[logfromblammo]

Start with zero. Add an infinitesimal epsilon an infinite number of times. Now go back to zero and subtract the same epsilon an infinite number of times. You have now traversed all the real numbers.

The decimal representation of the epsilon has an infinite number of zeroes after the decimal point and before the last digit, which is '1'. So if you were to just chop off the leading zero and decimal point to make an equivalence with natural numbers, the epsilon is as much a representation of the natural number 1 as 0.1 or 0.01 or 0.001 or 0.0001 . Infinite real number equivalents to one natural number.

[kmill]

That infinitesimal is not a real number. To simplify a little, a real number is something which is the limit of a sequence of rational numbers. Or, given an error bound 1/n, you can write down a rational number within 1/n of the real number. Two real numbers are the same if the difference between their approximations converges to 0 as n gets arbitrarily large.

A number with infinitely many zeros after the decimal point is within 1/n of zero no matter the n. Therefore the number is zero.

This is like how 0.9999... is 1. The reason is that 1-0.999... is within 1/n of 0 no matter the n.

Suppose you had an infinitesimal epsilon (outside the real numbers --- this is fine, and people do this). How many times are you planning on adding it to itself? To get any actual real number, you are going to have to add it to itself well more than countably many times, though I'm not sure this makes much sense.

[logfromblammo]

Fine, then. Just use the smallest positive nonzero real number, instead.

Edit: I really hope I'm not the only one laughing.

[mturmon]

"...only one laughing..."

You are illustrating the reason that mathematicians generally disparage the concept of an "infinitesimal", when it's used as proof rather than conceptual aid. (Yes, I know about https://en.wikipedia.org/wiki/Non-standard_analysis)

Not only do you get wrong conclusions, you get tedious, hard-to-adjudicate arguments.

[jtc1983]

I agree with your thought, and that of the majority of mathematicians for many years, that infinitesimals are better construed heuristically than literally.

You mention that you know of non-standard analysis and indicate that it's irrelevant. Though I don't know why you think this, I agree with you. I just wanted to plug non-standard analysis as both mathematically interesting and also very useful. One can jettison the philosophical thought that NSA "vindicates" the historical use of infinitesimals (as I think we should), while still seeing NSA as the wonderful and deep piece of math that it in fact is.

[logfromblammo]

Thanks for your support, but I never said anything about non-standard analysis. I made one post in support of its parent, and people jumped on it to say how wrong I am, as apparently I accidentally stumbled over a sore spot in mathematics. I don't know why infinitesimals apparently aren't allowed, but the tone around here has convinced me to not care.

And now every post I have made in this thread tree is getting downvoted. So I'm out. Y'all can argue about nothing--and nearly-nothing--by yourselves.

[kmill]

What would that be?

If c>0 were smallest, then c/2 would be smaller and still positive!

[lisper]

> before the last digit, which is '1'

No. There is no last digit. That's the whole point. If there were a last digit your argument would be correct, but there isn't, so it's not.

[logfromblammo]

I don't see the problem with having a first digit and a last digit and an infinite number of digits in between.

Edit: Infinitesimal divided by two is infinitesimal, in the same way that infinity multiplied by two is infinity. So 0.000...0001 / 2 = 0.000...0001 . Infinitesimal multiplied by any finite number is infinitesimal. Infinitesimal multiplied by infinity is every number in the interval from infinitesimal to infinity. Or none of them. Or just one. Or all of them except one. Infinity just throws common sense out the window, then catches it as it tries to sneak back in through the chimney and sets it on fire. I'm not convinced that any sane person can adequately grasp the concept.

Don't confuse the limitations on mathematical notation with a limitation on imagination. 0.9 repeating is not exactly the same as the infinite sequence of ( 0.9 + 0.09 + 0.009 + ... ). The repeating notation indicates to use nine's complement for that portion of the fraction instead of ten's complement. 0.9 repeating is literally equal to one, by notation convention, but the infinite sequence of 9 digits is one minus infinitesimal, which is equal to one in every calculation that does not involve an infinity.

You have to have an infinite number of infinitesimals to make any number that isn't zero, but when you do that, you can get all of them.

[kmill]

> Don't confuse the limitations on mathematical notation with a limitation on imagination

Good luck proving or calculating anything.

You can define "infinitesimal/2 == infinitesimal", but nothing good will come out of it. A definition is no good unless it lets you do something.

Letting e=infinitesimal, you have e/2==e, so e==2e so 0==2e-e so 0==e. This definition is inconsistent with being able divide by non-zero integers and subtraction.

[logfromblammo]

That's not the definition, that's just what it does.

The definition of infinitesimal is "the smallest-magnitude number that is greater than zero". If you divide a finite number by infinity, infinitesimal is what you get, but don't go thinking that if you multiply it by infinity again that you will get the same number back, because you won't.

The floating point standard does not include a representation for infinitesimal, but an underflow now hints at its existence, instead of just going to zero.

