Can someone prove to me how .999...=1

I've read over lots of arguments on this and I still think it doesn't.

 

Round up.

 

1/3 = .333
2/3 = .666
3/3 = .999


But wait... 3/3 is 1

.999 = 1

 

Wrong...

.999 doesnt equal 1, just like 2 isn't equal to 1. Found a short video that pretty much sums it up:

 

.999 doesn't equal 1, or else there would be no such measurement as .999

People just usually round up. It all depends on how precise you want to be.

 

It does not equal 1 unless you are rounding. Now if you are not rounding then there is no way that .999 = 1 thats just like if you have a job using a cash register and the change is .99 you will not give them a dollar you would give them change or your drawer will come up 1 cent short ...... just as 1 cannot be 2 and 2 cannot be 3 .999 cannot be 1.

 

0.999... = 1
It's just a different way of writing it.

Read the proofs from wikipedia that Darkest Dream posted and let us know if you still have any questions.

 

Did you watch the video until the end?
It was an April Fools video.... >_> Vihart (and yes I am a Vihart fan and so have seen this video before) agrees .999=1 as much as every other mathematician...

Here's her original video (not the April fools):
http://www.youtube.com/watch?v=TINfzxSnnIE

 

lol, posts april fools joke trying to prove his point.

 

hurr durr apparently I got trolled. No I didn't watch the video until the very end; only like the first 2 mins of it. Regardless, my stance hasn't changed and I still believe .999 cannot equal 1, just like 2 or any other number cannot equal 1.

 

I agree with that.
Just like how 2 doesn't equal 1, so does .999 not equal 1.
Unless you round up ofcourse, but that's no proof ^.^

 

What needs to be made clear here is that .999 != .999...
Of course .999 != 1
But .999... = 1

Yet another argument that you can add (in addition to the ones on wikipedia):

.999... = 9 * 1/10 + 9 * (1/10)^2 + 9 * (1/10)^3 + ...
So basically, we find that .999 is just the sum of an infinite geometric series.

Now, i'll derive the formula for the sum of a geometric series:

Let S = a + ar + ar^2 + ...
Sr = ar + ar^2 + ar^3 + ...

S - Sr = a + ar + ar^2 + ... - (ar + ar^2 + ar^3 + ...)
S(1 - r) = a
S = a/(1-r)

In our case, S = .999, a = 9/10 (Because that is our initial term), and r = 1/10 (because each term is 1/10 of the previous term.)

So we have:
.999 = (9/10)/(1 - 1/10) = (9/10)/(9/10) = 1.

 

Oh shit.
I assume you've never been to college or uni? If you source your information from wiki, university lecturers will rape you.

It doesn't equal 1.
They are different numbers.
If you work with angles, the difference between 0.999 and 1 is extraordinary.

1000 x 0.999 = 999
1000 x 1 = 1000

The outcome is different therefore they cannot be the same.

Edit: before anyone says 0.999 isn't 0.999! You can simply put the equation as 1000! x 0.999! And the outcomes will still be different.

 

Shoop your example is wrong!
.999 is not equal to .999...

1000 x .999... = 999.999... = 999 + .999...
1000 x 1 = 1000

So the issue is still down to whether .999... = 1. (Which it does, you can look at my proof if you want).

 

.999 is 1 round up (dont know if i say that correct (dutch)) but 1 isnt .999 lol

 

You get a shot from the doctor. The prickle in your skin is .000001 of your whole amount of skin. Do you only have 99% of your skin?


Give my statement a little time.

 

You guys don't know what you're talking about. (The ones trying to use physical examples especially)
.999... is not equal to .999...9
(Where ... is a finite number of 9's)
The first one has an infinite number of 9's. The second has a finite number of 9's. I agree that the second is not equal to one, but the first is.

By definition, x > y in the reals if and only if there exists some real number d such that x = y + d. (Think about this for a second if it doesn't make sense).

Now, find me that d for .999... and 1.

 

1 = 1.00000000000
1/9 = 0.111111111...
(1/9)*9 = 9/9 = 1
0.1111111111...*9 = 0.99999999999....

HOWEVER since 1/9 = 0.1111111111.... that means that (1/9)*9 must equal 0.111111111*9. Therefore, 0.9999999999... = 1.

 

When did this turn into math class?