well, i have a couple things to say. I did not read the whole forum, but my proffessor told us how .999~ does in fact equal 1.

we got into this discussion after talking about summation notations, and how in fact you CAN add up an infinite amount of numbers, as long as the r is less than one.

Also, something that i have always thought about. Numbers are fake. They are a manifestation of our minds to describe something, similar to words. Just because we say "red" doesn't mean something is red. what is red? Languages and math are very similar. Math is universal...at least for our planet though. .999~ does not = 1. But what .999~ repeating represents, does in fact equal what 1 represents.

No-one will ever be able to comprehend infinity, os its time to stop trying. Think of space and the universe. IT IS GROWING. how can it continue to grow with no stop? what is there to contain it? WE need something to contain it in order for us to understand it. We need a stopping point, but there is none.

So stop stressing out what 1 equals, or what 1.8999~ equals. Go party and have fun

Well he is sort of right (though I think he just decided to say you were wrong for #s and giggles). You both agree, even if you don't realize it. It's about limits to describe the behavior as something approaches something else. Thus as the denominator, x, in y/x, approaches infinity, the term approaches 0. Likewise, if y approached infinity, the term would approach infinity. But if they're both infinity, you have to play around with it, since you don't really know what infinity you're talking about. See, you can't really see infinity as a number - rather, it's best to look at it as a set of numbers. In that example, if both y and x were infinity, it would approach 1, due to l'Hospital's rule (you take the derivative of both the top and bottom as long as both the numerator and the denominator equal infinity or zero).

[edit on 12-12-2008 by Johnmike]

Originally posted by Sublime620
You start with reviewing limits. Then you have to master derivatives, a pretty large equation involving limits. Once you get that equation down forwards and backwards, the class gets a bit easy and you can start learning reverse derivatives.

Once you compute derivatives using its definition, which isn't really hard at all, lim x-->0 [f(x+h)+f(x)]/h, you learn the derivative rules, implicit differentiation, and some of their applications such as in optimization problems. I find integrals to be much harder than derivatives, and their simple definition using Riemann sums to be pretty confusing compared to the derivative formula. I think it then goes into much of the integral rules and series, which I haven't done yet.

Whoa just read this and laughed.

This is my pet project as it happens becouse 9 is the same as 3 or any odd number, you are correct that our mathis is flawed but not in a way we think. We need odd numbers why? becouse We are here.. not that may sound very silly or simplistic but lets not talk about the structure of maths so much but understand what it is!

Now Maths has evovled much with humans as it comes from our understanding of the universe. Now in time maths will change with our undestanding of the universe but untill then Logic dictates we have odd numbers to make sens of the "the bit we dont understand" or as some people call it Infinity!!..

like my sig says 1+1=3 realy is not a direct sum "in maths" as (1+1) = 1
and the outcome of that is 2 ie 1+1=2 ! but in order for you to DO the maths requires one calculation we tend to leave out "the important part"

us!! so the real "hidden" sum if you will is 1+1=3 (1+1) being me ect


in short YOU ARE THE ODD NUMBER cool or what

[edit on 12-12-2008 by theresult]

These aren't flawed assumptions.
.9 repeating is equal to 1,
and .3 repeating is equal to 1/3.

.333 * 3 = 1/3 * 3 = 1
and .999 = 1, so it all works out.
any infinite repeating decimal number of the form:
.xxx
in other words, "point x repeating",
where x could be any single digit
will be equal to x divided by nine. Hence:
.xxx = x/9

If you don't believe me, just try long division for a number over nine.



Hope this clears some stuff up!

mm

i want to cry

I too come out with 0.999 in the end. I just smacked this into my calc and I don't find any wrongs.

And what? Subtraction/addition first? Multiplication as well division goes first from what I've learned. In what instance is the opposite used? Tell us.

Right...

let me start out by simply answering ur question

YES ITS FLAWED

now i shall tell you why unlike all the other posts i laugh at...

MATHMATICS is a mesurment of a givin object

let me point out 2 things that happen in maths that SHOW why infact its flawed

infinity / PI / symmertry

MATHMATICS LOOPS

understand that? IT LOOPs

all my god damm life i have asked the very question WHY we have mathmatics

i may not be rich i may not be famouse but please please understand one thing

I UNDERSTAND WHY AND HOW WE USE MATH

ITS A TOTAL JOKE

the reason you get 909999999999999999999999999. or 333333333333

is becouse ur asking a question that leads into infintiy.. sounds oodd? lol why would it?

or shall i show u some stupid equation? !STUPID!

mathmatics is here becouse our small little minds can not deal with the fact the universe is infinante becouse we are live..

try this out if you want:

22/8
22/7

do you understand my question?

we apply maths to everything other than US

i may sound crazy... but so did the guy who told u the world is flat

and in years to come i shall be proven right.. that is my legercy

the universe mad me to show u WTF math is



my sig is there for a reason

... ohhhhhh dear...