It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions. Their behaviors can be defined separately, such that whatever rules you choose for them can produce different types of math, perhaps useful for different purposes (or none beyond cranking out the dissertation), in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries.

[kmill]

> The definition of infinitesimal is "the smallest-magnitude number that is greater than zero"

That is hardly a definition. The real numbers are defined either as Dedekind cuts or as equivalence classes of Cauchy sequences of rationals. If you say "e is defined to be a real number such that e>0 and for all c>0, c>e", you would get a contradiction purely from the definition of the reals since "for all c>0, c>e" implies "e <= 0".

The only consistent scenario I can think is that you are actually extending the real numbers with a new element called "infinitesimal." Go ahead, but don't pretend that it is an element of the set of real numbers. Also, don't get the idea that there is some "true" set of numbers that we are trying to approximate with better accuracy. Modern mathematics has blown this idea wide open by introducing a wide array of mutually-inconsistent number systems.

> If you divide a finite number by infinity, infinitesimal is what you get

So you say. This would need to be part of the definition, or at least provable from it. Quoting Timothy Gowers, a mathematical object is what it does. How was I supposed to know that twice infinitesimal is equal to infinitesimal?

Elaborating extension: there is a way to add ("adjoin") an infinitesimal element to the real numbers. Let R(e) be the set of rational functions in e, a formal constant. For instance, 2+3e or 5e^2. I think there is a way to give R(e) a total order by saying 0<e<c for all positive real c. You also have things like 1/e=e^{-1} ("infinity") is greater than all real numbers. Don't make the mistake that R(e) is the real numbers, however.

> It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions

In what way? In a vector space, I can divide by two, and I am apparently not able to divide infinitesimal by two (in the sense that e/2=e implies e=0).

> in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries

At least right now, there is a rather large difference: many interesting theorems follow from hyperbolic geometry.

What interesting things follow from asserting that there is a smallest-magnitude real number greater than zero? (If you say "I never said the number was real," then there has been no point to this discussion, because it started when you claimed the real numbers were countable by multiplying "infinitesimal" by other numbers.)

[lisper]

You can do that, but then the "last digit" doesn't behave the way you intuitively expect it to. For example, 0.0...1 is exactly equal to 0 for the same reason that 0.999... is exactly equal to 1. So you can't add them up to get a non-zero number.

The reason that adding numbers with a finite number of zeros before the first non-zero digit works to give you any number is that that carries go off to the left. But if there are an infinite number of zeros before the first non-zero digit then there will remain an infinite number of zeros after every addition. The carries can only "overcome" a finite number of zeros.

[kmill]

There's not a problem if you can answer this: What is 0.000...1 + 0.000...9? (The 1 and 9 are digits after infinitely many zeros.)

You are not allowed to say "undefined" if these are real numbers, because you are supposed to be able to add any two real numbers.

You are also not allowed to say 0.000...10 since that changes the place values.

Well, you are allowed to say 0.000...10 if you are imagining a real number is a pair (normal real number, natural number). If this is what you are imagining, then there's not a problem if you can answer this: What is the value of 0.000...1 divided by two?

[dragonius]

He's already answered you:

If you are going to allow this kind of notation, then

0.000...1 = 0 is equivalent to 0

0.000...9 = 9 x 0 = 0

0.000...1 + 0.000...9 = 0 + 0 = 10 x 0 = 0.000...10

(0.000...10) / 2 = 0/2 = 0

You're getting hung up on notation and missing the concept.

[kmill]

Neither has logfromblammo answered me nor am I hung up on notation. The notation is only incidental.

They are claiming that it is a well-defined number system with numbers "having a first digit and a last digit and an infinite number of digits in between." I say show that it works.

You are saying the way this works is to disregard the digits after the infinitely many digits. Sure, that would make a consistent system.

They seem to be saying something distinct from your interpretation. It's possible they mean to take the real numbers and adjoin a new "infinitesimal" element, for instance.

[logfromblammo]

I'm not going to answer you, as I haven't made any claim that I care to defend. I made one little post in support of its parent, and people crawled out of the woodwork to tell me how wrong I am, and apparently try to convince me that infinitesimals are not allowed in serious mathematics, or at least not allowed in the way I was trying to use them.

And now every post I have made in this thread tree is getting downvoted. So I'm out. Y'all can argue about nothing--and nearly-nothing--by yourselves.

[yorwba]

> Now go back to zero and ...

You can't ever finish adding epsilon infinitely often, so everything after that is dead code. There is no "now" to speak of.

The same thing comes into play when you speak about an infinite number of zeroes after the decimal point and before the last digit. There is no last digit. You could have numbers with two "ends", but the so-called real numbers are different. They stretch infintely to the right, without end in sight.

There are also p-adic numbers, which stretch infinitely to the left, but they behave very different from real numbers. https://en.wikipedia.org/wiki/P-adic_number