I have no idea what you’re talking about, to be honest. “Infinity” has a clear definition, and you can either use it or not. I have neither claimed it as a number nor defined it as such.
…Define …999. Define the term. Tell me what it means. Because if it’s not just a roundabout way of saying “infinity”, then I don’t know what you mean by it. I literally have no idea what you mean by “…9999999”. It doesn’t appear to be a meaningful concept.
For the same reason I don’t think a passed-out drunk is capable of explaining quantum physics to me.
If you read a couple of pages ago I mentioned a document written by James Tanton,
where he proves that 1=0.9999999…In his proof he uses …999999=-1.
While I disagree with 1=0.9999999…I agree that …999999=-1.
You should try and read the document on my post #1197.
I have no other proof that …999999=-1. I am relying on other people’s work.
And I did not claim that you said they were equal. In fact, I explicitly said that you do not think they are equal. I was exploring that assertion while keeping in mind that you previous have said two decimal representations are different numbers if they have different digits. I asked which digits are different in 0.333… and the decimal representation of 1/3.
You ignored my question, so I will ask again. What digits are different between 0.333… and the decimal representation of 1/3?
OK 7777 you win the internet. As others have said, all of mathematics relies on definitions, and axioms and it is perfectly possible for you to say that for you the notation 0.999… represents something that is not equal to one. Just as it is perfectly possible for someone to claim that whales are not mammals because they are using a different definition of mammal than the rest of biology. The question is what use is your definition.
Since your definition of .999… doesn’t conform to the usual understanding of this notation we should probably give it a different name so we don’t confuse the two. So lets call it say X9.
What properties does XX9 have? I assume that you want XX9<1. Also I assume that you want XX9 > 0.99999…9 for any finite collection if 9’s?
So what about 1-XX9? You claim that this is 0.0…01 > 0. Again this is not the normal way mathematicians define this concept, but we can give this value a new name so that we are all on the same page. Call it Epsilon. Now with these two concepts you can start working on other aspects of the resulting system. For example does 1/Epsilon make sense? and if so what properties does this have? But as you answer these questions you need to be sure that you are consistent. If ever end up with a contradiction than that indicates that your definitions have problems.
For example what can we say about 0.9 + XX9/10?
From one point of view it seems like 0.9+XX9/10=0.9 + 0.0999…=0.99999…=XX9
but from basic algebra, if 0.9+XX9/10=XX9 then 0.9= XX9-XX9/10 so 0.9 = XX9(1-1/10) so 10.9 = XX90.9
So XX9 = 1. Which contradicts the claim that XX9<1.
So presumably in order to keep your new system you are going to have to give up some properties of algebraic theory that have proven very useful to mathematicians over the years. Just like someone who proposes that their definition of mammals that excludes whales, is going to have to give up some of the nice properties of that taxonomic system. Unless you can demonstrate some serious advantages that your definition has that warrants getting rid of all of calculus and much of algebra, I think its going to be a hard sell.
Check my post above. Towards the end, Tanton reveals the direction he wants to go when “Albert” says, “IF …999 has a meaningful answer”. That’s the entire point. You first have to establish whether or not it has a meaningful answer.
For the conventional real number system, it doesn’t.
This is further corroborated by “Cuthbert” almost immediately after when he states:
[QUOTE=Cuthbert]
I bet mathematicians have created all sorts of arithmetic systems in which quantities like …9999 are defined. And we’ve proved here that in every one those systems …9999 equals -1 every time! (As long as the same basic laws of algebra still hold.) Context! Context!
[/quote]
CONTEXT! CONTEXT! The quantity can be defined under alternate arithmetic systems but certainly is not defined under our good old conventional system.
What you haven’t done is shown why anybody should accept it exists in THIS PARTICULAR SYSTEM, i.e. the standard real number line.
As we’ve now repeatedly stated, it’s ok to have your own system where it exists and where 0.9999… is not equal to 1. But what you haven’t really appreciated is that none of that applies to the standard real numbers we’re talking about. It’s a different system. That is, not the same one.
…999 = -1 comes from a sort of non-standard summation that can be used for divergent series. It’s a real thing used by real mathematicians. They are usually up front about it being non-standard though. That said, I think if you assign some meaning to notation like …999, it’s understood that you’re being non-standard.
For example, 1 + 2 + 3 + . . . = -1/12. I’m no expert on this but I think if a series can be expressed as 1 + n^1 + n^2 + . . . then you can use the formula 1/(1 - n) [same formula you use when -1 < n < 1 and the series converges] so 9 + 90 + 900 + . . . = 9(1 + 10^1 + 10^2) = 9/(1-10) = -1.
Another informal argument is: how do you add 1 to …999? Adding 1 to the rightmost 9 gives 10 so write a 0 in the ones column and carry 1. Write 0 in the tens column and carry 1… Gives …000 or 0.
That’s why I asked**7777777 **what argument convinced him/her that …999 = -1. I’m almost certain that no matter what argument it was, you could use analogous reasoning to show that .999… = 1.
OK, but this Tauntaun person, I don’t know him. Why don’t YOU show ME why you think …9999 = -1?
Let’s put it another way. Suppose I show you a proof that .999… = 1, and part of the proof is a step where …999 = -1 is used.
Then you say, “AHA, if …999 = -1, then infinity = -1, and it doesn’t so 0.999…<>1”.
Except you haven’t proved that. All you’ve proved is that a proof that includes as one of its steps the value …999=-1 is a bad proof.
