I'm not really into math, but a friend brought something up to me today that really seemed very strange. (For the duration of this post, .999 will mean .9 repeating unless otherwise specified- just for the sake of ease)
.999=x
10x=9.999
10x - x = 9x
9x=9
1x=1.
.999 = 1.

And 1/3=.333
.333 x 3= .999

The first one didn't phase me too much, because the obvious conclusion in my mind is that you can't substract an infinite number. If you ask what happens when you do the impossible, you're bound to get a strange number.

The second one however involves a more interesting flawed assumption it would seem. 1/3 does not equal .333 repeating. You check an equation by reversing it- if the reverse doesn't work, how can it be true? It would seem, that we don't have the numbers to accurately express the answer.

As far as I can tell, we can't have a perfect system of maths unless there is a "lowest number" which can not be divided, subtracted from, or inverted to a negative.

I was wondering if anybody else had any insight on the matter.

[edit on 24-3-2005 by The Vagabond]

A logical system can be both complete and consistent, however it doesn't necessarily have to be both, and, in fact, it rarely is both. A logical system that is both cannot be large or else it will not meet one of the criteria.

So now let's describe what I mean by consistent and complete:

A consistent system is one that always produces the same and predictable results from the same equations. Meaning, it's always the way that it is. You'll never get 5 from 2+2 no matter how many times you do it. Multiplication always produces the same results when the same numbers are used. Results don't vary when the same set of data is used by the same operator.

A complete system is one that has a proof for every single equation or relationship. A complete system is not possible when things become complicated, such as involving numbers. Some relationships are just how they are. Everything in math is made up by humans. Everything. But that doesn't mean that it isn't consistent. Therefore, the equations we don't have a proof for are still consistent, so we can still use them.


There is absolutely no way for a system to be both complete and consistent, unless it is stripped down to a point at which it can be, but at that point it isn't as robust as it once was and has limited usefulness. Arithmatic is not one of these systems. It is far too complicated.

So in conclusion, I'd have to say, well, deal with it

i hope that helps

Cheers,
nufan



[edit on 24-3-2005 by nmuxfpaxn]

.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.

The main reason you cannot find the logical result for your question is because you and I cannot completely grasp infinity. Sure we both have a good idea, but it is nothing that can truly be comprehended. Our brains would go on forever to fully understand it. Because we must have a finite answer. It is like trying to fully grasp a four-dimension object while living in the third. And on a personal matter, I think that it makes many physicist go nuts.

And your first problem has a mistake.

10 * .999 = 9.99

and

10x - x = 8.991 not 9

EDIT: wow....some grammar issues...blah!

[edit on 3/24/2005 by OXmanK]

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

And your first problem has a mistake.

10 * .999 = 9.99

and

10x - x = 8.991 not 9

As i mentioned before actually going into the math, it was to be understood that .999 was .9 repeating.
.9 repeating, times 10, would in fact be 9.9 repeating right?

[edit on 24-3-2005 by The Vagabond]

In advance, it has been some time since I have had to do any type of math (other then the usual BS), though I was decent in my day, please correct me if I am wrong


the problem with your 1/3 x 3 is that is not how it should be stated. If you are representing in fractions of 3, the number "3" is not stated correctly.

thus, 1/3 x 3 should be expressed 1/3 x 9/3, or 1.

For 10x (where x=.9 repeating), you have:

10x - x= 9x

then you have:

9x=9

this is wrong.

You are solving the multiplication first instead of the addition (or subtraction).

It should be:
10x(where x=.999) - x = 9x, at this point you move the "- x' to the other side by adding "x" to both sides.

thus, 10x - x becomes 10x - x + x (and the two "x's" cancel out)
on the other side, 9x becomes 9x+x or 10x

Thus:

10x=10x

Now you devide both sides by 10, so you have 10X/10 on both sides. This simplifies to "x"

Thus:

x=x

And since x=.999,

.999 = .999 = x

Now, I could be completely wrong, I haven't had to do any math since I was a high school calc student

EDIT: for better explination

[edit on 24-3-2005 by American Mad Man]

.999=x Inits
10x=9.999 Multiplied by 10 on both sides.
10x - x = 9x Opps.... While the statement is true it is a non-connected system not in relation to the former operation...

alternate yet legal manipulation:
10x -9.999 = 9.999 - 9.999 (subtract by -9.999~ on both sides)
10x - 9.999 = 0 (simplified above operation)
(9x + x) - 9.999 = 0 (disunioned 10x into (9x+x))
9x - 9.999 = -x (subtracted both sides by x)

