IAalsoNAM, and I’m still trying to figure the non-believers out on this one.
How the heck do you notate 1/3 then, in the decimal system, such that when you multiply it by three it equals exactly 1, and not .999… repeating?
BTW, I like Kabbes’s base-12 explanation.
First of all, pi is nothing like .999… Pi is an irrational number, which means it cannot be expressed by an integral fraction. .999… on the other hand is rational, and can be expressed as 3/3 or 1/1 or just 1. You’re still thinking in terms of a finite (if very long) sequence of 9s, judging by statements such as:
This is true when you round off, such as your calculator does when you ask it to tell you what 1/3 is. However, when you write the notation “.999…” you most certainly are expressing it absolutely accurately. You spoke of people having difficulty understanding infinity (and by extension, infinite series) and you are correct. You are one of them.
OK, in 0.9999… as the decimal places -> infinity, 0.9999… -> 1, Therefore if we specfically define 0.9999… as having infinite decimal places then 0.9999… = 1.
The problem with the people who have tried to prove that 0.9999… does not equal infinity is they have not definitely defined 0.9999… as having infinite decimal places.
sorry read: " who have tried to prove that 0.999… does not equal infinity" as " who have tried to prove that 0.999… does not equal one"
I hate to point this out, but mathematics is NOT about how you feel. Numbers don’t give a rat’s behind about your feelings. Mathematics is all about precision and definiteness. The terms used in mathematics are precise and well-defined.
The point is, .999… = 1, not because it is convenient to say so, it is because it is so. Period.
Yes, that is hard to understand. Mostly because it’s impossible. When dealing with real numbers, any two numbers that are not equal must have an infinite number of real numbers between them. There is no such thing as two real numbers that are the closest to each other. Real numbers aren’t like integers; Such a concept does not apply to real numbers. If two real numbers have no numbers between them, then they must be equal.
lucwarm,
You are correct!
Like pulykamell I have a hard time getting the people who don’t get it, especially considering the enormous amount of contrary evidence.
For me, grapsing the concept of a number (and that the same number can have different notations in different systems, like, say, 1 and 0.99~ :D) in a different thread on the same subject was the eye opener. I don’t know what works for “other” non believers. Keep trying.
What I meant by “feel” is that it is my opinion.
You can feel the tension in THIS particular thread, just as all “.999999… = 1” threads that I’ve seen here before. Sickening.
Well, again, opinions have no place in mathematics. The reason there is tension in threads like this is because those who understand the concept are frustrated wth those who don’t.
lucwarm is most certainly not correct, nor do I think that his post was entirely serious. 1/infinity, if it exists, must be the unique number a such that infinity * a = 1. There ain’t no such beast in the real number system; in fact, infinity is not a member.
criminalcatalog: As was stated, mathematics has fuck all to do with anyone’s opinion. What’s true is true; what ain’t, ain’t. And never, never cite dictionary definitions in a technical debate. That’s basically admitting that you don’t know what you’re talking about.
Finite time is not an issue. If the result of the computation is well-defined, we don’t care how long it takes. Hell, sometimes we even consider what would be true if something false were true!
I said it before, and I’ll say it again: a bounded monotonically increasing sequence of real numbers converges to its least upper bound. That’s all that matters here. If you don’t know what that means, you need to learn what it means before you try to argue that .9… is not equal to 1.
Q.E.D.,
In order for (0.999…) to be accurately expressed in decimal form, you must have an infinite number of 9s on the number. The “…” refers to the abstract idea of infinity, and so is not decimal notation.
Ultrafilter,
Infinity must be part of the real number system. How many numbers are there in total?
Phage, here’s the thing. Mathematics isn’t something handed down from heaven, it is something (IMHO) that humans have invented. Mathematics is about inventing rules and examining the consequences of those rules.
You are perfectly welcome to claim that 0.99999… is not equal to 1. What you have done is invent an mathematical axiom for what I’ll call Mathematics-P, for Phage. This is a perfectly fine mathematical axiom. Perfectly normal, perfectly healthy. Those other guys arguing with you have another axiom, that 0.99999…=1. And they are perfectly within their rights to claim that. Let’s call their branch of mathematics Mathematics-L, for Liebnitz. Newton gets all the credit for calculus, and I’m sick of it, let’s give Liebnitz some face time.
OK, we have two competing versions of mathematics here. But, which one is correct? In my opinion, neither is correct and neither is false. But we can look at the consequences of what happens with either axiom. It turns out that if we accept Mathematics-P, then we get some very interesting results. Under Mathematics-P, it makes a difference what number basis you use. If you use base-10 to do math, you get different results than if you use base-12. But why is that? It doesn’t make sense. If you use Math-P you quickly run into all sorts of contradictory statements. It turns out that Math-P is not internally consistent.
