Moderator: Flannel Jesus
Actually \(\infty\) - \(\infty\) = \(\infty\)surreptitious75 wrote:
\(\infty\) + I = \(\infty\)
\(\infty\) - I = \(\infty\)
\(\infty\) + \(\infty\) = \(\infty\)
\(\infty\) - \(\infty\) = 0
All the results can be derived from the starting equation \(\infty\) + I = \(\infty\)\(\infty\) + ANY NUMBER = \(\infty\)
\(\infty\) - ANY NUMBER = \(\infty\)
If you accept that \(\infty\)+1= \(\infty\) then none of those three equations makes sense. That's because they all depend on some sort of definite value/quantity for "each" infinity.surreptitious75 wrote:
when \(\infty\) = \(\infty\) then \(\infty\) - \(\infty\) = 0
when \(\infty\) < \(\infty\) then \(\infty\) - \(\infty\) = < 0
when \(\infty\) > \(\infty\) then \(\infty\) - \(\infty\) = > 0
Those equations deal with infinity as a numeric concept. They are not about sets or the number of elements in a set.surreptitious75 wrote:The size of an infinity is relative to how many members it has
For example the infinite set of integers is smaller than the infinite set of irrationals
So even though they are both infinite one is demonstrably larger than the other one
Magnus Anderson wrote:Silhouette wrote:Btw your notation made me laugh
That's probably because you're deeply insecure and have a strong need to see flaws in people around you in order to feel good about yourself. And you're looking for any kind of flaws, so as long they are flaws -- big or small, significant or insignificant, etc.
Normal people don't do that.
phoneutria wrote:I stand corrected.
Ecmandu wrote:I meant to type 9 instead of 8 Silhouette, glad you caught it.
Magnus Anderson wrote:Are you saying that my logic is invalid?
Are you saying that it's not true that we can know that an infinite product of \(0\) is \(0\) if we know that \(0\) raised to any number (whether finite or infinite) is equal to \(0\)?
How about an infinite product such as \(1 \times 1 \times 1 \times \cdots\)? How do you know the result of this product is \(1\)? Is it because we know that \(1\) times any quantity (whether finite or infinite) is \(1\)? Or is it because we know that \(1\) raised to any quantity (whether finite or infinite) is equal to \(1\)?
Magnus Anderson wrote:We know from \(\prod_{n=1}^\infty{\frac1{10_n}}\) that indeed any partial product has a different value - so that's no help
It's of no help if what you're doing is looking for a number that does not exist e.g. a finite number that is equal to the result of the infinite product \(\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots\). But if you're merely trying to figure out whether such a number exists, then it's quite a bit of help. It tells you that such a number does not exist.
Magnus Anderson wrote:The limit of an infinite product is not the same thing as its result. They are two different concepts.
Magnus Anderson wrote:It tells us that the result of the infinite product is smaller than every real number of the form \(\frac{1}{10^n}\) where \(n \in N\). Most importantly, it tells us that no matter how large \(n\) is, the result is always greater than \(0\).
Your argument is basically that there are no numbers greater than \(0\) but smaller than every number of the form \(\frac{1}{10^n}\) where \(n \in N\).
That's one of our points of disagreement.
Silhouette wrote:the argument remains that whilst \(\forall{n}\) where \(n\in\Bbb{N}\), \(\frac1{10^n}-0\gt0\), \(\prod_{x=1}^\infty\frac1{10_x}\not\gt0\)
Magnus Anderson wrote:It does not merely "look like". It will never ever get to zero for the simple reason that there is no number \(n\) greater than \(0\) that you can raise \(\frac{1}{10}\) to and get \(0\).
You can say that \(0.\dot01\) is approximately equal to \(0\), and that is true and noone disputes that, but that misses the point of this thread. We're asking whether the two numbers are exactly equal not merely approximately equal.
You can say that \(\frac{1}{0}\) can be substituted with \(\infty\) for practical reasons (given that \(0 \approx \frac{1}{\infty}\)) but you cannot say that \(\frac{1}{0} = \infty\) given that there is no number that you can multiply by \(0\) and get anything other than \(0\).
So from your point of view, the only conclusion that should make sense is that \(0.\dot01\) is a contradiction in terms, and thus, not equal to any quantity. By accepting such a conclusion, you'd have to agree that \(0.\dot9 \neq 1\). So at least one point of our disagreement (really, the main point of disagreement) would be resolved.
Still, one point of our disagreement would remain, and that would be your insistence that \(0.\dot01\) is a contradiction in terms based on the premise that there is no quantity that is greater than \(0\) but less than every number of the form \(\frac{1}{10^n}, n \in N\).
Magnus Anderson wrote:You have yet to explain where's the contradiction.
Statement 1: "At some point in time at some point in space, there exists an infinite line of apples."
Statement 2: "At some other point in time, no apples exist anywhere in space."
How do the two statements contradict each other?
Certainly, the word "infinite" does not mean "not being able to be something else at some other point in time".
Magnus Anderson wrote:Silhouette keeps insisting that \(0.\dot01\) is equal to \(0\).
