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So does this mean that an infinitely small number is zero? As in 1/∞ ?

thaumasiotes 7y ago
You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
selectodude 7y ago
In calculus, yes.
GolDDranks 7y ago
In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is. That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
brlewis 7y ago
There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
- gspr 7y ago
  > There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
  - brlewis 7y ago
    You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
    - gspr 7y ago
      > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
      - brlewis 7y ago
        If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
        gspr 7y ago
        > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
        brlewis 7y ago
        They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
        gspr 7y ago
        > They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?
gspr 7y ago
> So does this mean that an infinitely small number is zero? What does "infinitely small" mean? > As in 1/∞ ? What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.

You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)

selectodude 7y ago

In calculus, yes.

GolDDranks 7y ago

In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is. That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.

brlewis 7y ago

There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.

gspr 7y ago
> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
- brlewis 7y ago
  You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
  - gspr 7y ago
    > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
    - brlewis 7y ago
      If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
      - gspr 7y ago
        > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
        brlewis 7y ago
        They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
        gspr 7y ago
        > They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?

gspr 7y ago

> So does this mean that an infinitely small number is zero? What does "infinitely small" mean? > As in 1/∞ ? What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.