maths

[georgeous]

Who here believes that in maths, when a sum to infinity does actually equals a number and isn't never ending. For example when you keep halving one then you will never get to 0 EVER! I think this is true but my teacher and other people say that it isn't , but logically it can't because you can keep halving every number. Obviously you wil get close to 0 but never actually get to it. I understand the formula a/(1-r) but come on who can prove it that it works.

[Marsh]

Hi georgeous (I hope my wife's not looking),
If you rephrase your question to make it easier to understand you might get more replies. 
For example I don't know the formula a/(1-r)" or how it relates to an infinite series.
However, I'm killing time waiting for my holiday taxi so I'll try to answer what I think you are asking.
I think you are saying are having a problem with being given a total for a series before the addition gets to the final term (which it never will).   
You're not the first. This link (http://www.mathacademy.com/pr/prime/articles/zeno_tort/i ndex.asp)
takes you to "Achilles and the tortoise" an infinite series problem posed in ancient Greece.  I hope the explanation helps.
However, to use an analogy, a good batsman knows where the ball is going to go.  He has to, in order to start his stroke in time.  He can't wait for the ball to actually arrive.   Mathematicians do the same.  Other mathematicians check the logic and, if it holds, the conclusion is added to the building blocks of Mathematics. Some of the concepts seem close to the limits of common sense (or my idea of it) but they work out in practice. 

[QmunkE]

There are different concepts of infinity as far as I understand it. The use of infinity in mathematics would seem to contradict the "philosophical" or "common-sense" view - i.e. that infinity can't really exist.

In mathematics, the use of infinity does make sense, although only within the bounds of the situations to which is is applied. Until this year, I never really understood the concept of integration (I assume you're doing some sort of A-Level maths, you'll come across this) but I have recently had to study Engineering Maths for a computer science degree and it does, in fact, make sense. Weird, but true.

[bernardo]

I'm not sure what your question is, but if you mean "is it possible for a sum of an infinite number of terms to be a finite number?" then the answer is yes.

[georgeous]

thanks for the answers. And i have heard the achilles and tortoise problem and it seems logical but, besides that if you simply think about it, it isn't possible, whatever is said, don't try and think about about it how you've learnt it or how mathematicians have told you it is true. Even when using formulae it isn't true. I see what your saying with the point that if its to infinity then it must come to a finite number because infinity isn't a fixed number. But just imagine dividing a number by 2 for the rest of your life then its logical to think that it never reaches 0 or in any other case a fixed number. cheers for the view though

[QmunkE]

bernado, surely that statement is only true when a lot of those terms are, in fact, zero. The logical definitition of infinity will not allow an infinite number of non-zero terms to add up to anything less than "infinity".

[bernardo]

You are wrong, QmunkE.  It is possible for an infinite number of non-zero numbers to add to a finite number.

[QmunkE]

I think perhaps our definitions of infinity are somewhat different then. Although I can agree that mathematically it may be correct that you can have an infinite number of divisions within a finite length, in terms of the actuality of "infinity" then it is logically impossible to go back the other way. Either infinity is infinite, or it isn't.

Two definitions of "infinity" from dictionary.com

Unbounded space, time, or quantity.
An indefinitely large number or amount.

Unbounded, i.e. not finite.

[bernardo]

I don't understand your point.  You said "The logical definitition of infinity will not allow an infinite number of non-zero terms to add up to anything less than "infinity". " , which is incorrect.  It is possible for an infinite number of non-zero items to add up to less than infinity. e.g.

 

1/2 = 1/2

+ 1/4 = 3/4

+ 1/8 = 7/8

+ 1/16 = 15/16

+1/32 = 31/32

+1/64 = 63/64

[etc]

+1/16384 = 16383/16384

[etc]

+1/2^loads = 1-(1/2^loads)

[etc]

 

which adds up to 1.

[QmunkE]

I know the maths. I know it seems to work. But it doesn't acutally use "infinity", only exceedingly large numbers. By that logic, the sum can never actually reach 1, it would appear asymptotic to it on a graph, with the infinitieth term being 1 - (1/infinity). Unless you believe 1/infinity is 0.

My argument is that you can't actually do maths with "infinity" since it is a concept, and not a number. If there are an infinite number of terms, then clearly by the definition of "infinity" the sum of those terms is infinitley large, unless they are infinitessimaly small i.e. 0.

[georgeous]

I agree with QmunkE, although you are right bernardo, with the definition of infinity it is not possible to use it as a number. When i was told to multiply a formula with a very high power it would obviously display the finite number i.e. 0 or 1 becuase the calculator can only display a certain number of digits. By using the definition that QmunkE gave it is unarguable that infinity isn't a number and therefore, the problem isn't true. It's confusing, but if you have different definitions of infinity then your bound to have a dispute or argument. THanks for the replies though