lets see....

Damned mathematicians thinking there is a diffrence, us engineers know it doesn't matter.

[quote name='psyclo' date='Sep 17 2005, 01:43 PM']Damned mathematicians thinking there is a diffrence, us engineers know it doesn't matter.


[/quote]

mathmaticians know theyre the same.



lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1

0.9999... = 1



Thus x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999...

9x = 9

x = 1.

You're implying that 10x has an extra figure, 10x - x should really be 8.999999....1



Mark

WTF that's just sad. Next time I'm going straight to bed when I get home.



Mark

thats only as it approaches infinity. It never truly gets to inifinity its just an approximation.

http://mathforum.org/dr.math/faq/faq.0.9999.html



and also:



http://www.straightdope.com/columns/030711.html

[quote name='7lufnis' date='Sep 17 2005, 03:43 PM']thats only as it approaches infinity. It never truly gets to inifinity its just an approximation.


[/quote]

its not constantly approaching it. its a set value.

Allright one last time.



in math we usually use base ten



hence 121 is read as 1*10^2 + 2*10^1 + 1*10^0 = 100+20+1 = 121



and 0.1 = 1*10^(-1)



However 10 is a somewhat arbitrary decision. Some ancient people did math in base 6. For my example we will use base 3.



121 in base 3 is 1*3^2 + 2*3^1 + 1*3^0 = 9+6+1 (base 10) = 16 base 10





Now consider 1/2 in base 10 this is written as 0.5. It has a definite value. In base 3 however it is written as 0.111~ because it cannot be represented with finite digits. The value is still exactly 1/2 not approaching 1/2 but exactly 1/2 but is symbolized by 0.111~.



Similarily in base 10 1/3 is written 0.333~. This is exacltly 1 part of 3 however one cannot write it in finite decimal expansion in base 10. In base 3 however the same number is 0.1



Finally 0.9~ is just one way of representing 1 in base 10. It's value is well defined like 1/2 is 0.11~ in base 3 is well defined. We do not do a function on 0.9~ nor is it rounded or an approximation. It is simply a valid representation of 1 in base 10



By the same argument 0.2222~ = 1 in base 3.

It means there's a squigly and you should have a margarita.

Infinity is not a number it is a concept. You can't technicaly do algebra with infinity.

[quote name='psyclo' date='Sep 17 2005, 06:38 PM']Infinity is not a number it is a concept. You can't technicaly do algebra with infinity.


[/quote]

were not doing any with infinity.

just apply it to something tangible...



will .9~=1 give me more HP?

You don't need modular arithmetic to solve this one. If you want to take that path, we know .333~ is a third, and .666~ is 2/3 - but on that basis we can't assume that .999~ is 3/3, i.e. if we try to perform operations on a number with recurring decimals because you're always going to be out by 1/infinity.



If you multiply something by 10, you're effectively moving the whole number across the decimal point one digit. Unless you convert .999~ to a fraction/different base etc. you're going to be out by 1/infinity when you try to do this, because you can't multiply a number and get a result that's more accurate than what you started out with. So the best you can do is the following:



0."infininte amount of 9's" x 10 = 9."(infinte - 1) amount of 9's"



so what you have is:



9.999...99

0.999...999 -

8.999...991



and 8.999...991/9 = 0.999~



Now your arguement states that "infinity" and "infinity - 1" are the same figure, which is essentially right - since they are both undefined, you have the problem psyclo talks about. Think of it in terms of distance, where we have 0.999~ millimeters, but instead of the 9's recurring to a negligible distance, have it the other way round where you have 1 millimetre as the negligible distance and the recurring figure is the units. So instead of the millimeter tending towards nothing, you have the units tending towards light years and multiples of that, you're always going to be out by 1 millimetre.



Mark

infinity - 1 is still in finity. theres always an infinite amount of 9s to the right of the decimnal, regardless of how many of them you move to the left of it. and this number only exists because we use base10 in our number system. look up to where i posted the argument in base3 and it works out just fine.

Your still treating infinity like a number, which its not.

no, he isnt.

it doesnt matter what u do .9~ does not = exactly 1 it is the closest you can get and there is no real number that can express infinite so it has to = 1 but rally it doesnt actually = 1 its just the closest you can get to 1.

infinity has nothing to do with this equation.

.999... is not infinite. and does not approach anything. It is finite and has a very well-defined, fixed value.



It is a nonterminating repeating decimal. This means it is a rational number, which means it can be expressed in the form a/b where a and b are integers and b != 0.



The only possible way to satisfy those requirements for .999... is if a=b, and a/b = 1.





Proof 1:

The set of rational numbers is a member of the set of real numbers.

Any infinitely repeating decimal is a rational number.

0.999... is an infinitely repeating decimal.

0.999... is a real number.