Magnus Anderson wrote:So which one is it? Is \(N = \{1, 2, 3, \dotso\}\) the same size as \(2N = \{2, 4, 6, \dotso\}\) or is it actually smaller?
The answer is that \(N = \{1, 2, 3, \dotso\}\) does not specify the size of the set. The size of the set is something that is specified separately (usually merely assumed, without any kind of explicit specification.)
MagsJ wrote:..the exact point at which infinity becomes self-defining, so yes.. anything infinite is not bounded within a defined measurable set.
MagsJ wrote:Silhouette wrote:The term infinite defies definition by the definition of "definition" and of "finite". Either something is infinite or it is not - of course. Infinite is only "definable" insofar as we can easily define finite... and then saying "not that". This says what you don't have and not what you do have. The analogy I used is that this "defines" what's in a hole by defining the boundaries of the hole and what's outside of it (i.e. it doesn't define what's inside the hole at all).
A good definition.. it’s not what you’ve got, it’s what you ain’t got. I like it.
MagsJ wrote:..but then wouldn’t that simply mean that something is either infinite or not? which I ‘think’ Silhouette (I don’t want to put words in his mouth) is also saying.
Ecmandu wrote:I realized that silhouette’s argument hinged on 1/10*1/10... (repeating) hinged upon 0.0...1 being equal to zero.
Ecmandu wrote:So anyways,
No response yet from Silhouette, so I’ll just post it.
Silhouette and I agree that 1/10*1/10 (repeating) never expresses the 1 at the tail end!
Ok, so far so good.
That means the only way that you can shift the decimal is not from right to left, but from left to right!
That means that:
0.999... + 0.111... must equal 1.111... if 0.999... equals 1
There’s a problem with this!
0.999... + 0.111...
Equals: 1.1...0
And we know that 0.0...1 is the number that makes 0.999... equal one.
That means that there is a discrepancy of 0.0...2 which makes not the smallest possible number (equal to zero) that can possibly be made!
Thus, Silhouette’s argument thus far, has been falsified.
I made edits.
Silhouette wrote:Ecmandu wrote:I realized that silhouette’s argument hinged on 1/10*1/10... (repeating) hinged upon 0.0...1 being equal to zero.
As I just explained to Magnus, "0.0...1" is a contradiction so can't even exist, never mind be equal to anything.
The intended implication of the contradiction is indistinguishable from zero though - yes.
As you say, you never ever ever get to any terminating "1". It never comes into existence, leaving you only with the "0.0... = 0"Ecmandu wrote:So anyways,
No response yet from Silhouette, so I’ll just post it.
Silhouette and I agree that 1/10*1/10 (repeating) never expresses the 1 at the tail end!
Ok, so far so good.
That means the only way that you can shift the decimal is not from right to left, but from left to right!
That means that:
0.999... + 0.111... must equal 1.111... if 0.999... equals 1
There’s a problem with this!
0.999... + 0.111...
Equals: 1.1...0
And we know that 0.0...1 is the number that makes 0.999... equal one.
That means that there is a discrepancy of 0.0...2 which makes not the smallest possible number (equal to zero) that can possibly be made!
Thus, Silhouette’s argument thus far, has been falsified.
I made edits.
Concerning this "0.999... + 0.111... = 1.1...0":
in the same way that you never ever ever reach the "terminating" 1 in "0.0...1" leaving you with only "0.0...",
you never ever ever reach the "terminating" 0 in "1.1...0" leaving you with only "1.1...".
I'm sure you have no objection with \(\frac{9}9+\frac{1}9=\frac{10}9\), which is the fractional representation of the decimal sum you're demonstrating.
Perhaps you'd like to quantify the difference between \(\frac{9}9\) and \(0.\dot9\), \(\frac{1}9\) and \(0.\dot1\) and \(\frac{10}9\) and \(1.\dot1\) individually?
I get that you're trying to accumulate these suggestions of "terminating" digits in non-terminating decimals such that they seem to become significant, but the problem is in the inherent contradiction of trying to do so. There are no terminating digits to accrue into something significant.
There is no concept of time in mathematics.As I was trying to explain to Magnus, there’s two ways to look at this:
1.) completed infinity (on this you are 100% correct)
2.) process towards infinity (on this I am 100% correct)
So... I posit this to you:
Does endlessness ever become complete?!?!
I think you’ll like that train of thought!
phyllo wrote:There is no concept of time in mathematics.As I was trying to explain to Magnus, there’s two ways to look at this:
1.) completed infinity (on this you are 100% correct)
2.) process towards infinity (on this I am 100% correct)
So... I posit this to you:
Does endlessness ever become complete?!?!
I think you’ll like that train of thought!
Ideas like process and completion don't make any sense.
Magnus Anderson wrote:If you agree that \(\sum_{i=1}^{n} \frac{9}{10^i} < 1\) for every \(n > 0\), and if you agree that \(\infty > 0\), then it necessarily follows that \(\sum_{i=1}^{\infty} \frac{9}{10^i} < 1\). Everything else is irrelevant. The only condition is that \(n\) is greater than \(0\). No need to satisfy Gib's definition of the word "quantity".
