(0.111… )*9=0.999… Equation 2 (ordinary multiplication by 9)
(1/9)*9=1 Equation 3 (ordinary division and multiplication)
Substitute for (1/9) in Equation 3 with the equivalent (1/9) from Equation 1 :
(0.111…)*9=1 Equation 4
The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.
I’m writing this post for Carleas to link to (not because I’m reporting anyone here - that’s absurd!) but because I know he loves numbers and has participated . I love numbers too! Any of you can take a stab at it:
I have two number theory positions:
1.) series don’t converge
2.) there are no orders of infinity
1.) series don’t converge! If a series converges, this means that every counting number is an infinite sequence (proof through contradiction).
2.) there are no orders of infinity!
I use this order symbolically (not using numbers) to disprove this!
A.) rational number
B.) uncounted infinity
C.) different rational number
D.) different uncounted infinity
Etc…
This places all the “lower” infinities in 1:1 correspondence with the “highest order of infinity”, thus there are no orders of infinity.
Abstract concepts in mathematics not using numbers are fair game!
Thus my proof sits.
Magnus, for you personally, if you don’t understand that, I’m not crazy, and it’s not my fault.
I know Carleas doesn’t have the time for this thread, but I know he’s a math pervert (we all should know that) and I don’t think he’s seen these arguments before!
You moved out of the decimal system to make your argument, you switched back to whole numbers, which is the same as never using decimal expression in the first place.
1.00/9.00= 0.111… Equation 1 (ordinary division by 9)
(0.111… )*9.00=0.999… Equation 2 (ordinary multiplication by 9)
(1.00/9.00)*9.00=1.00 Equation 3 (ordinary division and multiplication)
Substitute for (1.00/9.00) in Equation 3 with the equivalent (1.00/9.00) from Equation 1 :
(0.111…)*9.00=1.00 Equation 4
The left side of Equation 2 is exactly the same as the left side of Equation 4. Therefore whatever is on the right side of equation 2 is exactly equal to whatever is on the right side of Equation 4.
Nice try. You may as well as typed the proof without using decimals at all (which is how standard Mathematicians argue it). At some point, you dropped the decimals and went to only whole numbers.
Yes. That proof always works with only whole numbers.
First, we’d have to define how one even does sums with non-integers, how to increment the value of i. Obviously you start with the initial value (1 in the cases we’ve been dealing with), but what’s the next number? Do you go:
1, 2, 2.5?
Do you go:
1.0, 1.5, 2.0, 2.5?
Do you go:
1.0, 1.1, 1.2 … 2.4, 2.5?
Do you stop at 2 (because you can’t jump by a whole integer amount from 2 to 2.5)? Do you stop at 3 (because maybe n = 2.5 means don’t stop as long as you’re under 2.5)?
But for argument’s sake, let’s say you go: 1, 2, 2.5. Then the sum is:
I’m not just saying (\infty) does not refer to a specific number, I’m saying it doesn’t refer to any number. It can’t. You’ve heard me say it before and I’ll say it again. Infinity means endless. It refers to a property of sets. It’s not that we’re not being specific about which number it refers to, it’s that we (or I) are not referring to something that could be a number.
Like I said, (\sum_{i=1}^{\infty} \frac{9}{10^i}) is not a special case of (\sum_{i=1}^{n} \frac{9}{10^i}). It’s a case that lies outside all specific cases of (\sum_{i=1}^{n} \frac{9}{10^i}) (because (\infty) is not a valid value for n). (\sum_{i=1}^{\infty} \frac{9}{10^i}) is saying: don’t pick a value for n. Let the sum run forever.
So I agree that there is no n such that (\sum_{i=1}^{n} \frac{9}{10^i} \geq 1), but (\infty) is not a valid option for n. Plugging (\infty) into the upper bound of the sum isn’t picking a value for n. It’s refusing to pick a value for n. Only in that case does the sum equal 1.
You’re so close to biting, Magnus. Wanna talk about hyperreals?
Apologies in advance for the drive-by, I haven’t followed this thread but Ecmandu asked for my response.
This seems question begging of the subject of this thread: if .999… = 1, then every counting number is an infinite sequence (.9 + .09 + .009 + .0009 + …)
I don’t follow this proof. Are you rejecting the idea that each rational number represents a different infinity? Can you cast the proof in terms of Hilbert’s Hotel?
It’s totally on topic for the thread (thanks for stopping by btw!)
Think about what this means. It means that 1 (a definitely finite number)
IS!!! ( bolding important )
An infinite sequence (which is a disproof through contradiction of definitions)
As for the second part. I’m sorry it didn’t make sense.
All I’m trying to say is that ALL infinities can be paired in 1:1 correspondence. The argument that they cannot, is put forth that you cannot enumerate all infinities in one single list.
So I CHEATED =)
I just list them (symbolically (not numerically)) as “uncounted numbers”
Math uses abstract symbols constantly that people complain about!! But, from my perspective, my CHEAT is fair game!
As far as Hilbert hotel is concerned, this is already a 1: 1 correspondence, you’re not doing “dimensional flooding” in that thought experiment. “ Dimensional flooding” is when every number in a single list has an infinite amount of numbers that cannot be put in each slot of the list.
Lots of people think number theory is dry, boring and useless.
Little do they know, I’m actually making a theological argument.
If there are no orders of infinity, then that means that no being is greater than another being (including hypothetical “god”)
I think everyone in this thread can all agree that they are “math perverts”. Math always has philosophic implications, in fact, it is the purest language we have for discussions.
That’s what I wanted to add. Again, thanks for joining the thread.
I think “is” may be ambiguous in this context. I would say that natural numbers can be expressed as the sum of an infinite series. I think that is synonymous with the mathematical ‘is’. (Aside: my understanding is that category theory is an attempt to rebuild math without using the ‘=’ operator (or without using it the way it’s traditionally used. Unfortunately, I don’t understand it any further than that, and I don’t know its implications for questions like these.)
You need a contradiction to create a proof by contradiction. You’ve assumed that a finite number can’t be expressed as the sum of an infinite series, and I agree that if we take that as a given, we could use it to create a proof by contradiction. But we don’t have to take that as a given: there is no contradiction between something being both a finite number and the sum of an infinite series.
This assumes something that isn’t true for uncountably infinite sets. You can’t list them, even in theory. There is no next largest real number.
Before you answer my last post. Consider that I pointed out that the best English word for you in argument 1 was “represents”
It’s the same in argument 2. I misspoke and said “substitutes”. However, if I use the word “represents” as well, it changes the meaning and ability to refute it substantially
If (i) must start with (1), end with (n) and increase by (1), then (2.5) is not a valid value for (n).
On the other hand, a sum of natural numbers (1 + 2 + 3 + \cdots + n) is equal to (\frac{n(n+1)}{2}). If we plug (2.5) into that, we get (4.375) as a result. We might also be able to find a sum with an index that starts with certain number, ends with (2.5) and increases by certain value.
Have you considered limiting (n) to numbers that have no fractional component but wihtout limiting it to integers?
But you want to put (\infty) in there. So let’s talk about (\infty) as a fuzzy blob of impossibly big numbers, not as limitlessness, as the spectrum of numbers you would find if you transcended the limits of the limitless.
