Came upon a page via another Twitterer today (@blogofinnocence) that made me go a lit­tle cross-​​eyed. Here’s the site quoted in full:

0.999… is the same as 1. Not just very close, but pre­cisely identical:

a = 0.999…
10a = 9.999…
10a – a = 9.999… – 0.999…
9a = 9
a = 1

There’s no trick here. It’s just a math­e­mat­i­cal fact that most peo­ple find deeply counterintuitive.

No, there’s no trick, but there is a fail­ure to parse.

Lines 3 and 4 are quite bad, and it looks to me like there’s an implicit assump­tion being made in line 3 that is the source of all the confusion.

In line 3, the poster seems to be try­ing to say (in a short­hand way) that 10a — a is the same as 9.999… — 0.999… This is indu­bitably true, but it also mud­dies the issue, since it allows the next line to seem sensible:

9a = 9

If that were true, then to solve for a all you’d have to do is divide both sides by 9:

9a /​ 9 = 9 /​ 9

Which would make the con­clu­sion, a = 1, make sense. However, the line 10a – a does not make 9a = a. I’m not sure where that asser­tion even came from, but it doesn’t belong in this algorithm.

What we have here is an exam­ple of trun­ca­tion, basi­cally (which elim­i­nates every­thing to the right of the dec­i­mal), pre­ceded by a mul­ti­pli­ca­tion by 10 to bring one digit to the left. (This is some­thing that could actu­ally work with any single-​​digit infi­nite repeater from .111… to .999…)

Truncation is some­thing occa­sion­ally done in soft­ware, but not too often and usu­ally for very spe­cific rea­sons, because it’s sim­ply false to claim that any trun­cated dec­i­mal is equal to its whole num­ber equiv­a­lent. Put another way, 3.1415927* is not equal to 3.

Passing the above exam­ple through an accu­rate analy­sis, we see the prob­lem more clearly:

a = .999…
10a = 9.999…
10a — a = 9

If we solve for a now (10a = 9), we find that a = .9

Try it again:

a = .222…
10a = 2.222…
10a — a = 2 <— Truncation
a = .2

So, long and short, this really is why it was so impor­tant to your alge­bra teacher that you always showed your work when you did all those math assign­ments. That’s the only way to step back through an equation’s solu­tion and find where it went off the rails.

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* Yes, I actu­ally have mem­o­rized pi to the sev­enth dec­i­mal place. More than that would be unnec­es­sary pre­ci­sion, and a waste of time. (This is irony, kids.)

One day I hope to have a phone num­ber equal to pi, within cer­tain lim­its. Dialers couldn’t miss it, but they might not know when to stop enter­ing num­bers. So it goes.

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