[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.

[kmill]

For what it's worth, I spent time arguing with you because the ideas you were proposing were interesting, but the problem with math is you have to make rigorous definitions and such. The ideas as stated cannot be defended, but as I said elsewhere, you can make something like infinitesimals work with some more effort, but they aren't the real numbers anymore.

"Define it that way, but show me the theorems" is an important organizational philosophy of math. It also helps remove ego from everything. It can be painful creating math without realizing this, and communicating the philosophy was the main point I was trying to make. (I also had some hopes you would show interesting consequences!)

It's not like what you were saying was obviously wrong. It wasn't until the mid-1800's that people really sorted out the real numbers. I myself spent some time thinking I "solved" the 1/0 problem and thought about "numbers" like 0.000..infinitelymany...01, but the nice thing about math is that performing experiments isn't too expensive.

(I didn't downvote you. Sorry for the full-contact lesson in math philosophy, and don't get the idea this is a "sore spot in mathematics," rather the non-existence of infinitesimals in the real numbers is easily defended.)

Edit: Beyond the algebraic way of making infinitesimals work, there is also the real analysis version: limits to 0. The idea of infinitesimal there is that no matter what positive real number you give me, I can give you a smaller one. This concept of infinitesimal isn't a number per se. Similarly, one of the many ways infinity shows up is that no matter what number you give me, I can give you a larger one.

The metaphor of infinity also shows up in: cardinality of sets, as the added point in a one-point compactification, the extended real line, the Riemann sphere, arbitrarily large numbers (limits), and that's all I can think of at the top of my head.

[logfromblammo]

Proofs and explanations are not always the same. Proofs depend on logic and rigor, while explanations depend on the audience. Mathematics and pedagogy are not usually considered to be closely related areas of study. And yet universities make mathematicians teach mathematics.

Are they the best teachers? No. No, they are not. But they are the only ones that understand the subject matter well enough to do it. And that leads to the vicious cycle where you have to think like a mathematician in order to learn math from one, because they have difficulty explaining anything to any other type of person. A student that needs an explanation gets a proof, which is technically correct, but still fails to elucidate.

I think some kinds of math are fun and interesting, but proving the math is [currently] less than 1% of my job, and I have never had to worry about precision that would underflow a 64-bit floating point double.

Take another look at the whole thread tree, originating at https://news.ycombinator.com/item?id=15236430 , and look at the posts by "zelah". Realize that all the responses made them realize that they got something wrong somewhere, but it looks like they are still as confused as ever, and probably net negative karma from being wrong on the Internet and not knowing why.

My original goal was to help zelah understand, and I failed. My secondary goal was to play the game alluded to by lisper, who essentially said I cheated. There is nothing left for me to accomplish here. I wasn't trying to be pissy and storm out the door in a cloud of drama, but rereading, it seems like that's probably the simplest interpretation of my last post. So... sorry for that. I'm still not writing you any theorems.