.9999... equals 1?

[FritoSanchez]

I was in my math class the other day, and we were talking about a special problem, and it ended up with my guess being 2. And so it begun. The math teacher says, no, it's about 2. I said, oh, so it's like 1.9 repeated, it isn't 2, but it almost is. He says, no, that's different, 1.9 repeated is 2.

So me and my friend, we both say, no it isn't. He says, yes it is. No, we say, it almost is. The difference is so minuscule that it doesn't matter, but it doesn't equal 2. He starts writing on the board:

x = .999...
10x = 9.999...
9.999...
-.999...
______
9
9x = 9
x = 1

So he begins the problem with x being .999... and ends it with x being 1. How can x equal two things, for one thing. I see how the problem makes sense, but he only solved it backwards, not forwards.

So we go home, and I look it up, turns out that that is a famous equation used to prove that .999... equals 1, and that concept is accepted by most mathematicians. But... It doesn't equal 1.

I have yet to find a majorly convincing argument that .999... = 1.
Do any of you agree with that concept, and if so, could you explain it further?

[GATOONA]

.9999999 does not equal 1.
It will never equal 1.
It is like saying. 1.000000000000000000000000000000
equals 1.1

[Kidiri]

0.999...=1 deal with it.

[newnerdproductionsTM]

yeah, there is always that .000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000000000000000000 0000000000000000000000000... ...0001 left over.

[andhination]

No matter how many equations exist, if something is missing, no matter how small, it is missing.

[GATOONA]

At 2/7/09 03:02 PM, Kidiri wrote: 0.999...=1 deal with it.

LIES!

[Be-Easy]

My math teacher told me 2 was an odd number

[FritoSanchez]

At 2/7/09 03:02 PM, Kidiri wrote: 0.999...=1 deal with it.

Oh, I guess in that case, I agree with you.

Genius.

[Conspiracy3]

That is because a never ending decimal is a falacy. What you would really have is

X=.99999
10X=9.9999
10X-X=9.00009 NOT 9

[MonkeyV]

Maybe inductive reasoning will work better for you.

9/5 = .5555...
9/6 = .6666...
9/7 = .7777...
9/8 = .8888...
9/9 = .9999 = 1

[Timmy]

Once you learn limits, you'll understand the concept of a number that goes on forever, always getting closer to a certain number, but never actually reaching it (like how 1.9999999999999999999999 etc never actually gets to 2). It's important to know that even small differences between two numbers are still differences that need to be acknowledged.

[HeartbreakHoldout]

1/3 = 0.3...
3/3 = 1 = 0.9...

It's widely accepted. There's a proof to do with limits of sequences, but I can't be fucked to look it up. Especially since you probably won't be convinced regardless.

[FritoSanchez]

At 2/7/09 03:04 PM, Timmy wrote: Once you learn limits, you'll understand the concept of a number that goes on forever, always getting closer to a certain number, but never actually reaching it (like how 1.9999999999999999999999 etc never actually gets to 2). It's important to know that even small differences between two numbers are still differences that need to be acknowledged.

Yes, exactly, and I do understand that concept.

At 2/7/09 03:04 PM, MonkeyV wrote: Maybe inductive reasoning will work better for you.

9/5 = .5555...
9/6 = .6666...
9/7 = .7777...
9/8 = .8888...
9/9 = .9999 = 1

My problem with that one is the same as my problem with the 1/3 proof. 9/5 does not equal .5 repeated, but .5555... is the best way to represent the decimal form of 9/5. In the same way, .333... is the best way to represent the decimal form of 1/3. They don't equal their fraction forms exactly.

[Conspiracy3]

The basic point is that repeating decimals are not exact numbers but approximations.

1/3 =/= .333333333333333333333333333333333333333 33333333333333333...

That is an approximation NOT an actuall number. That is just hte closest you can get in base 10. That is why in high level math classes almost every teacher will tell you to use fractions/the pi/e symbol instead of approximated decimals.

[BananaBreadMuffin]

At 2/7/09 03:03 PM, newnerdproductionsTM wrote: yeah, there is always that .000..0001 left over.

No, there isn't, because there's no way to quantify those ...s inbetween either side of your decimal.

[InsertFunnyUserName]

There's a difference between theoretical math and hard numbers.

9.9999999... will always be 8.999999... more than 1, no matter how many funky equations you put it through.

In theory, yes, 9.9999999... equals one, but in practicality, it does not. In the real world, something that weighs 9.9999... ounces is heavier than something that weighs 1 ounce.

[Joe]

Rounding, and approximations, and assumptions! Oh my!

[SteveGuzzi]

At 2/7/09 03:00 PM, FritoSanchez wrote: So we go home, and I look it up, turns out that that is a famous equation used to prove that .999... equals 1, and that concept is accepted by most mathematicians. But... It doesn't equal 1.

I have yet to find a majorly convincing argument that .999... = 1.
Do any of you agree with that concept, and if so, could you explain it further?

1/3 = 0.3333333333333333...
2/3 = 0.6666666666666666...
3/3 = 0.9999999999999999... = 1

You don't need any magic equations to prove it. It's just a limitation of base-10 counting.