Because I haven't done it yet: 0.999... = ?

[packbat]

Point nines recurring [1] equals one.

I understand that this claim is intuitively displeasing to many intelligent people. "Point nine recurring doesn't equal one!", they say, and give various reasonable accounts of why the two should be distinguished. However, as an engineer, I submit to the reader that we should equate them, however displeasing it

Comments

[active_apathy]

Ah, you're using that proof. My favourite for this is:

Let x = 0.9
10x = 9.9
9x = 9.9 - 0.9
= 9
x = 1
∴ 0.9 = 1

[packbat]

That's a good one - I mainly used the one-third proof because it was dead easy to transition to. Edit: Also, because it emphasized my point - which is that allowing recurring decimals at all implies one equals point nine recurring.

Another one for your viewing pleasure:

Let x = 0.999...
= 0.9 + 0.09 + 0.009 + ...
= 1-0.1 + 0.1-0.01 + 0.01-0.001 + ...
= 1

[jeffreycwells]

I had the same negative visceral reaction to this idea when it was first shown to me in high school, but it's hard (if not impossible) to counter the proofs.

[packbat]

Actually, curiously enough, I didn't have that reaction - not exactly sure why. I think I even thought it was really cool, as in, "Hah! The carry is infinitely far away!" (I even remember coming up with some trick for calculating fractions based on using 0.999... instead of 1.) I think it's some sort of brain-wiring thing.

...but it's hard (if not impossible) to counter the proofs.

Some proofs more than others, of course - although it's a good reference, I count Sam Hughes's "Quickest Proof" (and, in fact, his Preliminary Note) as dubious.

[jeffreycwells]

Hm. Yeah, I don't like how that one almost seems to dip its toes into linguistics in the formation of its argument. I prefer the more strictly numerical proofs.

which is bigger, e**pi or pi**e ?

[anonymous]

without using a calculator, please: which is larger, eπ or πe? (from a Russian math problem collection, seen a few decades ago ...)

- ^z - http://zhurnaly.com

Re: which is bigger, e**pi or pi**e ?

[packbat]

eπ - smaller to the larger is bigger than larger to the smaller with 2 and 10, so it should continue. What's the right answer?

Edit: Merle did 2 and 3, which are the other way 'round, so he guessed πe.

Edit 2: If you take the log, you get π and e ln π - taking the Taylor series, that's π and e * (π-1), and the latter's larger. So I guess I should switch too, to πe?

[zwol]

But if you deny that 0.999... = 1, and instead define ε = 1 - 0.999..., and work out the consequences from there, you get the hyperreals, which are fascinating and useful (for instance, they make a way of motivating differential calculus that is IMO better than the standard).

[packbat]

I do like infinitesimals - that's a good point. Making the conscious choice to define it that way is the important bit, though - as it will imply, for example, 1/3 = 0.333... + ε/3. As long as you know what you're doing, it's good.

Totally irrelevant to your point

[baxil]

I like your footnote notation - the superscript numbers and the return-up-arrow. Be aware, though, that if you're going to continue using it, your links will break on your journal page (and possibly on other people's friends pages). When I've linked footnotes, I've tried to insert dates in the links to prevent that; for full compatibility, links not buried beneath a cut tag may require the anchor tag be something like username_date_footnotenum.

Re: Totally irrelevant to your point

[packbat]

Ah, excellent point! I'll be sure to correct it appropriately. (What about [month][abbreviated subject line] as a salt?)

Edit: Make that [yymm][keyword] between [foot/ref] and the number - that way, digits and numbers alternate.

(Also: I cannot take credit for the footnote notation - I've seen it many places, I merely borrowed it as good.)