Flannel. Once you understand, and I hope Obsrvr does too.
Every number equals every number.
It’s funny to watch you all have these debates.
We live in a temporal reality right now.
Obviously there’s a difference between 7 and 1.
But when you see reality as it is (and I know you can understand this)
3=0.555… 0.444… etc…
So you don’t “feel the need” - so no need to talk about it.
And you don’t care whether it is true.
The meaningless - “Our polls show…” – the opposite of science, math, and logic
I “don’t feel the need” to prove that I have proven it - especially since you cannot tell me what is wrong with the proof I offered.
Only that you agree with what “10 times” something means - and it doesn’t mean “shift the decimal” – that is merely a short cut that is usually sufficient.
It’s kind of ridiculous to say that I made up the rule that to multiply - you have to multiply every digit – I thought you knew at least something about basic maths.
I know how easily you get offended and what you do from there – so I’ll leave it here.
What I find interesting is that, if you read the wiki page underneath the image I posted, it actually somewhat agrees with you observr:
Students who did not accept the first argument sometimes accept the second argument, but, in Byers’ opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals.
conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.
And then the wiki goes on to lay the groundwork for proof using infinite sums, and a handful of other approaches.
Let’s try it with x equaling a different number and see what happens:
x = 9.999…
10x = 99.999…
10x = 99 + 0.999…
10x = 99 + x
9x = 99
x = 11
So according to the logic of the “proof” of 1.0 = .999…, then 11.0 = 9.999…
Heck, I’ll go one step farther and say that 1 = 2, that 3 = 4, and that 99 pennies = 100 pennies!
…and why stop at numbers? Blue = Green, and Yellow = Purple! Ford = Chevy, and Up = Down!
Flannel.
Shit. Now you’re bringing completed infinities into the thread. I guess it’s perfectly on topic.
Shit. How do I explain this to a human?
Umm… infinity just by being itself can’t be itself because it never ‘stops’.
That’s what creates the finite.
It doesn’t sound logical to a human that infinity creates the finite. If infinity were finite, existence wouldn’t exist. 1=0 in this equation. Another way of explaining it is everything would be exactly the same!!
Which equals nothing at all.
I have different ways of explaining this too… but let’s move on.
Because infinity can never be itself… we exist.
This is just the cliff notes version by the way.
So… you’re talking about infinite sums.
What about infinite subtractions?
Oh man. You folks are pushing my limits.
How do I explain this to a human.
I know what to say…
Existence is not a noun.
motor, I don’t think your post demonstrates anything other than that you don’t understand the algebra in the original proof that you’re trying to parody. You make a pretty glaring mistake before you get to X = 11, where you make a substitution without the proper context.
…
10x = 99 + 0.999…
10x = 99 + x
…
You didn’t set X = .999… at the start, so this substitution here doesn’t make any sense.
Flannel is right. In order to be a sentient being… you need context to do transitions.
Let me explain more Motor.
No matter what I say. No matter how helpful it is. You can always say “no” or “I agree to disagree”
No matter what it is.
Does that sound like a decent way to move through existence?
As you’re eating a meal in front of me, you tell me you’re not eating.
Iambiguous has this problem too.
It can be funny. But you’ll lose friends very fast by repeating this joke. Especially when it hurts someone.
So what? You started with x = 0.999… and turned it into x = 1.0. What’s good for the goose is good for the gander!
Here, I continued your “proof”:
x = 0.999…
10x = 9.999…
10x = 9 + 0.999…
10x = 9 +x
9x = 9
x=1
10x = 9.999… (as stated previously)
x = 0.999… (as stated previously)
He’s trying the convergence argument by talking about infinite sums.
I get a bit annoyed when mathematicians do this.
Flannel and Motor…
A reply from my post above. People not understanding implications 10 steps ahead annoys me.
Flannel. You’re making the god argument.
My life sucks.
When someone is making a god argument, it disturbs me.
And motor. That’s called context.
I even want to reply to my own reply of my reply…
The math discussions I have and describe are not god arguments… they’re everyone arguments.
I understand everyone’s trying to be the king or queen, or they want a king or queen.
I’m not interested in any of that.
What you see in math reveals a tremendous amount about your psychology.
But you’re not doing what’s good for the goose. You’re creating a brand new scenario, where you’ve started out with the assumption that X is = to 9.999… and just randomly decided to substitute X for 0.999…
It wasn’t random in my example, it was the initial condition. It is random in your example, because it has no context in your example.
You can substitute one symbol for another one when you’ve established their equality. I established the equality of X and 0.999… at the start of my scenario, so I can substitute one for the other whenever I please.
