Current Events > I can think of three ways to prove .999.. = 1 to the layman

(+) Yeah! (+)

I understand that most people who don't think this is true probably don'r even know what division is or how to divide and it's much easier to explain division in-person

but divide 1 by 3

3 goes into 10 3 times with remainder 1
3 goes into 1 .3 times with remainder .1
3 goes into .1 .03 times with remainder .01
3 goes into .01 .003 times with remainder .001

and so on. This goes on forever. The point is 1/3 = .333...
So if
1/3 = .333...
then
1/3 + 1/3 = 2/3 = .666...
and
1/3 + 1/3 + 1/3 = 3/3 = .999...
therefore 1 = .999...


Now if for some reason you don't know how to do division, but know how to do Algebra

try 10n - n = 9

10n - n = 9n
so 9n = 9
n = 9/9
n = 1
so we know n = 1 and only 1
but plug in .999... into n

10(.999...) -1(.999...) = 9
9.999... - .999... = 9
9 = 9
true

therefore .999... = 1

But suppose you couldn't do algebra or division
and could somehow solve infinite series and find limits

let's look at .999... as a geometric series

.9 + .09 +.009 +.0009 ...

also seen as
9/10 + 9/100 + 9/1000 + 9/10000 ...

we can represent this as the geometric series
n= 1 to infinity Sigma 9/(10^n)

where a = 9/10
and r = 1/10

The geometric sum is a * (1-r^n)/(1-r)

so 9/10 * (1-(1/10^n))/(1-(1/10))



find the limit as n goes to infinity
1/10^n goes to 0 as n approaches infinity
so 1/10^n = 0
so
9/10 * (1-0)/(1-1/10) = 9/10 * 1/(9/10) = 9/10 * 10/9 = 90/90

therefore the sum of the series (which we already know has a sum .999...)
is 1
therefore .999... = 1


wow, some truly mind blowing basic math.


Are there any basic proofs I'm missing?

.999 /= 1 they are two different numbers.
sorry tc you did this for nothing

9 is a whole just like 10. Want proof? multiply any combination of 3, 6, and 9, then add the sum either left to right or right to left and it will always equal 9. No matter what any combination of 2s or 6s (doesn't even have to include a 9) will equal 9 and not 3 or 6 or anything else for that matter.
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NintendoFanGirl posted...

.999 /= 1 they are two different numbers.
sorry tc you did this for nothing

1/3=.333~
2/3=.666~
3/3=Profit
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https://www.youtube.com/watch?v=LsvUho4na_o
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NintendoFanGirl posted...

.999 /= 1 they are two different numbers.
sorry tc you did this for nothing

oops forgot to say no shit posting

It definitely makes sense mathematically, just not physically.
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.999 doesn't actually equal 1

It's just a rule we accept to make algebra true, similar to dividing by 0 being "undefined". If you want to say that algebra being true is more important than actual truth then yeah .999 does equal 1
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RedWhiteBlue posted...

The problem is that people think a decimal is an integer. Makes me wonder if they've even reached calculus and done the proofs.

Most people haven't taken calculus.
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0.999...=/=1

0.999...~=1

lol game faqs and caps
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What I never got is that if this is true, then what are asymptotes?
The equation approaches a certain number, but never reaches it, just like this problem.
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If .999 = 1 then what does .9999 equal

Hey posted...

What I never got is that if this is true, then what are asymptotes?
The equation approaches a certain number, but never reaches it, just like this problem.

0.999__ doesn't approach anything. any number that is a fixed value is a perfectly horizontal line. basically, 0.999__ is the asymptote and a hyperbolic equation would approach it, not the other way around.

It's also equal to 1.

Think about it this way.

How many numbers are between 0 and 1? well there are a lot, there is 0.1, 0.2, 0.3, 0.11, 0.12, 0.111, etc. infinity numbers.

what about how many numbers are there between 0.9 and 1? well there is 0.91, 0.911, etc. another infinity numbers.

But how many numbers are there between 1 and 1...? well, there aren't any. they are the same number.

So now ask yourself this, how many numbers are there between 0.999__ and 1? Because if you can't think of any, how is it different from 1? How can two different numbers not have a different amount of numbers between itself to the same constant?
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