It's because you're thinking about "adding nines" to the decimal number.

I was imagining decimal fractions in what I suppose is a constructivist manner, yes. I was imagining decimal points, followed by strings of 9's of various lengths, from one 9 to an infinite number of 9's.

Maybe that's because when you get taught how to do long division, you do it in steps.

I don't understand the reference to long division.

But mathematically, 1/3 is **equal** to 0.333..., which is an infinite sequence of 3s

Or at least that's what school kids are told, when they are being taught fractions and decimals.

The question in this thread seems to be whether there's an actual mathematical identity between 1/3 and a 0.3333... (a decimal point followed by an infinitely long string of 3's), or whether the infinitely long string of threes is a infinitly close approximation of 1/3.

I can understand why school teachers don't get into that kind of discussion with their pupils (they probably don't understand it themselves in many cases), but I can also understand how a few of the brighter and more mathematically talented school kids might think of it themselves. (Students thinking critically and creatively about what they are taught is a GOOD thing.)

because the division is a single operation. That is, all those 3s are "added" in parallel, not sequentially. You get taught to do it in steps, but that's just a convenient algorithm, the mathematics doesn't actually say anything about how to divide a number by another number, just that the operation is defined, or well-defined.

I don't entirely understand that.

Again, the question seems to me to be whether the decimal

0.333333... (with an infinite number of 3's) is equal to 1/3, or whether it's an infinitely close approximation of 1/3.

Pretty clearly it isn't equal to 1/3 if the string of 3's terminates at some finite length n, no matter how large n is. And pretty clearly making that terminating string longer by another 3, so that it's now (n+1) 3's long, won't make the string equal 1/3 either. That observation would seem to hold true indefinitely, by mathematical induction.

I'm not too embarassed to write that, even if it's wrong, since Isaac Newton and Gottfried Leibniz seem to have had similar ideas. So it's probably not totally stupid.

But today, and certainly here in this thread, it's being asserted that 0.3333... is literally identical with 1/3,

I'm just speculating now, but it looks like it might have something to do with how infinity is conceived. If a finite string of 9's obviously doesn't equal one, but an infinite string supposedly does, then it seems to be the infinity that's responsible for closing the gap somehow. My constructivist idea of infinity, in which some constructive process (adding 3's or 9's in these examples) is continued forever with no termination, doesn't seem to provide the desired result. The gap will always remain, it will just get forever smaller. So perhaps infinity is being conceived in some philosophically different (mathematically realist?) way that (somehow) leads to different results?