I’m sure that all the people who do a lot of math could list out all sorts of other consequences that show why Math-P is inconsistent. If we used Math-P to try to figure out things in the real world we’d be unable to do anything, not even add 1 and 1 to make 2.
Now, if we use Math-L, we do get consequences that are hard to understand. But Math-L is internally consistent. As long as we use the same definitions, we are able to do things like add 1 and 1, or convert from base-10 to base-12 or base-2, or figure out the volume under curves. The reason everyone is yelling at when you say that 0.9999…!=1 is that you are right then mathematics is internally inconsistent. They are just using shorthand when they say that internally inconsistent mathematics is false, when they really mean it is illogical.
Of course, if you claim that the rules of logic don’t apply to the real universe, then there is no way to logically convince you that logic MUST apply. Contrariwise, if logic doesn’t work then there is no way for you to convince the rest of us that it doesn’t work. Does that help? Maybe someone with better math skills than me could compile a list of things that prove that Math-P is internally inconsistent, maybe that would help.
C, whatever that is. But it’s most definitely not a real number, unless you can find a sequence of rational numbers that converges to C. You can’t.
And I would like to see a cite backing up your claim that infinity is a real number, preferably from a math textbook. Don’t spend too long looking–you’ll only get frustrated.
Everyone is being too smart by half. The thread began (paraphrased):
Since 1/3 = .33333…;
And 2/3 = .6666…;
And .3333… plus .6666… = .9999…;
And 1/3 plus 23 = 1, then .3333… plus .6666… must also = 1.
The fundamental statement is incorrect. 1/3 does NOT = .3333…
Rather, .3333… is the closest approximation one can write to the exact sum 1/3 using decimal numbers. The difference is generally negligible, but unlike .25 which exactly represents 1/4, .3333… is still only a very close approximation.
The same with .6666… and 2/3.
If you add the close approximation .3333… to the close approximation .6666… you will get a close approximation to the correct answer of 1/3 plus 2/3, which is 1.
No gcarroll, because we have defined that 0.3333… has infinte amount of decimal places and is not therefore just a good approximation to 1/3 but is equal to it.
Yes, it does. And I’m not saying it’s approximately equal or that it’s so close it might as well be equal, I’m saying it’s exactly equal.
.d[sub]1[/sub]d[sub]2[/sub]d[sub]3[/sub]…, 0 < d[sub]i[/sub] < 9 for all i [symbol]Î[/symbol] N is a shorthand for the infinite sum whose ith term is d[sub]i[/sub]/10[sup]i[/sup]. As you may recall from your analysis classes, the value of the infinite sum whose ith term is s[sub]i[/sub] is the limit as n increases without bound of the sum from s[sub]1[/sub] to s[sub]n[/sub]. You may also recall from your algebra classes that the series with ith term ar[sup]i[/sup], 0 < |r| < 1, is a/(1 - r). Take a = 3 and r = 1/10, and you get that .3… = 1/3. Take a = 9 and r = 1/10, you get that .9… = 1.
And before you ask, .25 = .250000000… We just omit the trailing zeros by convention, but there are an infinite number of them.
Ah, but it does, and you don’t need anything more than grade school math to prove so.
Try this:
1/3 = 1 divided by 3 (by definition)
OK, divide 3 into 1. Simple long division, not on a calculator (ah, the feeling of power). (Though you’ll have to do this on paper, since I can’t get the symbols right here.)
As you learned in the third grade, you can’t divide 3 into 1 directly. So you add a decimal and a zero after the one. Now you are dividing 3 into 10. The answer is 3 with a remainder of one. Put the three to the right of the decimal point.
Now, to get the next decimal place, you bring down the next zero and put it to the right of the one remaining. Now you are again dividing 3 into 10. The answer is again 3 with a remainder of one. Repeat. Repeat. Repeat infinite times. You keep coming up with a 3 with a remainder of one. It should be clear that 1/3 has an infinite number of 3s after the decimal place.
For 1/3 not to equal 0.3333… it would require one of two things:
or
Both, needless to say, are impossible.
It’s one thing to miss the concept on 0.9999… = 1, but to claim something that violates the rules of simple division is just plain ridiculous. You might as well argue that 2 + 2 = 5.
Lemur866: I like your Math-P example.
For my part, I always like to think of limits as a “legal showdown” in an evidentiary courtroom.
If Phage thinks that 9-Bar (.9999…) is not equal to 1.0, then he must believe there is a difference. So: name the difference? Is it .1? .01? .000001? Name it. And every time he names a tiny little number, like 10 to the power negative 20,000 I am able to say, .999…9, containing 20,001 9s, is closer to 1.0 than that.
Epsilon/Delta proofs are a form of “burden of proof” game. Since no matter what number Phage chooses, we can easily better it by just adding another 9, he can never define the difference. Thus, in the “court of law” model, he loses.
Trinopus