Magnus Anderson wrote:\(\sum_{i=1}^{n} \frac{9}{10^i} < 1\) holds true for every \(n > 0\). It does not only apply to integers. It literally applies to anything greater than zero.
That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.“T” (time) is a variable in all branches of mathematics.
Humans exist within time and therefore need time to understand things.I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
Ecmandu wrote:As I was trying to explain to Magnus, there’s two ways to look at this:
1.) completed infinity (on this you are 100% correct)
2.) process towards infinity (on this I am 100% correct)
So... I posit this to you:
Does endlessness ever become complete?!?!
I think you’ll like that train of thought!
phyllo wrote:That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.“T” (time) is a variable in all branches of mathematics.
I can have a variable "U"(unicorns) but it does not mean that unicorns are a part of mathematics.Humans exist within time and therefore need time to understand things.I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
What is there to explain?Ecmandu wrote:phyllo wrote:That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.“T” (time) is a variable in all branches of mathematics.
I can have a variable "U"(unicorns) but it does not mean that unicorns are a part of mathematics.Humans exist within time and therefore need time to understand things.I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
Nothing in mathematics requires time to complete ?!?!?!?!?!
Really!?!?!?!? Please explain!
phyllo wrote:1/3=0.333...
3*(1/3)=1
3*(0.333...)=0.999...
Therefore 1=0.999...
If that's not true, then simple division and multiplication don't work.
You can even take out the multiplication: 1/3+1/3+1/3= 0.333... + 0.333... + 0.333... = 0.999... = 1
It's not rocket science.
If you say they are equal, you’re making the claim that EVERY counting number is equal to an infinity of infinities (contradiction)
There seems to be a lot of confusion here about what "infinite" means. "All finite numbers are infinite" doesn't make any sense as a statement.Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)
That's simple enough. All you need to do is to restrict yourself to whole numbers, natural numbers or integers.Let’s say we hypothetically live in a world where fractions don’t exist... it would be unfathomable that 0.999... = 1.
In a decimal world. 0.111... * 9 equaling 1 is impossible.
gib wrote:Fixed CrossFixed Cross wrote:The first part of the disagreements comes down to whether infinite also means indefinite.
If it does, then it is not a quantity but rather something like a condition, of a set or whatever.
As far as I know, "indefinite" means something like "we don't know when or if it will ever end." Infinite means "we definitely know it won't end."
Infinite is the property of being endless, it's not a quantity. Quantities are the things you find on the number line. Infinity is a property of the number line itself. It is where the number line extends to (or more accurately, the property of its extension being unlimited).
It's LaTeX.
You can see how we do it by quoting our posts and looking at in the text editor.
This board doesn't seem to have all LaTeX features enabled though. I know the mars symbol ♂ can't be posted in LaTeX.
Magnus seems to be the real LaTeX guru. He uses it even to say \(n > 0\). I'm not that hardcore. I'd rather just type out n > 0.
Fixed Cross wrote:So: "keep adding 9s after the decimal point but you stop at some number n of 9s, then you will still be below 1" is wrong. The formula doesn't provide for a "stop at some number", it rather says to keep going indefinitely.
That's a bit ambiguous, turning on what exactly is meant by "indefinitely".
Fixed Cross wrote:But, not to keep going indiscriminately. You have to keep going with a specific task which by definition precludes any step from altering the result of the previous step. Which is what would have to happen for 1 to be reached.
What \(\sum_{i=1}^{n} \frac{9}{10^i} < 1\) for any n means is: pick any integer from the number line. You are completely unlimited in which number you pick. But it does not mean: pick infinity. And not just because infinity isn't a number on the number line, but because substituting \(\infty\) for n actually means: don't pick a value for n. Just keep adding forever.
Who the heck knows what you mean by "INFINITE number". I certainly don't.Ecmandu wrote:Phyllo,
I’ll press you on this for now.
Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”
EXACTLY!!!!!!!!
That’s my whole point. It makes no sense!
It makes no sense that 0.999... EQUALS 1!!!!
In this formulation, 1 by definition is an INFINITE number!! By equality!!!
phyllo wrote:Who the heck knows what you mean by "INFINITE number". I certainly don't.Ecmandu wrote:Phyllo,
I’ll press you on this for now.
Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”
EXACTLY!!!!!!!!
That’s my whole point. It makes no sense!
It makes no sense that 0.999... EQUALS 1!!!!
In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Infinite digits doesn't mean infinite number.
Sure. It looks counter-intuitive but the math equations show that it must be true.Ecmandu wrote:phyllo wrote:Who the heck knows what you mean by "INFINITE number". I certainly don't.Ecmandu wrote:Phyllo,
I’ll press you on this for now.
Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”
EXACTLY!!!!!!!!
That’s my whole point. It makes no sense!
It makes no sense that 0.999... EQUALS 1!!!!
In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Infinite digits doesn't mean infinite number.
Doesn’t matter how you word it... you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! *To make this rated g, the pebble never hurts the squirrel*
You’re still making the claim that 1 is ....
EQUAL!!!
To “infinite digits”
EQUAL!!!
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