Curse

Sol_Vie · Feb 09, 2006, 11:24 PM // 23:24

Err is this an actual debate? .9999 does not equal 1. It equals .9999

When someone rounds up in such cases it's for convenience and for the fact that it's not entirely practical to always be so exact.

PieXags · Feb 09, 2006, 11:27 PM // 23:27

Quote:

Originally Posted by Sol_Vie

Err is this an actual debate? .9999 does not equal 1. It equals .9999

When someone rounds up in such cases it's for convenience and for the fact that it's not entirely practical to always be so exact.

That's one way of looking at it, Sol.

Heheh, I seem to recall saying it's a matter of acceptance.

Sol_Vie · Feb 09, 2006, 11:30 PM // 23:30

=\

This semester I'm taking my third stats course (same stuff all three classes, all for different credits... why must I live through this crap again?) and we had a similar discussion that I abruptly ended (once I started paying attention) with a similar argument.

It's practicality that is the cuplrit here.

Rayndeon · Feb 10, 2006, 12:40 AM // 00:40

I:

Quote:

Originally Posted by PieXags

Heh, first of all nowhere in the original post of this thread does it state that we're dealing entirely within the standard system of mathematics. So don't try to tell me that simply because PieXags enjoys new means of thinking that it doesn't work or is incorrect, when it's in this thread my options are entirely valid---whether they work within our standard system or not.

The standard system is used far more than nonstandard system. Thus, it has become convention that a mathematician state that he is working within a nonstandard system. Otherwise, it is uniformly assumed to be in the standard system.

II:

Quote:

Originally Posted by PieXags

I could write on and on about how yes, my presented idea is valid and I could sit here and take my time to form a system around it. But for someone you cares so much about logic, you're not being very logical. Why would I sit here and take up ANY of my time thinking of examples and more ways to represent my idea just to please you, because you'll never dare to move from the standard system we've put in place. Thinking in such "dangerous" ways that contrast existing systems have in the past benefited mankind in so many ways I couldn't begin to list them all, so please refrain from trying to tell me such ways of thinking are bad. For the purpose of this thread, they are entirely valid. I'm free to move from the standard system all I want, to represent new ideas and theories so much as I can. Some people can accept them, and some people can't. Personally I think something much MUCH more pointless than a relative sense of thinking is spending time trying to convince someone it's not dangerous at all, so I won't bother. Trying to get you to accept the possibility is like trying to convert someone from their way of life to a new one after 60 years, it's not going to happen and it's a waste of time to try, even if if the new cultures just as acceptable as the old. So I'll consider this a finished matter, and both possibilities acceptable.

I'm not asking you to "please me," I am asking you to substantiate your claims. Your mode of thinking however is dangerous in the fact that all possibilities are equally valid. 0.9 recurring = essentially anything. Without a firm axiomatic system to support this, the statement is meaningless. I am, in the end, just asking for you to substantiate your claims.

III:

Quote:

Originally Posted by Sol_Vie

Err is this an actual debate? .9999 does not equal 1. It equals .9999

When someone rounds up in such cases it's for convenience and for the fact that it's not entirely practical to always be so exact.

The question is not whether 0.9999 equals 1. The question is if 0.9 recurring equals 1. Several proofs were given as well.

The first proof many make is the following:

Let x = 0.9 recurring
x = 0.9 recurring
10x = 10 x 0.9 recurring
10x - x = 9.9 recurring - x
9x = 9.9 recurring - 0.9 recurring
9x = 9 + 0.9 recurring - 0.9 recurring
9x = 9
x = 1
0.9 recurring = 1

Also, there is the fact that 0.9 recurring is in fact an infinite decimal expansion, like any other decimal numeral. 0.9 recurring is in fact 0.9 + 0.09 + 0.009 + 0.0009... This can be converted into 0.9 + 0.9 x 0.1 + 0.9 x 0.1 x 0.1 + 0.9 x 0.1 x 0.1 x 0.1... Which can further be converted into 0.9 x 10^0 + 0.9 x 10^(-1) + 0.9 x 10^(-2) + 0.9 x 10^(-3)... Which is summarized as Sigma k = 1 limit -> infinity 9 x 10^(-k) Which is further summarized as a/(1-r) where a here equals 0.9 and r = 0.1, thus 0.9/(1-0.1) = 0.9/0.9 = 1. Thus, Sigma k = 1 limit -> infinity 9 x 10^(-k) = 1. However, earlier we said Sigma k = 1 limit -> infinity 9 x 10^(-x) = 0.9 recurring. Thus, as A = B and B = C and A = C, 0.9 recurring = 1.

