Infinitely small number added infinitely?

What is the sum of an infinitely small number added to itself an infinite number of times? Do the infinities cancel out? Take an infinitely small number. Now, imagine adding that number to itself an infinite number of times. Wouldn't the infinite reduction and expansion cancel each other out? What would be left over?

 

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This is quite simple...

The sum is Infinity!!!

 

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Umm...an "infinitely small" number is zero, so I believe you would still have zero if you add your infinitely small number an infinite number of times.

If you had an infinite series of non-zero numbers you can get either infinity or some finite number, depending on the series. For example the infinite series 1/2 + 1/4 + 1/8 + 1/16... adds up to 1. But you'll notice that every number being added is non-zero, and therefore not infinitely small. If you add any non-zero number to itself infinitely many times, you will get infinity - but again, since the number is non-zero it's not infinitely small.

 

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Thanks Nasor for introducing some actual logic into this discussion. Even an infinite sum of finite numbers can be finite, as in geometric series like the one you showed. And an infinite sum of infinitesimally small numbers is the sum \(0+0+0+\ldots+0\) which still adds up to zero. That is all.

 

If you are correct, then tell me what is the first fractional number past zero heading towards 1? Surely, such a fractional number must exist, right?

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No such number exists. There is no non-zero fractional number which can be said to be "the closest fraction to zero". There are infinitely many numbers in any interval of non-zero size.

 

Well, 1/2 exists, so does 1/4, 1/8, 1/16, 1/32 et cetera. Now, you appear to be telling me that at some infinite point in that regression, the next fraction in the series suddenly becomes zero. O.K. But what is the fraction right before that infinite point? Obviously, there must be such a fraction just shy of infinity, or all the other fractions in the series would also be zero. So, what's the worlds next to smallest number, CptBork?

 

This is one of the problems people have with conceiving infinity. No matter how far you go down the chain, you'll always have non-zero fractions and there will always be a smaller non-zero fraction. An infinite set of numbers does not have to include a smallest number.

 

Consider the set of all positive numbers, i.e. the set of all real numbers greater than zero. This set does not have a smallest element.

 

You are being inconsistent. You say that an infinitesimally small number is equal to zero, and yet, you admit that a regressive series of smaller fractions toward zero are postive numbers and that series extends infinitely -until you come to the point of the infinitesimally small number. How is one suppose to tell the difference than between the infinite regressive series of smaller fractions toward zero (which you claim are positive numbers) and the infinitesimally small number (which is supposedly zero)?

 

No, as captbork has already said, no such number exists. There is no "smallest number" that is closest to zero. No matter how small a number is, or how close it is to zero, you can always divide it by 2 and get a number that's even smaller.

No. If you ever reached a number in the series that was "at the end," that was "just one place away from zero", the series would not be infinite. By definition, an infinite series in endless. If there is an end number, it's not an infinite series.

Again, there is no fraction that's "right before" zero. If there was, the series would have an end point, and it wouldn't be infinite.

This is just false.

 

You're mistaken. I said you will never reach this infinitesimally small number no matter how far you go down the chain. No fraction is infinitesimally small. All you can do is produce a sequence of fractions which gets smaller and smaller, and you can find fractions as close to zero as you want, but you will never actually reach zero.

 

This is the core of your mistake. It is an infinite series, so you will never actually come to that point. The series is endless. If it had an end point, it wouldn't be endless.

It is easy to tell the difference. Each number in an infinitly regressive series will be non-zero, no matter how far into the series you go. An infinitly small number, on the other hand, is exactly equal to zero.

 

So I take it that you deny the existence of the infinitesimally small number? But, I thought you said it was equal to zero? Which is it? Is the infinitesimally small number undefined or is it zero? And if it's zero, why wouldn't the infinitesimally small number be positive?

 

No, if that is how you interpret me, then you would be wrong. The only infinitesimally small, non-negative number is zero. As a number, zero exists, hence an infinitesimally small number exists. A positive number is anything greater than zero, and any such number would not be infinitesimally small. You'll never actually reach an infinitesimally small number by taking a sequence of fractions. You can find smaller and smaller fractions, but none of them will be zero or infinitesimally small. When I say you can find fractions as small as you want, I am saying that if you have a positive number such as 0.000000001, you can always find a fraction which is smaller than this number.

 

O.K. Another question for you CptBork. What do you get when you add together an infinite number of the infinitely small number? Now, I know you're going to say zero. But, why don't the infinities cancel each other out and leave you with something other than zero? Indeed, wouldn't canceling the infinities out leave you with the closest positive number to zero, without actually being zero?

 

Draw a line that represents the distance between 0 and 1

0----------------------------------------1

Divide this line in half

0--------------------0.5.........................1

Now divide by 2 again

0---------0.25...........0.5

Continue this process mentally and you will find that successive divisions will get you as close to 0 as you wish without ever reaching 0. Whatever number you think of, it is always possible to divide it by another number to yield a smaller one. At no point can you divide one number into another and get 0 as the answer.

 

The sum of infinitely many zeros is still zero. The reason is because of the way in which infinite sums are defined. Let \(N\) be a natural number, i.e. \(N=1000000\). If I add \(N\) zeros together, I still have zero. This holds true no matter how large I choose \(N\) to be, i.e. \(N=1000000000\), \(N=1000000000000\), etc. That is what we mean when we say infinitely many zeros still add up to zero. Your problem is you need to learn the mathematical definition of infinity, and that's something they won't usually teach you in high school.

 

What do you mean by "infinitely small"?

 

This is the concept at play in Zeno's paradox. I am quite familar with what you are demonstrating. However, what happens when you add an infinite number of the infinitesimally small number together? Don't the ascending and descending infinities cancel each other out? Thus, you are left with the smallest number that is not itself zero. If you consider that 0+0=0, and that zero is equal to the infinitesimally small number, than an infinity of 0's added together would cancel out the infinite regression in size, thus creating the first positive number nearest zero. Consequently, an infinite number of nothingnesses, which is possible because nothing=nothing + nothing = nothing + nothing + nothing et cetera, will create something, i.e. the closest positive number to zero. From nothing comes something.