Something that needs to be made clear here is that just because something is comprised of infinitely many pieces does NOT mean that it is infinite. I'm calling on Zeno's Arrow Paradox here. Just because a number has the mystical "..." attached to the end of it does not render it unusable in standard mathematics - it simply means that it has an infinite decimal expansion.

Furthermore...

ALL NUMBERS ARE INFINITE DECIMAL EXPANSIONS.
This is true and has been since the day you were born, and most likely well before that. "1" is shortened notation for "1.00000...", 4/5 is shortened notation for "0.80000...", and "pi" is shortened notation for "3.14159...". These are three different numbers answering to three differing number sets - whole, rational, and real, and all THREE of them follow this rule. No matter what set your number belongs to, there are several ways to write it - in convenient shorthand (such as 1, 2/2, .5*2,) or with its infinite decimal notation (good luck writing out something comprised of infinitely many parts, like 1.00000000000... [and by 'write it out, I mean write out an infinite number of zeroes. Good luck.])

Now, to business...
.999... = 1.

It's true, I'm sorry if it disturbs or disappoints any of you. .999... is another way of writing the shorthand form of 1, and there are many mathematical proofs that prove this.

Take the Additive Property, for example. It states that any real number "a" plus the number zero will equal "a." Therefore, [a+0=a]. By rearranging this equation, we get what I like to call the Subtractive property - any number minus itself equals zero [a-a=0]. NOW, remember that "0" is actually the shorthand of 0.000... SO, when we plug the two different forms of writing the number "1" into the equation (1 and .999...), we get this:

1-.999... = 0
0.000... = 0
Therefore .999... = 1.

Imagine that.
But there should be a "...001" at the end of that "0.000..."!
The entire point of the "..." is that the expansion goes on forever, without end. There IS no end to tack the "...001" onto.

The Limit Argument
A sequence can have ONE and ONLY ONE limit.
Let us observe the sequence:
.9
.99
.999
.9999
...

therefore, the sequence gets closer and closer to .999... infinitely close.
It also gets infinitely close to 1. Therefore,
.999... = 1.

I cannot find for the life of me why THIS algebra is being disputed:
let x = .999...

Therefore,

10x = 9.999...
Subtracting x from both sides (legal) gives us

9x = 9.999...-x
Substituting ".999..." for 'x' on the right side (which is completely legal) gives us:

9x = 9.999... - .999...
9x = 9
x=1.
Above is pure, completely correct algebra.
But we already said that x=.999... therefore 0.999... equals one.

FOR THOSE OF YOU THAT DISPUTE THE ALGEBRA:
Assume that .999... is less than or equal to 1.
Let us next assume that .999... does NOT equal 1. (INCORRECT ASSUMPTION)
Therefore, .999... < 1.
Therefore, there should be some positive number N such that
.999... + N = 1.
HOWEVER, for ANY positive number N,
.999... + N > 1.
Therefore, one of our assumptions is wrong. I'll give you a hint: It's the second one.


".999... = 1" is INDISPUTABLE. There are NO PROOFS ANYWHERE THAT OPERATE UNDER STANDARD MATHEMATICS that can make this equation false. I've checked, but if you'd like to yourself, go ahead. Religious reverence of the concept of infinity or plain old disbelief does not make you correct - neither does the fact that 'so and so' agrees with you. Proof is everything in mathematics - and there are no proofs to the contrary. I rest my case.

Sam Hughes has an excellent article on this subject at his website "Things of Interest." [qntm.org]. I advise anyone who disputes this truth to look up the arguments posted there.

[edit on 14-9-2009 by ARandomGuy]

[edit on 14-9-2009 by ARandomGuy]

[edit on 14-9-2009 by ARandomGuy]

Some people in this thread are a little bit confused.
There are more than one of systems of measurement because some are more appropriate for measuring than others are in certain situations. The decimal system is _GREAT_ for dividing things by 10, by 5 etc.
It is not good at all for dividing things by 3s.
For dividing things by threes you use a 12 digit system (eg. 1 foot = 12 inches)

1/3 can not be represented in decimal form. It's impossible unless you use the repeating symbol. So .33333333 repeating does equal 1/3. However, .99999999 repeating does _not_ equal 1. 1/3 + 1/3 + 1/3 = 1/1. Whoever said .333repeating + .333repeating + .333repeating equals .999999 repeating did not add up all of the decimal places!
But I can see how it is easily confused.