The set of integers is a member of the set of real numbers.

1 is an integer.

1 is a real number.

0.999... and 1 are both real numbers.

A property of the real numbers is that between any two distinct real numbers A and B, B>A, there are an infinite number of real numbers greater than A and less than B.

If 0.999... and 1 are distinct real numbers (not equal to each other), there are an infinite number of real numbers greater than 0.999... and less than 1.

There are no numbers greater than 0.999... and less than 1.

Therefore, 0.999... and 1 are the same number.



Proof 2:

1 = 1

1/3 = 1/3

1/3 = 0.333...

3(1/3) = 3(0.333...)

3/3 = 0.999...

1 = 0.999...



Proof 3:

x = 0.999...

10x = 10*0.999...

10x = 9.999...

10x - x = 9.999... - x

9x = 9.999... - 0.999...

9x = 9

9x/9 = 9/9

x = 1

0.999... = 1



Proof 4:

0.999... is an infinite geomtric series of the form:

Code:



∞ ∑ a * r^n n=0







The value of an infinite convergent series is its sum.

An infinite geometric series of the above form converges if r < 1.

0.999... in series form is:

Code:



∞ ∑ .9 * (1/10)^n n=0





a = 0.9 and r=1/10.

Because r < 1, the series converges.

The sum of a convergent geometric series is a/(1-r).

The sum of this series is 0.9/(1-1/10) = .9/(1-.1) = .9/.9 = 1.







A repeating decimal can be converted to its fractional form like this (it's a shortcut derived from proof #3 for repeating decimals where everything to the right of the decimal is part of the repeating sequence) :



First, find the period of repetition and call it n:



.142857142857142857... has six digits that repeat, so n=6

.123451234512345... n=5.

.234234234... n=3.

.090909... n=2

.333... n=1

.999... n=1.



Take the repeating sequence and call it a:



.142857142857142857... a=142857

.123451234512345... a=12345

.234234234... a=234

.090909... a=09

.333... a=3

.999... a=9



Take n nines and call it b (mathematically, b=10^n-1):



.142857142857142857... n=6, 10^n=1000000, b=999999

.123451234512345... n=5, 10^n=100000, b=99999

.234234234... n=3, 10^n=1000, b=999

.090909... n=2, 10^n=100, b=99

.333... n=1, 10^n=10, b=9

.999... n=1, 10^n=10, b=9



Form a fraction with a as the numerator and b as the denominator:



.142857142857142857... = 142857/999999

.123451234512345... = 12345/99999

.234234234... =234/999

.090909... = 09/11

.333... = 3/9

.999... =9/9



And reduce:

142857/999999 = (3*3*3*11*13*37) / (3*3*3*7*11*13*37) = 1/7

12345/99999 = (3*5*823) / (3*3*41*271) = 4115/33333

234/999 = (2*3*3*13) / (3*3*3*37) = 26/111

09/11 = (3*3) / (11) = 9/11

3/9 = 3 / (3*3) = 1/3

9/9 = (3*3) / (3*3) = 1



But some people seem to think .9 repeating is a special case - the ONLY repeating decimal in the entire set of rational numbers where this doesn't work.



and like ive said probably three times already, if you change the number from base10 to base3 and do the problem, then it works out just fine. proofs are things that are 100% true, and here are four of them.

^^ some kid tried to prove that .9~ =1 in one of my math classes and the teacher just shhut him down by mathmatically disproving him... and i.9~ isnt a special case .x~ =.x~ nothing else.

In the end it doesnt realy matter, given enough digits the diffrence is so small as to be of no practical use. Its a mind game for board mathematicians.

One of my math teachers and I had a discussion aboot this. There are actually different types of infities, and I may have to take a class that is like mathematics of infinity. You can do some weird calculations with it apparently. But yes Eric, by the laws of limits .99~ is 1.





I also had a discussion with one of my engineering professors who said that it is impossible to ever get to exactly the right position with a hole or something of that sort. That you will always be at .99~ or .000001. I tried to reason that at some point you MIGHT hit exactly 0. But he's still pretty much right.



- Hand

0.999... = 1 - 1/∞

since 1/∞ does not equal 0 (however close it may come),

0.999... does not equal 1.



QED.



Mark

you cant divide a number by infinity.



if you cant prove that the proofs are wrong then they are correct, and if they are correct then .9~ = 1

Get a damn job.

he does, he sells knives.

I would like to take this moment to say that our little boy, Eric, is growing up. He has developed reasoning skills. Honey I'm so proud of you. Just letting you know that mama loves you.



- Hand

https://www.nopistons.com/forums/pub...1047683549.gif (i just wanted to use this because i dont know what else you would use it for)

It's just a representation of a number that tends towards 0.



Mark

yay i can vote!



i thought i couldnt because i'm a minority