Does 0.999... equal 1?

[verryberry10]

I discussed this with shredder on msn.

 

[Dorito]

Meh, if you want to be politically correct, if you get my point, yes, .999... does equal 1.

Regardless, many people don't agree or question it widely. If you go through a college course you'll almost always be taught they are equal and that every non-zero, terminating decimal has a twin with trailing 9s.

Whether this can be called true, is up for you to decide, because this is almost like a discussion about the existence of God, in other words, there is no definite answer.

Personally, no, I do not see them as equal, as no matter how many 9s, infinite or not, there will always be an infinite number of zeros followed by a 1.

 

[WhereIsTheGod]

Zero is NOT infinitely small. Zero is void. It is nothing. Therefore it cannot be big, small, anything. Zero cannot be expressed by a fraction with a numerator besides zero. .0(bar)1 can be expressed in multiple ways through a fraction. Not a rational one, but a fraction nonetheless.

Do it.

In such a way that you don't get 1/infinity = zero

 

[Annex]

I posted this on offical forums ages ago but it died.
This is how i learned it.

X = 0.111111
10X = 1.111111

so, 9X would be 1.
Divide them out and it's 1/9

So, carrying on.
0.9999 is 9/9 which is 1/1, which is 1.

No its not you idiot. Divide 9 by 9, and you don't get .99999, you people are honestly essential level math, now its time to debunk your idiocracy

X = 0.111111
10X = 1.111111

No it wouldn't, you don't add another .000001 when multiplying by 10, the anwser would be 1.111110, again we can't properly express 1/9 in decimals, so we round it for essential people like yourself who don't know how to divide numbers. Ill make this loud and clear for everyone else

9/9 = 1, BECAUSE THE DECIMAL OF A FRACTION IS OBTAINED BY DIVIDING THE NUMERATOR BY THE DENOMINATOR, IT WILL NEVER = .9999999999 SO STOP WITH THE DAMN GRADE 3 LOGIX. THE SAME THING GOES FOR ALL OTHER FRACTIONS THAT ARE 0.########### TO INFINITY. NOW ANYONE ELSE MAKING A STUPID STATEMENT LIKE THAT WILL BE LAUGHED AT FOR THEIR LACK OF ABILITY TO DO SIMPLE GRADE 3 MATH.

Also another note, this isn't a debate, no matter how much we debate, 0.9999999999 will never equal one, equal is a mathematic function that means the exact same, not within x thousandths or millionths, IT MEANS THE EXACT SAME.

[/thread]

 

[WhereIsTheGod]

Annex, contrary to popular belief... you aren't any smarter when you're louder.

x=.1111...
10x=1.11111...
9x=1
x=1/9

Don't /thread him because he forgot to put the dots, or the bar.

 

[fatass1]

As Shredderbeam has already said, as well as countless other people, it is an infinitely small number, but a number none the less.

I'll add more in a minute once I get a trade out of the way.

EDIT:

What is the number that is infinitly small? Just so you know you cant have anything go past infinity so you nothing can come after anything repeated for infinity. You cant say that .000...0001 because you are effectively making infinity not infinite. It just doesnt work like that.

If .999... is equal to 1.

Then .989... is equal to .999.

If that's true, then .979... is equal to .989....

No, your logic is flawed. if .999... is equal to 1
Then .989999... is equal to .99.
Then .979999... is equal to .98.

And so on. Your logic is flawed, as is math. I said it before, and I'll say it again. Give me 3 equal numbers in decimal form that add up to 10. IT IS IMPOSSIBLE.

3.333... & 3.333... & 3.333...

NOT ALL VALUES CAN BE REPRESENTED EAXACTLY in decimal form. The decimal system works in bases or powers of 10. Things like 1/3 or 2/3 CANNOT be represented in decimal form.

You CAN represent them in decimal form. However, you CANNOT represent them in decimal form without a repeating number.

FRACTIONS ARE DIFFERENT then decimals, some of you don't seem to grasp that concept very well.

All Fractions can be perfectly represented by decimals, even though many must use repeating decimals.

You can get the fraction value of any repeating decimal by using the same algebra that is proving
that .999... = 1:

Take for instance the repeating decimal .123123123...
what fraction value does it represent?

x=.123123123...
1000x=123.123123123...
999x=123.123123123...-.123123123...
999x=123
x=123/999
simplify
x=41/333

Therefore, 41/333 perfectly equals .123123123123... just like .999... perfectly equals 3/3.

 

[The Supreme Intelligence]

This is a very, very, stupid and illogical debate. .999 is simple nine thousandths, nine hundredths, nine tenths, of 1 whole part. If they were equal, why would they be written differently with entirely different numbers.

Imagine this: 3/3- 3 apples divided into 3 different sections, you would have to put ONE apple into each section. Therefore, your answer must be 1, not somewhere close to it.