(9x-9.999)/-1 = (-x)/-1 (divided both sides by -1)

-9x+9.999 = x (simplified above operation) (result sign change)
(-9x+9.999)/9 = x/9 (divided by 9)
(-x+1.111) = x/9 (simplified above operation)
(-x+1.111)+x = x/9 + x (added x to both sides)
1.111 = x/9 + x (simplified above operation)
(1.111) * 9 = (x/9 + x) * 9 (multiplied both sides by 9)
9.999 = x + 9x (simplified above operation)
10x = 9.999 (re-added 9x + x which would be 10x)
(10x)/10 = (9.999) /10 (divided by 10)
x = 0.999 (final result matches *inits*)

still comes back in one peice.. the gotcha was in the above (marked opps) operation...

[edit on 24-3-2005 by Crysstaafur]

Originally posted by American Mad Man
the problem with your 1/3 x 3 is that is not how it should be stated. If you are representing in fractions of 3, the number "3" is not stated correctly.

thus, 1/3 x 3 should be expressed 1/3 x 9/3, or 1.

Unfortunately, I don't seem to have a divided symbol on my keyboard, and I was using the slash for that, i was not trying to take a fraction over to decimal.
So 1 divided by 3 equals .3 repeating (or so my calculator claims)
but .3 repeating multiplied by 3 equals .9 repeating.
Therefore either 1 and .9 repeating are equal, or 1 divided by 3 does not equal .3 repeating. There is apparently a strong school of thought in mathematics, or so I'm told, which chooses to believe that 1=.9repeating.

For 10x (where x=.9 repeating), you have:

10x - x= 9x

then you have:

9x=9

this is wrong.

You are solving the multiplication first instead of the addition (or subtraction).

I've never heard of solving addition and subtraction first. Are you sure that's right? I'm no math major, but that's sure as heck not what we did in highschool.

I do find it interesting that reversing the order of operations makes it work. I'm going to think that one over to see if that's whats going on. (I'm confident that to do so would be a deviation from standard proceedure though, because I got this problem from a friend of mine who has been arguing at length with a math professor who assures her .9 repeating equals 1.

nah.. 1 and 0.99999~ (tilda to represent an eternal chain of digits) are still different.

However, I do remember on a math related site some time ago that had a goofy yet correct proof that 1 = 0. lol

Originally posted by Crysstaafur
.999=x Inits
10x=9.999 Multiplied by 10 on both sides.
10x - x = 9x Opps.... While the statement is true it is a non-connected system not in relation to the former operation...

I'm going to run your explanation by my friend so she can pass it back to whoever gave us this little headache. I'm interested in seeing what they say.

I could use a better explanation of the gotcha there. Why exactly can that true equation not be applied? I don't see what causes the disconnect.

Like I said, it has been 10+ years since I have had to do any kind of math other then figuring out if a piece of property is worth the intrest I would have to pay to buy it

Crysstaafur seems to have a good grasp on this, perhaps he (she?) could clarify.

This is how I recall doing algebra, and I very well be mistaken.

(In retrospect, maybe I would have been better off taking some math courses in college rather then excepting their pass )

.999=x Inits Legal
10x=9.999 Multiplied by 10 on both sides. Legal
10x - x = 9x Opps.... While the statement is true it is a non-connected system not in relation to the former operation.. Not Legal

In the above, you have completely disregarded the repeating 9.
The only thing achieved here is a separate equation that says
(Unknown multiplied by 10) subtracting an unknown of simular scale should yield
Unknown multiplied by 9. Unrelated.

If you were trying to include an x on the repeating side, you would wind up
multiplying 10x by x which would be 10x^2
ergo
10x=9.999 Multiplied by 10 on both sides. Legal
10x^2 = 9.999x Multiplied by x on both sides. Legal

The big gotcha was the disregard for the repeating 9. simple.

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.

Originally posted by Crysstaafur
.999=x Inits Legal
10x=9.999 Multiplied by 10 on both sides. Legal
10x - x = 9x Opps.... While the statement is true it is a non-connected system not in relation to the former operation.. Not Legal

Bear with me here please

Please explain (in lamens terms if you can :lol why this is not legal.

In the above, you have completely disregarded the repeating 9.
The only thing achieved here is a separate equation that says
(Unknown multiplied by 10) subtracting an unknown of simular scale should yield
Unknown multiplied by 9. Unrelated.