Originally posted by politHowever, .99999999 repeating does _not_ equal 1. 1/3 + 1/3 + 1/3 = 1/1. Whoever said .333repeating + .333repeating + .333repeating equals .999999 repeating did not add up all of the decimal places!
But I can see how it is easily confused.

Can you please post your proof that .333... + .333... + .333... is not equal to .999...?

I do remember reading a while back a good discussion that attempted to prove that there is nothing in this existence with a value less than 1. I'm not exactly in the position to argue this point, so I'll try to dig up the thread/link.

Originally posted by MrDead
.999999 is so close to 1 that it's really not important right? You'd go on for infinity and be 1 digit away. Same as .333333333333333 etc.
It's such a tiny amount that you could never reach, and nothing is perfect anyway.

Your fallacy is that you are not accounting properly for infinity. There is no 'final digit' that will be 'one away'. It never ends. That is why mathematics requires the use of "limits" to handle situations involving infinite reputation. Utilizing limits, it's easy to see that since the .999... is repeating infinitely, there is no difference between .999... and 1.

Originally posted by OXmanK
And your first problem has a mistake.

10 * .999 = 9.99

and

10x - x = 8.991 not 9

You are not working with the proper formula for the explanation. It is not .999, it is .999... which means that the 9s repeat to infinity.

10 * .999... = 9.999... Since the nines repeat infinitely, there is no reduction of the value after the decimal point. It is still .999... to infinity.

Using that, you find that .999... does equal 1.

Let x equal .999... (repeating to infinity)

10 * x = 10 * .999... = 9.999...

10x - x = 9x.

In this calculation, that is the same as: 9.999... - .999... = 9

Now, we've shown that 9x = 9.

Divide both sides by 9 and you get: x = 1.

Therefore, .999... = 1

QED.

Originally posted by American Mad Man
Wow - this actually has my attention now - and I usually HATE math

OK so taking 10x - x = 9x, how did you get to 9x = 9?

As I recall, you can manipulate both sides, so long as you do the same to both (although I do recall there being exceptions to this rule).

If you were to devide by x, that would give you:

(10x-x)/x = 9x/x

this would then become 10x/x - x/x = 9x/x, each "x" cancels the other out (x/x = 1), thus:

10 - 1 = 9

so I don't think that does anything.

Your argument is flawed based on the mathematical Law of Equality.

10x - x = 9x.

This is true in the same way that 3x - x = 2x.

Example: Let x = 2

3(2) = 2 + 2 + 2 = 6
3x - x = 3(2) - 2 = 2 + 2 + 2 - 2 = 4
2x = 2 + 2 = 4

This relationship will be constant for any value of x.

Want another way to think of it? yx - x = (y-1)x.

To prove that, all you have to do is multiply through the right side.

Therefore, 10x - x = 9x.

QED ^.^

Edit to add some more information.

The reason we get 9x = 9 is simple as well. For reference, I'll write the entire proof here and put in some notes where applicable.

PROOF
---------
Let x = .999~

//We are declaring the variable x to be equal to .999~ where the nines repeat to infinity.

10x = 9.999~

//We multiply both sides by 10 which shifts the decimal one place to the right. It is important to note, however, that because the 9s repeat to infinity, they continue to do so after the multiplication. In essence, the 9s following the decimal are equivalent in both .999~ and 9.999~ due to the properties of infinitely repeating sets.

10x - x = 9x
//Now, as I described above, yx - x = (y-1)x through the Law of Equality.

9x = 9
//Now, we've shown that 10x = 9.999~ and x = .999~. If you subtract .999~ from 9.999~, you get 9. IE: 9.999~ - .999~ = 9

9x/9 = 9/9
//Divide both sides by 9.

x = 1
//Result shows that x = 1.

.999~ = 1
//Since we started out by setting x equal to .999~ repeating to infinity and have shown that x is equal to 1, by the Transitive Property, we have proven that .999~ is equal to 1.

QED ^.^

[edit on 14-9-2009 by BriggsBU]

Bleh. How come it always seems that when I make a knowledgeable, informative post, the thread goes dead? It's really troublesome. No one ever wants to discuss my delightfully insightful posts.