Also, by the Density Property of Real Numbers, between any two unequal real numbers, there is another real number. If we assume 1 to be not equal to 0.9 recurring, than there should be a real number between them. But the problem is that there is no real number between them! Thus, by contradiction, 0.9 recurring = 1.

Also, if 0.9 recurring does not equal 1, then 1 > 0.9 recurring. Thus, 0.9 recurring + k = 1 Thus, k = 1 - 0.9 recurring. We get 0.0000.... it is an infinitesmial! Which is equal to zero! Thus, 0.9 recurring = 1.

Also, if we know that 0.1 recurring = 1/9. Then, 0.1 recurring x 9 = 0.9 recurring. But, 1/9 x 9 = 1 Thus, 0.9 recurring = 1. The same argument may be used on 0.3 recurring and both can be proved by Sigma k = 1 limit -> infinity 1 x 10^(-k) or Sigma k = 1 limit -> infinity 3 x 10^(-k)

Could you please refute the proofs?

tosyazul · Feb 10, 2006, 12:41 AM // 00:41

Quote:

Originally Posted by Rayndeon

...Mr. Style, 0.999(...) is 1, as shown by several proofs. It is a convergent infinite geometric series, converging to 1.

Also:

Let x = 0.9999999999* (assignment of a variable)
x = 0.9999999999* (given)
10x = 9.99999999999* (Multiplication Property of Equality)
10x-x = 9.9999999999* - x (Subtraction Property of Equality)
10x-x = 9.9999999999* - 0.9999999999* (Substitution Property of Equality)
9x = 9 (Subtraction Property of Equality)
x = 1 (Division Property of Equality)
0.99999999999* = 1 (Substitution Property of Equality)

Also:

1/9 = 0.111111111.....
2/9 = 0.222222222.....
3/9 = 0.333333333.....
4/9 = 0.444444444.....
5/9 = 0.555555555.....
6/9 = 0.666666666.....
7/9 = 0.777777777.....
8/9 = 0.888888888.....
9/9 = 0.999999999..... (by inductive reasoning)

I found an interesting website discussing this: http://qntm.org/pointnine He makes strong, valid arguments.

1-0.999(...) must equal zero; otherwise it is in violation of the intermediate value theorem and thus by reductio ad absurdum, it must equal 1. Also, to say that 1-0.999(...) = 0.0(...)1 is in violation of mathematics, such as the intermediate value theorem and the axiom of infinity. By deductive reasoning, it is 1.

IX:

You cannot use infinity in arithmetic operation, strictly speaking. It rarely happens when one is allowed to and generally there are special circumstances. Also, 0.0(...)9 is again in violation of the intermediate value theorem and the axiom of infinity.

Also, mathematics is as close to truth as possible. It attempts to be as objective as possible, assuming fundamental truths and deductively reasoning from there. It is certainly the most immortal field we have at least. Aeschylus may be forgotten in time, but the mathematical statements of Archimedes and others will remain immortal, or at least they should.

The argument in bold is incorrect. It ignores the fact that the difference between 10x and 9.9999* is ten times greater than the difference between x and .9999*. To correct your argument's mistake, observe the following:

1 - 1/x = .9999*
[multiply by 10]
10 -10/x = 9.9999*
[subtract 1 - 1/x from both sides]
9 - 9/x = 9
[divide by 9]
1 - 1/x =/= 1

therefore, .9999* does not equal 1. There is still an infinitesimal quantity that does not get cancelled out.

Rayndeon · Feb 10, 2006, 12:44 AM // 00:44

Infinitesmials in the standard system are zero. http://mathworld.wolfram.com/Infinitesimal.html

Tur713 · Feb 12, 2006, 10:27 AM // 10:27

Here's an easy way to put it 1/9 = 0.11111(ect.) correct? Multiply that by 9 and you get. 0.99999(ect.). But if the problem is set up as 9(1/9), logically, that would lead to it being 9/9, thus the answer is 1. I have no idea why we even need the complicated 10x-x thing to prove this.