It seems to me that it should still hold true. For instance if you were to put any fraction in for X (even one that repeats), 10X-X would still equal 9X

If you were trying to include an x on the repeating side, you would wind up
multiplying 10x by x which would be 10x^2
ergo
10x=9.999 Multiplied by 10 on both sides. Legal
10x^2 = 9.999x Multiplied by x on both sides. Legal

The big gotcha was the disregard for the repeating 9. simple.

Oh good lord, I knew that HS math was going to come back to haunt me

(a)10x=9.999 Multiplied by 10 on both sides.
(b)10x - x = 9x Opps.... While the statement is true it is a non-connected system not in relation to the former operation

Where did the 0.999 go to????? Where did the surplus of unknown's come from on the 9's side??
I hope both questions above give a gentle hint.


that is the key that makes them (a&b) disconnected and unrelated.

[edit on 24-3-2005 by Crysstaafur]

Actually, I think people should understand that no number system is going to be flawless but it can still be very functional.

The fact that .999... is equal to 1 makes sense when you look at it correctly.

1.) Similar to the concept of Time, the number line has NO BREAKS, we divide it up however in order to make it functional. There is no break point between Yesterday and Today aside from the one we intentionally agree upon in our minds. Yesterday and Today are in fact connected and flow seamlessly together. Numbers like Time flow together like a stream not like drops or puddles.

2.) Numbers such as .999..., .333..., .142857..., Π-Pi, Φ-Phi, √2 and so on are also not "Real" numbers in the same way we've defined "Real Numbers" like 1, 2, 3 etc... (Personally, I think they are actually much more "Real" than our official "Real Numbers", but that's a much too detailed to get into now.)
Even 0-Zero isn't a "Real" number that can be used for an exactness like what you're looking for. Therefore you should expect some problems when mixing different sets of Numbers together that don't always play well together.

3.) Even though for most people our Number line would seem Infinite since it goes on and on, which can easily be demonstrated, there is a subtle trick being used. For example:

1,2,3,4,5,6,7,8,9...goes on and on forever, right? Or does it?

100, 101, 102 etc...while being a hundred points beyond our start, when added together as single digits still produces 1+0+0=1, 1+0+1=2, 1+0+2=3 etc... 245 = 2+4+5 = 11 or 1+1 = 2 & 244 equals 10 or 1 as well.

So as you can see by this very simple example which is self evident to most everyone, which is also why it's missed by many, Infinity doesn't exactly mean it's as big as one imagines. Like fractals, it may be that it's just a whole bunch of "Self Similar Cycles" that make it seem that way.

[edit on 24-3-2005 by mOjOm]

Ok, if a pattern that cycles through a chain of seemingly (or directly) infinite sequences of values, how many cycles could potientially exist? lol

Numbers are in and of themselves fundamentally linked to quantity, therefore anything that we try to quantify such as Time will behave simularly

Digital concatenation is not within the boundaries of accepted technique when one has to consider amounts known and unknown as finite quantities.

Infinity will always be larger than what we can imagine.

When one is inclined to round 0.999~ to the nearest whole finite number (1), then yes. If 0.999~ is to be treated as is with no rounding at all, then we have a different story.


Although there is a distinct difference between whole and real numbers, one is a sub-set of the other.

(Infinity one way).................(Infinity in two ways)...............(infinity in all but 1 direction)
Wholes(0,1,2,3,4,5,...) -> Integers(...,-3,-2,-1,0,1,2,3,...) -> Reals(all except those who are involved with a square root of negative one in some way)

(infinity in one direction??)
Imaginary (all off on it's own, based on the impossiblity to simplify square root of -1)

I will admit that 0 can be a weird construct at times in that it is simply a placeholder for nothing literally. Can also be a *neutral* location in a given field of integers. ergo, it is possible to find logic glitches between 0 and some other value...


[edit on 24-3-2005 by Crysstaafur]

Well I thought i answered 0 pretty well above in my post... it doesn't exist nothing can be nothing because in order for it to be nothing, it becomes something..nothing

there is no 0 the first is 01

[edit on 24-3-2005 by Mayet]

[edit on 24-3-2005 by Mayet]

Imaginary (all off on it's own, based on the impossiblity to simplify square root of -1)

Did you mean it could not be simplified as a digit? Because the square root of -1 can be simplified into the letter i.