Show me the proof. That’s all… show it to me.
You condescending remarks only reflect your own insecurity regarding the issue. I would venture to have read far more on the subject than you, but maybe not. So enlighten me with your wisdom please. Tell me how you know .999… = 1 even within the real number system.
erik, you say that you’ve studied math through calculus. You also say that you’re approaching this intuitively. I can’t understand how those two statements can possibly work together. Even in high school math, you surely have been told infinitely often that you have to have precise definitions and rigorous procedures before you can do anything at all.
High school geometry, which is Euclidian geometry, depends on starting with five very carefully chosen axioms. None of the axioms can be proved by the others. As a group, though, they can be used to prove everything else. But you have to be careful on two points. One is that you can start with a different set of five axioms and still have a completely rigorous, complete, and useful geometry - the non-Eurclidian geometries. The other is that how you choose to prove things matters greatly. There are many things, like the trisecting of an arbitrary angle, that can be accomplished with a ruler and compass but are completely impossible with a straightedge and compass. (There are some specific angles that can be trisected with a straightedge and ruler. That’s why proofs have to be generally applicable, not just to one specific example.) Definition is everything.
Intuitively, people have always felt that the so-called parallel postulate doesn’t belong and should be provable from the others. Hundreds have tried to derive it from the other four. It can’t be done, but people feel it should be able to so they keep trying. Same with a straightedge and compass proof of the squaring of a circle. You can tell them a million times that it has been proved impossible, but they don’t like proofs that conflict with their intuition.
How to define and manipulate infinities fell into the intuitive category even among mathematicians. Finally Cantor came along and stated forthrightly that all intuition has to be thrown out the window when dealing with infinities. The only way to make them rigorous enough to work is to do things that are counter-intuitive but can be mathematically proven. Forget all notions of infinity being a very big number, he said. The only way to define infinity is to say that it is never ending. That solves everything.
Take the most basic problem. Intuitively, it is obvious that there are half the number of odd integers as the total number of integers. Cantor says no, they are the same. Why? Because they can be put into a 1 to 1 correspondence with one another.
1 3 5 7 9 11 13 15 17 …
1 2 3 4 5 6 7 8 9 …
You can say that 1,000,001 is only the 500,001st in the sequence so intuitively the odds must be smaller, but you can never find a number to prove that. No matter how far out you go you can pair an odd number with a unique place in the number line. There is no end. If you stop, all you’re doing is solving a specific problem rather than the general one.
1 to 1 correspondence solves everything. There are as many reals between 0 and 1 as there are points in the entire three-demensional (or n-dimensional) infinite universe. How is this possible? They can be put into 1 to 1 correspondence if you align them correctly. All infinities of the same type are equal.
Some infinities are not equal. You can’t put the integers or irrationals into 1 to 1 correspondence with the reals, so the reals are a different infinity. You can consider one larger than the other but the proper rigorous definition lies in the lack of 1 to 1 correspondence.
Notice something else. I haven’t mentioned the word limit at all. It is not necessary here, although it becomes incredibly useful when trying to do sums or other math on infinities.
So how does this apply to whether 0.999~ is equal to 1? Directly.
let x = 0.999999~
Multiply both sides by 10:
10x = 9.999999~
Subtract.
10x=9.999999~
x=0.999999~
9x=9
x=1
That’s a proof. It’s as sound as 9/9 = 1. You are never dividing by infinity. All that is required is that the 9s are in a 1 to 1 correspondence with one another after the decimal point. They are, as part of the definition of what a countable infinity is. That means we are left with 0 when we subtract. The terms after the decimal point cancel, to use some high school terminology. You asked for a proof without limits. Here it is.
What you’ve been arguing all along has been the equivalent of saying that since - intuitively - odds are half of the odds plus evens there must be twice as many entries in the total number line. If you start out with a statement as wrong as that nothing you say after can be true.
You’ve started by giving your intuition about infinities, ignoring all the definitions, axioms, procedures, and proofs, and then insisting that the definitions, axioms, procedures, and proofs must be wrong because they aren’t intuitive. Nope, they’re not. But they are math. And math is rigor and solely rigor. The Principia Mathematica doesn’t get to proving that 1 + 1 = 2 until somewhere past p. 379. But “1 + 1 = 2 is intuitively obvious,” you shout. The only proper answer to that is, no, it isn’t. If you can’t accept that, if you can’t accept that every single mathematical statement has to be undergirded by rigor rather than intuition, you can’t do math at all.
Nope and our friend the Great Antibob would agree with me on that I think.
Re:
let x = 0.999999~
Multiply both sides by 10:
10x = 9.999999~
Subtract.
10x=9.999999~
x=0.999999~
9x=9
x=1
Goodness, I destroyed that proof long ago in this thread.
What is .99 x 10? 9.99? or 9.90?
If you said 9.90 you are correct.
So where do you get off throwing the extra 9 in at the end of 10 x .999…?
What proof would you like to see? Mathematics has defined .9999… as equal to 1. Mathematics makes the rules for the system of numbers. It says that .999…is equal to one like the construction industry says that my 1.5 x 3.5 board is “2 X 4”.
This thread is the equivalent to you arguing that it is really a 1.5 X 3.5 and other posters arguing that, no, it’s a 2 X 4 because that’s what it is. The proof is the documentation in the construction industry. You say that’s circular.
The proof of .999..=1 is that it is close enough for government, private sector, and any other type of work such that there is no discernible difference between the two anywhere in the universe, so much so that one is in fact equal to the other.
If there is no discernible difference between two things, aren’t they equal?
