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.

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.)

> 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.

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

> 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.

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.

> 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

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...

> 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?

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.

> They have to know that 0.999... means you never stop writing nines.

How do they know that that's a real number?

They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.

You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).

Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.

And that is where it all goes wrong.

In calculus, yes.