You started your scenario establishing the equality of 9.999… with X, so you can substitute THOSE things with each other whenever you please. But you can’t just randomly decide to substitute other values that you haven’t established the equality of. That’s not how any of this works man.
I’m going to show you how you would do your math with the correct substitution, the substitution that you’ve enabled with your premises
x = 9.999…
10x = 99.999…
10x = 90 + 9.999…
10x = 90 + x
(Note here that we’ve replaced the symbol 9.999… with X, because we’ve established their equality)
9x = 90
x = 10
That’s what it looks like if you truly follow what the goose did.
I appreciate your effort. Your last two steps are ad hoc.
9 times 9.999… does not equal 90.
It equals 90.999… at best.
I’m not doing the math, but it’s probably 90.818181…
Obviously that’s not an INFINITE decimal because it’s just an online calculator, and there’s not really any way to explicitly specify an infinite decimal there. But I think you’re confused about the value of 9.999…, to think that 9x9.999… “equals 90.999… at best.”
In regards to this, no they’re not ad hoc. They’re the same last steps as the original proof screenshotted from wikipedia.
Subtract X from both sides.
Whatever number is now being multiplied by X – divide that number from both sides.
These steps are part of the allowable symbol maniuplations of math. They’re not ad hoc, they’re just math.
Sound or not, that sort of argument is hard to digest. The key problem are the terms finite and infinite quantities. What do they mean? When you say “infinite quantity”, you certainly do not refer to one that can be represented using an infinite series. "1’ can easily be represented using an infinite series e.g. it can be represented as “1 + 0 + 0 + …” or “1.000…”.
The best argument, in terms of its convincing power, I believe is the one that shows that regardless of how many 9’s there are after the decimal point, the number is always less than 1.
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
…
But what if the number of 9’s is infinite e.g. equal to the number of natural numbers? In that case, the result is 10 raised to the power of negative number of natural numbers. The result, in other words, is an infinitesimal, and thus, greater than zero.
One can observe the pattern where the result is equal to 10 raised to 0 minus the number of 9’s. 1 - 0.9 = 10^-1, 1.09 = 10^-2, etc. When the number of 9’s is equal to the number of natural numbers, it follows that the result is 10 raised to 0 minus the number of natural numbers.
A bit more convincing than this is perhaps this argument:
The problem with that proof is that when you take 0.999… and multiply it by 10, you end up reducing the number of 9’s that come after the decimal point. As such, “10 x 0.999…” isn’t equal to “9 + 0.999…”.
10 x 0.9 = 9.0 (1 vs 0 nines)
10 x 0.99 = 9.9 (2 vs 1 nines)
10 x 0.999 = 9.99 (3 vs 2 nines)
etc
You’re probably going to say now that “infinity - 1 = infinity” ignoring the fact that “infinity” is not a number but a category of numbers and that that statement is true only in the sense that “an integer - one = an integer” is true.
What I love about this argument is that it perfectly mirrors an argument that 1 = 0.999… from another angle.
If you read the wikipedia article, the section titled “Infinite series and sequences” approaches the problem from exactly this angle - an inifnite series of sums of 90% as you describe. The idea is based on geometric series sums converging on a value.
Now, I expect some to reject that geometric series like this actually do converge, that the result of the infinite sum is a finite value – which is fine, because the point here isn’t to convince you that it converges to a finite value, but just to show that assuming that it does converge to a finite value doesn’t produce inconsistencies with the symbolic manipulation systems of math.
In other words, you can choose to take a philosophical position that the infinite sum of a geometric series doesn’t converge, OR you can choose to take the philosophical position that the infinite sum of a geometic series DOES converge, and the second approach doesn’t break our symbolic manipulation system in any sense. What it does give you is extra tools that aren’t available to people who don’t make that assumption. It gives you extra tools, and it breaks nothing.
The symbol that is “0.999…” has no 0’s (except for the one immediately to the left of the decimal point). I agree with that. However, the number represented by that symbol can be represented with a symbol that has more zeroes than that e.g. “…000.999…”. You can also have zeroes AFTER all those nines. For example, every position with an index that is an infinite quantity can be filled with a zero. There is, however, no standard way to represent that using a simple symbol.
“Endless” does not mean “endless in every way”. When you say that a sequence of numbers is endless, you do not necessarily mean it is endless in both directions (let alone in every conceivable way.) Perhaps you mean it’s endless “in the middle” such as “1, 3, 5, …, 4, 2, 0”. That’s a perfectly valid concept.