If I tell you that 1+1=2, and as proof say 1 is chocolate and chocolate plus chocolate is pokemon, and pokemon is two, all I’ve done is give you a nonsensical proof. That doesn’t mean 1+1<>2, it just means I haven’t demonstrated 1+1=2. I might be able to demonstrate it in another way, or I might not.
Again, I’m not asserting that 0.999…=1. I can assert it, but that would be cheating. I’m just saying that if 0.999…<>1, then 1-0.999…<>0. You state that the answer is not zero, but an infinitesimal. The problem with that is that I don’t know what an infinitesimal is. Is 1 + infinitesimal = 1? Or something other than one?
Because it can’t be 1.000…0001, because if it is I can just add another zero in there and make it smaller. The point is, there is no number you can name that I can’t name a smaller number. So if there is a difference, it can’t be determined what the difference is. Which to my logic means there is no difference.
The flaws in this “proof” are in steps 2 & 3. …999 is defined as a divergent series that sums to infinity. But you can’t multiply a number with infinity the way you can multiply a number with another number. The same holds for subtraction. Multiplication and subtraction with infinity are undefined.
There are similar “proofs” that show that 1=2. (Here’s one.) They always involve either multiplying by infinity or dividing by 0 (which are the same thing).
Yeah, here the PDFthat **7777777 **linked to. It’s not intended to be a real proof. It’s intended to show students how the core concepts of calculus may violate the mathematical intuitions they learned in algebra.
I don’t think that 1/3 equals to 0.3333333333…so what you are asking is
not relevant to me.
You seem to be forcing me to admit that 1/3=0.333333333…so that you would see that all the decimals of 1/3 are the same as 0.333333…but no, I
will not do that.
Instead, I say that 1/3 = x, now there are your decimals, they may all be different than your decimal representation of 1/3. You see, it is you who has a decimal representation of 1/3, not me, therefore I cannot answer your question in
a more satisfactory way except that all the decimals are different.
In other words, we are just debating this very issue. It is not solved yet.
I said I have no a priori opinion as to what is 0.999999…equal to as
opposed to other people who think 0.999999…=1 as an absolute truth.
I am trying to see if the truth about 0.999999…is dependent on the context.
Sometimes numbers with every digit different can still be the same, but
it may lead to new problems, for example 2=1 depending on the context.
Why not then, if 1=0.999999…is true, also 2=1 isn’t true?
Which one you must accept? Either 2=1 or 1=0.9999999…as true?
This is the central inconsistency at the heart of mathematics.
Yes, and that context is our standard real number line. The one we teach in school.
No need for “opinions” a priori or otherwise. The equality falls out of the very way we establish the real numbers.
Did we have to build up numbers this way? Not necessarily. But that’s the context we settled on. And under this context, it just happens that 1 = 0.999…
No, if 1=0.999…, that has nothing to do with 2=1. There’s no inconsistency there.
I get that you don’t understand that. But it’s true. The equality 1 = 0.999… has nothing to do with 2.
But it IS true that 2 = 1.999… just as it is true that 3.5 = 3.4999…
That’s something that a lot of students have problems with - the notion that there is not a unique decimal representation for each real number.
x = 0.333…
10x = 3.333…
3 + x = 3.333…
10x = 3 + x
9x = 3
x = 3/9
x = 1/3
0.333… = 1/3
Note that unlike Tanton’s similar “proof” of …999 = -1 I never multiply by an an infinity. 0.333… is not infinite. It’s greater than 3, but less than 4.
What? I did no such thing. I explicitly acknowledged that you don’t think that 1/3 is equal to 0.333… I did that in both posts and you claimed I said the opposite of what I said in response to both of them. Is English your first language?
I am asking you what the decimal representation of 1/3 is. The decimal representation of 1/2 is 0.5. The decimal representation of 1/8 is 0.125. What is the decimal representation of 1/3?
…7777777, is this an accurate representation of your proof that …999 = -1? This is such a fundamental misunderstanding of the concept of infinity that it straight-up boggles the mind. And it only applies to infinite numbers, mind you - try this with any finite number, and it falls apart - for any finite number, it falls apart:
E = 999
E * 10 = 9990
9990 = 999 - 9 !!!
Now, maybe you could claim that this works if the numbers are infinite. But that’s the thing about infinity! It doesn’t work that way! Here it is again, stated correctly:
E = …999
E * 10 = …999
It doesn’t go up! Because, and here is the key point, …990 and …999 are the same thing. They are both representations of the same thing - a number which is infinite. Multiply an unbounded number by a real number and it’s still unbounded! And changing the representation doesn’t change the way that the number works.
To go back to my child’s example from before about the baseball team and the hotel, we can easily define two numbers as “equal” if we can match them - if 5=5, we can match 5 baseball players to 5 hotel rooms. If infinity = infinity, we can match infinity baseball players to infinity rooms. This is a basic concept in discussing infinites, numbers, and sets - cardinality.
Now imagine if instead of one infinite team in an infinite hotel, you have to place 10 infinite teams in that hotel. Well, that’s easy - the first ten rooms go to each of the players with the jersey number 1, the next ten to each of the players with the jersey number 2… In the end, you’ve multiplied “infinity” by 10 and it hasn’t changed - there are still an infinite number of players, who map to the same infinite number of rooms - thus, the number of players hasn’t changed, despite being multiplied by 10. Infinity just doesn’t work the way you want it to. Thus for you to claim that …999 = -1 is asinine.