Current Events > An informal proof that .99999... = 1

(+) Yeah! (+)

Let us consider a number .999 with finite digits.
.9 = 1 - 1/10
.99 = 1 - 1/100
.999 = 1 - 1/1000

And we can generalize this to say that for a number with .999 for n 9s (we'll call this function nine(n)) is
nine(n) = 1-1/(10^n).

As .999... explicitly has infinite digits, we can evaluate it as the limit of our nine function as it approaches infinity. lim(1/10^x) as n > infinity is 0, and thus lim(nine(n)) as n > infinity = 1 - 0 = 1 = .999...

If .999... had finite digits, this would not be the case, but is by definition non-terminating, and as such it must equal 1.

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.999...=x

10x=9.999.......
10x-x=9
9x=9
x=1

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I feel like the series sum proof is the most convincing.

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Anteaterking posted...

I feel like the series sum proof is the most convincing.

I mean, it's the same thing.

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Garioshi posted...

I mean, it's the same thing.

I've taught the proof that 1/(1-x) = 1 + x + x^2 + ... in a setting where the students didn't know limits at all though. For whatever reason, once you introduce a limit people are like "Sure it APPROACHES it but that doesn't mean it IS it".

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1/3 = .333...
1/3 3 = 1

.333... 3 = .999...

.999...=1

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Takes Juan to know Juan

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math is 4 nerdz

Anteaterking posted...

I feel like the series sum proof is the most convincing.

The conceptual proof is more convincing imo.

How many numbers exist between 0 and 1? A lot. 0.1, 0.2, etc. Infinite amount really. How many between 0.9 and 1? Again infinite, 0.91, 0.93, etc. Keep adding 9s if you want. 0.99999999 > x > 1 still has infinite solutions for x.

But how many numbers exist between 1 and 1 such that 1 > x > 1? None there aren't any because they're the same number and that's a false statement because 1 = 1.

So if you can't think of a single number that is between 0.999... and 1, that has to be because they're actually the same number.

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Anteaterking posted...

I've taught the proof that 1/(1-x) = 1 + x + x^2 + ... in a setting where the students didn't know limits at all though. For whatever reason, once you introduce a limit people are like "Sure it APPROACHES it but that doesn't mean it IS it".

Damn, what a nice proof. You may be right on this one.

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Doe posted...

1/3 = .333...
1/3 3 = 1

.333... 3 = .999...

.999...=1

This.

Now stop asking about it.

here's a proof that it equals one: it just does deal with it

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malenz posted...

here's a proof that it equals one: it just does deal with it

i mean yeah

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This topic is wrong because the laws of australia outweigh the laws of mathmatics

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Imagine thinking a lot of 9s was the number 1.

Go back to school kiddos

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You can give a proof with out even doing math...

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