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.
* 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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