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# .9 repeated is equal to 1.

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 Voting Style: Open Point System: 7 Point Started: 4/7/2011 Category: Science Updated: 7 years ago Status: Voting Period Viewed: 5,837 times Debate No: 15833
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39 comments have been posted on this debate. Showing 1 through 10 records.
Posted by gizmo1650 7 years ago
worst place for the indent to change.
Posted by gizmo1650 7 years ago
wow, I am actually going to have to vote con on this. For anyone interested, the simple algebraic proof is:
let x=0.(9) | an infinite amount of nines
x=0.9(9) | 9 followed by one less than in infinite amount of nines, or simply followed by an infinite amount of nines
10x=9.(9)
-{x= 0.(9)} | this was our definition of x
9x=9 |subtract the two equations
x=1
x=0.(9) | again, by definition
1=.(9) | transitive property
Posted by RoyLatham 7 years ago
twsurber, 1 is to the right of the number you referenced. However, 0.999... = 1 exactly.

I think you are up against the problem of accepting something that is true, but not intuitive. Modern physics is loaded with such things, such as the non-intuitive fact that nothing can exceed the speed of light. Einstein made the claim and it has been well proven, but it doesn't seem to make any sense. With many such things, it is better to accept the evidence over your intuition. In this case let x = 0.999... then 10x = 9.999..., 9 x = 9, and therefore x = 1. Q.E.D.

There are many other things in math that may not be intuitive. The four-color map theorem may not seem true, but it is. Or perhaps 1/3 = 1/4 + 1/16 + ... The tricks used to sum infinite series are like that used to prove 0.999... = 1. One you get used to applying them, you get to trust the result.
Posted by twsurber 7 years ago
Question for cowpie. I agree that 3/3 equals 1 whole. I am not seeing that .333 equals a third though, unless you are going off the infinity principle Roy spoke of.

Let's say that I have a dollar in pennies. If I count out 33 pennies I have .33, but I do not have a third. In order to have a complete third I would need 33 and 1/3 pennies.

If I had 3 groups of 99 pennies would be .99, therefore I would not have a whole dollar. I would need to have 3 groups of 33 and 1/3 pennies to make a full dollar.

To Roy & Cowpie:
I'm seriously not trying to be sarcastic, just sharing another view.

Should I accept a universally accepted mathematics principle as absolute truth if I still have a question that has not been answered satisfactorily? Do I not have the right to ask why? It's not exactly circular reasoning, but I am being asked to accept something as fact that another has presented while there still remains a legitimate question about it's validity. While .999 infinite does go on indefinitely, it can never actually achieve a full 1, but rather is merely "expressed" as 1.

I concur & accept that (.999 infinite) is "expressed" as 1 whole, in fact I would likely not say \$999,999.99 but rather would just say a million dollars for simplicity. However, don't both .999 infinite and the whole number 1 each have a separate and respective position on the number line?
Posted by twsurber 7 years ago
Roy,

I am arguing that every number has it's own place on the number line. The whole number 1 has it's place to the right of: 0.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
Posted by RoyLatham 7 years ago
twsurber, I have no idea what you are trying to get at. Yes, arithmetic works. It is a fact that every mathematics text says and every mathematician understands that 0.999... = 1, and that is true whether you agree or not. Your job is to prove they are all wrong. A Nobel Prize awaits.
Posted by RoyLatham 7 years ago
Gizmo, The problem 0.999... = 1 is posed with real numbers. "In the 1960s Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated." http://en.wikipedia.org... In other words, if you have discovered an inconsistency in real numbers, you also shoot down the hyperreal numbers. Your claim is that real numbers are inconsistent.

Your hangup is with repeating decimals converging. You agree that the repeating decimal 0.111... [base 10] is exactly equal to 0.1 [base 9]. You are correct that changing bases cannot change the result. Therefore repeating decimals converge.

Have you taken differential calculus yet? Do you agree that it is invalid if limits do not work?
Posted by gizmo1650 7 years ago
sorry, I meant |x| < 1/n for all positive n.
Posted by gizmo1650 7 years ago
@Roy
I agree that their is no real number x. The nearest real number to x would be 0, also referred to as st x, or x's standard part. As 0 and x differ only infinitesimally, and infinitesimals do not exist in the real number system, do you see how .(9)=1 in the real number system, and .(9)=1-x in the hyperreal system.

As far as bases go, changing bases should not change your result.
Posted by RoyLatham 7 years ago
Gizmo, The x for which |x| < 1/n for all real n is x = 0. You understand that if you say that lim 1/n is not zero, then differential calculus is invalid. So you are arguing that calculus is all a big mistake, right?

I was referring to notation systems. Real numbers have many alternate representations. For example, do you agree that 0.1111 ... [base 10] = 0.1 [base 9] ?
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