Zero and Infinity

ron_jay

We know that anything divided by zero is 'undefined' or equal to infinity. Is it not possible to define in anyway such indeterminate quantities? The concept of zero basically refers to 'nothingness' or 'void', but that indeed has utmost importance in writing numbers. If you consider a general number line, we can position the zero at any given reference point and the negative and the positive integers follow suit either way of the line till infinity. Is there actually any absolute zero quantity and does infinity ever tend toward finite grounds?

Blouge

>Is it not possible to define in anyway such indeterminate quantities?

Sure, as x approaches 0, sin x approaches 0 while sin x / x approaches 1.
sin x / (x*x) approaches is "undefined" (i.e. it approaches plus or minus infinity) as real x approaches 0.

So, 1 / (x*x) at x = 0 is a much larger infinity than 1/x at x = 0. The former has a double pole there while the latter has a simple pole. The magnitude of the former infinity is essentialy the square of the latter, within some complex number factor.

You can also say that d/dx (x*x) = ((x+dx)*(x+dx)-x*x)/dx = 2x + dx where dx is an infinitesimal quantity equal to 1 / infinity. Or you can round it off to d/dx (x*x) = 2x.

Usually intuition suffices, but if you want to do this stuff rigorously, look into non-standard mathematics where
one can use the compactness theorem to shoehorn an infinitesimal quantity into the real number system.

>Is there actually any absolute zero quantity

Yes, it's denoted by 0 and its the quantity that when multiplied by any finite quantity yields 0. If you are dealing with times/positions/temperatures where you can choose to call some arbitrary measurement "0" then you've left the realm of pure numbers.

>does infinity ever tend toward finite grounds?

Yes, the sequence 1/5, 1/4, 1/3, 1/2, 1/1, 1/0, 1/-1, 1/-2, 1/-3, etc. shows how a positive quantity can grow, become infinite and then suddenly become negative. In this case plus/minus infinity appears between the positive and negative numbers. Other interesting examples of infinity becoming finite include zeta(-1) = 1 + 2 + 3 + 4 + 5 + ... = -1/12 and also exp(-zeta'(0)) = 1 * 2 * 3 * 4 * 5 * ... = sqrt(2 * pi)

Hurkyl

This statement is not correct in the sense that you meant it. If something is undefined, then anything involving it is undefined. (In ordinary arithmetic) the statement "1/0=infinity" is undefined -- it is not true, nor is it false: it is nonsense. In ordinary arithmetic, there is nothing called "infinity", so that's another reason that assertion is nonsense.

On the projective real line, there is something called "infinity". And in this context, 1/0 is not undefined: it is defined, and equal to infinity. Just to emphasize the point, that division symbol has a different meaning than it does in ordinary arithmetic. (Though the two usually agree)

On the extended real line, there is something called "positive infinity" and something called "negative infinity". In this context, 1/0 is undefined.

In the hyperreals, there is nothing called "infinity". But there are numbers that are infinite; by definition, x is infinite if and only if |x|>n for every ordinary natural number n. In the hyperreals, 1/0 is not defined. But if e is an infinitessimal number, then 1/e is an infinite number.


No it doesn't. In (usual) arithmetic, the concept of zero is something that satisfies the identities:

0+x = x+0 = 0,
0x = x0 = 0.

It might be the case that in certain applications of arithmetic, there is a notion of nothingness, or of a void, but those notions are not a part of arithmetic.

quetzalcoatl9

infact, one of the marks of distiction with regard to the mathematics of ancient civilization is whether or not the concept of zero was developed. in other words did they understand the notion of:

12304 = 1*10,000 + 2*1,000 + 3*100 + 0*10 + 4*1

without zero you cannot have a positional number system, which is a major disadvantage (try multiplying in greek numerials, for instance).

Kummer

This should be moved to philosophy forum.

HallsofIvy

Any time anyone says something so vague as to be meaningless, it should be moved to the philosophy forum?

ron_jay

Pure maths shouldn't be confused with philosophy.Those who do (no insinuation) are philosophers themselves.

Dragonfall

What about the people who philosophize about the difference between pure math and philosophy?

matt grime

From post 1:
Anyone who confuses that statement with pure maths is not a pure mathematician.

ron_jay

Ok. Lets have a definition of 'philosophy', without which all relative arguments are bound to crop up.There is a subtle difference between 'philosophy' and 'truth'.What I describe about these two entities is not philosophy, but factual and essentially the 'truth'.

Check this definition of philosophy from Wikipedia:

All Philosophy may not be true, but the truth remains the truth and thus there is no confusion between all that 'being a pure mathematician or not'

As said when a number is divided by zero, on an extended real line it is equal to infinity, then if you multiply infinity by 0, should it yield the number back?If not, then what is its solution?

HallsofIvy

Does that make him an impure mathematician?

HallsofIvy

The problem is not a question of philosophy or mathematics but rather your vague understanding of mathematics. Certainly no one has ever said "when a number is divided by zero, on an extended real line it is equal to zero"! You may have meant "when a number is divided by zero, on an extended real line, it is equal to infinity" but that is not true either. You can extend the real line to include "infinity" (or in a different way to include "positive infinity" and "negative infinity") but that is purely "geometric" and you lose the "algebraic" properties of the real numbers. It makes no sense to talk about either division or multiplication on the extended real number line.

ron_jay

Slip of the hand...anyway, then what exactly does any number divided by zero or inifinity yield or what is the product of zero and infinity?

CRGreathouse

Huh? Surely we can multiply and divide most elements by analogy with reals when real You still can't divide by zero, and can't multiply infinity by 0, but the rest can be defined in the ordinary way... yes?

ron_jay

I understand that they cannot be defined in an ordinary way...but is there any way we could perhaps define these undefined quantities quantities and find a solution for them?