It’s an infinite number of 9s because I’m using the definition of what an infinite number is. That you don’t know the definition of what an infinite number is pretty much sums up what everybody has been trying to explain to you.
Any number with digits a.bcd…klm0 is equal to a.bcd…klm.
So 9.90 equals 9.9, as you know.
And 9.999…0 (if there could be such a thing) would be equal to 9.999…
In gradeschool arithmetic, the rule for multiplying by ten isn’t “shift to the left and add a zero,” it’s “shift to the left.” You only put in a zero if you need to fill in digits down to the one’s place. After the one’s place, you don’t put in zeros.
There is no “extra” 9 at the end of that sequence. There is no end to that sequence. There are 9s at the “end” of that sequence in perpetuity.
But anyway, as has been said, it’s fine if you don’t want to accept the limit theorem. You’ve stated that you consistently don’t believe any repeating decimal is equal to a fraction with only integers in its numerator and denominator. That’s fine–but it takes a lot of work which you haven’t put in to actually specify such a system. And you’re decieved if you think you’re getting at the “truth” of things–what you’re doing is building a formal system which may or may not apply to this or that purpose.
RE: And 9.999…0 (if there could be such a thing) would be equal to 9.999…
You don’t even know if there is such thing but profess to know what it would be equal to? That very impressive. Anyway to back that claim up?
You’ve been given several. You just don’t want to accept the logic.
In particular, Indistinguishable gave a perfectly good explanation of it earlier in the thread.
Everything stated in that post is valid in our number system.
If you don’t accept them, DEFINE (as I originally requested) your number system, numbers, and notation.
Of course, the catch is that if all you’ve done is create a different number system, it doesn’t actually contradict or disprove anything about our current number system.
Argument by expertise? I’m a card carrying mathematician, not that it matters. Your qualifications don’t matter, but your arguments do. I suggested you read an elementary analysis text (and still do) because you haven’t demonstrated even a basic understanding of where your arguments are failing.
I get equally exasperated by moon hoaxers and 9/11 conspiracy theorists. It doesn’t make them right about not landing on the moon or that the US government was behind 9/11, either, nor do they reflect my insecurity regarding the “official” explanation of those events.
I can profess to know what it “would be equal to” because I am just applying formal operations to symbols on paper. When I say “if there could be such a thing” what I mean is that, in mathematical practice, the string is not defined to represent anything at all. But the fact that it doesn’t represent anything doesn’t mean we can’t apply formal manipulations to it. And when we do apply those formal manipulations in a way consistent with the way we apply them to strings that do represent a number, we find that we end up with a string that represents a known value.
In the same way, I can say that 3.555&0, if there were such a thing, would be equal to 3.555&. I’m not naming values here, I’m talking about the formal manipulation of symbols on paper.
What part of the definition if infinity don’t you get? In your first post you wrote:
It seems that indeed, you do believe that infinity can’t exist for just this reason. But as explained time and time again, this is exactly what infinity is. There is always an addtional 9 at the end. Infinity is exactly the same as infinity + 1.
RE: You only put in a zero if you need to fill in digits down to the one’s place. After the one’s place, you don’t put in zeros.
Just because you don’t write them doesn’t mean they are not there. They are implied, i am just making the point that you a filling in another 9 at the end (if there was such thing). Now I don’t know that there is, but do you? Do you know you can put a 9 there?
We’re not just “accepting” that there is a 9 there.
Here’s an example of one of the “contexts” where infinity is actually defined.
Write 0.9999… in a different form:
sum(i = 1 to “infinity”) [9/10^i]
Note that the “infinity” in the index of the summation just tells us not to stop adding more terms ever.
Now, multiply this by 10:
10*sum(i = 1 to “infinity”) [9/10^i]
We can bring the 10 “inside” the sum:
sum(i = 1 to "infinity") [10*9/10^i]
Now simplify:
sum(i = 1 to "infinity") [9/10^(i-1)]
Written in a more ‘standard’ form, this is 9.99999…
There is no ‘0’ at the end at all. Nor are we “adding” any digits at all. We’re just multiplying by 10.
ETA: Francis Vaughn gave the same explanation earlier in the thread. I simply expanded the explanation a bit. Just wanted to give credit where it was due.
This and the other post I responded show that you don’t really understand the problem set out.
There is no .99 or .999999999 or .9990. They appear nowhere in the problem. We started out with a single question. What is the infinite decimal represented as 0.9999~ equal to? You cannot answer that question by looking at any finite truncation of that decimal. It has to be the whole unending series of 9s. No extra 9 is ever throw in. It was always there from the start. I didn’t put anything there. It’s part of the definition of what an infinite decimal is.
Your misunderstanding is that you keep trying to leap from a finite, i.e. truncated, series of 9s to an infinite series of 9s. They are two different things and must be handled differently. The question concerns infinity. You’re not allowed to change it to suit.
RE: In particular, Indistinguishable gave a perfectly good explanation of it earlier in the thread
He puts the concept of .999… = 1 in his definition of .999… I’m not sure where he got the definition. It is not part of the definition of real numbers. More or less it’s his interpretation.
I’m still waiting for that rigorous proof. Even serious mathematicians admit the 10x =9.999… thing is a trick, not rigorous at all. You can say I am not accepting logic, I can say the same of you. I don’t see the proof.
Next I suppose someone will post a geometric series which has already been discussed ad nauseum. It uses limits which presuppose the concept of .999… = 1 as well. If you don’t understand that then you don’t understand limits.
Once again, this is wrong. Utterly, completely, factually wrong. No mathematicians would ever describe this as a trick. It is a proof.
Your presupposing that 10 x .999… = 9.999… can you prove that?
Cite?