And for all that bullshit above me, simply ignore it.

 

[fatass1]

This is a very, very, stupid and illogical debate. .999 is simple nine thousandths, nine hundredths, nine tenths, of 1 whole part. If they were equal, why would they be written differently with entirely different numbers.

Imagine this: 3/3- 3 apples divided into 3 different sections, you would have to put ONE apple into each section. Therefore, your answer must be 1, not somewhere close to it.

And for all that bullshit above me, simply ignore it.

.999 does not equal .999...(repeated)

So for all that bullshit above me, simply ignore it.

 

[WhereIsTheGod]

There ARE multiple ways to represent a number. Especially in fractions and decimals.

All numbers can be represented like that.
50=49.9....
-1=-.9...
etc.

So yes, I take that position as well... AND SO CAN YOU.

 

[shawnvd]

no i dont think so, because there is a 0.0000000000000001 in between 1 and 1.999 repeated.

 

[fatass1]

no i dont think so, because there is a 0.0000000000000001 in between 1 and 1.999 repeated.

The zeros go for infinity. You cannot have anything past infinity. The difference would be 0.

 

[Dorito]

.999... cannot equal 1.

Like I said before, in any college courses you will see that every non-zero, terminating decimal has a twin with trailing 9s.

Just because it is "accepted" doesn't mean it's correct.

If you are asking if .999... = 1 is accepted by the professional field, then yes, it is.

If you are asking if .999... = 1 makes logical sense, then I'd be forced to say no.

To fatass1 who said 3.333..., 3.333..., and 3.333... equal 1. No, I add them together, I get 9.999.... Which, well, obviously isn't the number 10.

.999... goes on infinitively. But even after infinite, there will always be a .0...1.

How can something smaller, than 1, equal 1?

1 > .999...

Just because something goes on infinitively doesn't mean the number at the end randomly disappears and goes out of existence, it just means it gets pushed back an infinite amount of number places.

My opinion behind this will always be that no, they are not equal. But then again, this is a never ending discussion.

 

[The Force]

I am a high school freshman and it is to my knowledge that .999 = .999

Call me Crazy

 

[Dorito]

All proofs given that .999... equals 1 have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.

* The proof on fractions assumes that 1 / 3 = 0.333..., how do we know it's true? Why can't there be an infinitely small number ε such that 1 / 3 = 0.333... + ε / 3?
* The proof on digit manipulation says "To prove that the manipulations also work for infinite decimals, one needs the methods of real analysis." So the proof depends on something unproved.
* The proof on infinite series says: "The last step — that lim 1/10n = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
* The nested interval proof uses nested intervals theorem, which is just another form of Archimedean property.

In fact it is impossible to prove rigorously that 0.999... equals 1 with a naive intuitive approach to numbers. This gives a motivation to more serious mathematical constructions. If we wish to prove or disprove the statement, we must have a precise definition of real numbers.

Although I hate using Wikipedia as a source, I'm a bit lazy at the moment, and I think it explains my point well enough.

How can you say that 1/3 = .333... exactly, if it goes on forever, if something is infinite, it isn't defined, therefore impossible to be exact.

 

[WhereIsTheGod]

In mathematics, the recurring decimal 0.999… , which is also written as 0.\bar{9} , 0.\dot{9} or \ 0.(9), denotes a real number equal to 1

From wikipedia.

 

[bocagangsta9]

im not good at math but it seems to me as if u did do the math that it should equal 1

 

[Bartaggio]

0.999 does NOT equal 1 or 3/3. they are 2 different numbers. People just say 0.999 = 1 because they don't want to say/type all those 9's and round it up

 

[barcacl2006]

You wrote
Take the fraction, ⅓ for instance.

⅓= 0.333...the decimal that keeps going on.
⅔= 0.666... the decimal that keeps going on.
3/3= But would it make sense for it to be 0.999... as 0.333 x 3 would equal to 0.999.... and that's 3/3 which equals to 1.

But that is technically incorrect. 1/3 is NOT 0.33333, it is the closest thing to it as a decimal. Take pi for example. It is a decimal 3.14..... and the closest thing to it would be 22/7 but it is not the EXACT answer.

 

[Super Toaster]

two numbers that are different in value cant equal eachother
__
0.99 cant equal 1 because they have two different values

 

[Dorito]

I like barcacl2006's point about pi. That is a perfect way to argue this. Just as 1/3 and 2/3 cannot be exactly converted in decimal form, pi cannot be exactly converted into fraction form, ONLY ESTIMATED OR ROUNDED to 22/7.

I will ask again, how can you get an exact value if it is infinitive, meaning not defined, meaning NOT exact. If you are saying .999... equals 1, you are defining what infinity is, which is impossible to do. Infinity ISN'T a number, it's a concept.