finite is defined in mathematical terms to mean "not infinite". What troubles me is that one could say "a door is finite", which doesn't really make sense to me either.

The answers in that thread are misleading. The people answering played along with the not-so-sensible question. Numbers are neither finite nor infinite, not when considered atomic mathematical objects in their own right at least.

The answers in that thread are misleading. The people answering played along with the not-so-sensible question. Numbers are neither finite nor infinite, not when considered atomic mathematical objects in their own right at least.
That's what I suspected. However, this wikipedia page says all Real Numbers are finite:
http://en.wikipedia.org/wiki/Finite

So either it's wrong, or "finite" has a different meaning when applied to numbers. In this case, it would appear that they mean "not equal to positive or negative infinity". This seems to imply that there are numbers that are equal to infinity. Is such a number a Hyperreal? If there is a mathematical description of number that equals infinity, then I can understand how numbers themselves can be considered finite or infinite.

CRGreathouse
Homework Helper

So either it's wrong, or "finite" has a different meaning when applied to numbers. In this case, it would appear that they mean "not equal to positive or negative infinity". This seems to imply that there are numbers that are equal to infinity. Is such a number a Hyperreal? If there is a mathematical description of number that equals infinity, then I can understand how numbers themselves can be considered finite or infinite.
There are lots of mathematical structures with infinite members. Sets are the canonical example: {1, 2, 3} is finite but omega = {0, 1, 2, ...} is infinite.

The extended reals are another example. In the extended reals there are exactly two infinite members: +infinity and -infinity. (The representations of these numbers need not be infinite, just like their names have a finite number of characters; but *in the extended reals* they're infinite.)

The hyperreal numbers have many nonequal infinite members. You can think of a hyperreal as a sequence of real numbers (with certain conditions); if the sequence converges to a real number the hyperreal is finite, otherwise it is infinite. Once again, the representation is immaterial: every hyperreal can be represented by an infinite sequence, but as a hyperreal they could still be finite. (1, 1, 1, 1, ...) = (1/2, 3/4, 7/8, 15/16, ...) = 1 is a finite hyperreal (where "=" is the hyperreal equality).

It seems you're confused because of the "infinite number of decimal places" of Pi. But all real numbers have an infinite number of decimal places. 1/9 = 0.1111... where the 1s never stop, and 7 = 7.000... = 6.9999... where the 0s or 9s never stop.

Hurkyl
Staff Emeritus
Gold Member

Once again, the representation is immaterial: every hyperreal can be represented by an infinite sequence, but as a hyperreal they could still be finite. (1, 1, 1, 1, ...) = (1/2, 3/4, 7/8, 15/16, ...) = 1 is a finite hyperreal (where "=" is the hyperreal equality).
Actually, the hyperreal denoted by (1/2, 3/4, 7/8, 15/16, ...) is infinitessimally smaller than 1. (the difference being a positive hyperinteger power of 1/2, in fact)

It seems you're confused because of the "infinite number of decimal places" of Pi. But all real numbers have an infinite number of decimal places. 1/9 = 0.1111... where the 1s never stop, and 7 = 7.000... = 6.9999... where the 0s or 9s never stop.
I didn't ask that question, I just happened to stumble on the thread. I understand that Pi is not infinite, I just don't understand how a number itself can be finite.

I did some more searching and there doesn't seem to be much information online about what a finite number really is, and it appears that the answer to this question might be slightly trickier than expected.

So we can say that a number is finite if we don't have to do an
endless sequence of operations to generate it; or if it lies between
two finite numbers.

There are other subtleties, like how to deal with 'numbers' that have
components, like complex numbers or vectors. But if you ignore those,
does this definition make sense?
http://mathforum.org/library/drmath/view/62854.html

The first definition seems to makes sense, but the second definition he gives is circular ("a finite number is a number that lies between two finite numbers"). Also, how would one treat complex numbers?

The only bit of information I found comes straight from Russell's Principles of Mathematics, but it only seems to apply to integers.

http://fair-use.org/bertrand-russell/the-principles-of-mathematics/s120

Actually, he gives a definition of a finite number in the previous chapter:

Thus we may define finite numbers either as those that can be reached by mathematical induction, starting from 0 and increasing by 1 at each step, or as those of classes which are not similar to parts of themselves obtained by taking away single terms.
http://fair-use.org/bertrand-russell/the-principles-of-mathematics/s119

He repeats the first definition in Philosophical Essays on page 72 in the chapter "Science and Hypothesis (A review)"
A finite number means one to which mathematical induction applies; an infinite number means one to which it does not apply.

CRGreathouse
Homework Helper

Actually, the hyperreal denoted by (1/2, 3/4, 7/8, 15/16, ...) is infinitessimally smaller than 1. (the difference being a positive hyperinteger power of 1/2, in fact) Of course, you're right. I *meant* that the real shadow of both is 1. Gah; my statement would have made the reals equal to the hyperreals.

CRGreathouse
The answer to that question is entirely dependent on what you call a number. If numbers are integers (or real numbers, or complex numbers), then they can't be infinite. But if they're extended reals, projective reals, hyperreals, surreals, or cardinals then they can be infinite (respective examples $+\infty$, $\infty$, (1, 2, 3, ...), {1, 2, 3, ... |}, and the equivalence class of the reals $$\mathfrak{c}$$).