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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

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 9y ago
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 9y ago
  Fine, then. Just use the smallest positive nonzero real number, instead. Edit: I really hope I'm not the only one laughing.
  - kmill 9y ago
    What would that be? If c>0 were smallest, then c/2 would be smaller and still positive!
  - mturmon 9y ago
    "...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 https://en.wikipedia.org/wiki/Non-standard_analysis) Not only do you get wrong conclusions, you get tedious, hard-to-adjudicate arguments.
    - jtc1983 9y ago
      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 9y ago
        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.
yorwba 9y ago
> 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 https://en.wikipedia.org/wiki/P-adic_number
lisper 9y ago
> 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 9y ago
  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 9y ago
    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?

kmill 9y ago

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 9y ago
Fine, then. Just use the smallest positive nonzero real number, instead. Edit: I really hope I'm not the only one laughing.
- kmill 9y ago
  What would that be? If c>0 were smallest, then c/2 would be smaller and still positive!
- mturmon 9y ago
  "...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 https://en.wikipedia.org/wiki/Non-standard_analysis) Not only do you get wrong conclusions, you get tedious, hard-to-adjudicate arguments.
  - jtc1983 9y ago
    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 9y ago
      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.

yorwba 9y ago

> 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 https://en.wikipedia.org/wiki/P-adic_number

lisper 9y ago

> 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 9y ago
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 9y ago
  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?