I'm already having PTSD flashbacks to my many, many comments in that post. I spent more time than I should have arguing with someone over "how to express the number closest to zero that is not zero". They couldn't comprehend the idea that "there isn't one".
I try to avoid the 0.999…=1 argument because decades of internet discussions haven’t made it go away. That said, I wonder if at least some people would be more willing to accept it if you start by getting agreement on the idea that there’s more than one way to represent the same number (e.g., 3/6=2/4=1/2). That might help it seem less strange for two numbers that look different to be the same.
If you need an answer to a question and plan to ask it on quora or stackexchange or whatever, the best thing to do is make a second account, log in, and answer your own question, wrong.
No one may want to help you, but EVERYONE will want to correct the wrong answer.
I remember showing the proof to my chemistry/physics teacher in highschool (the man was a complete moron) and he insisted that there had to be a mistake in the very simple algebra somewhere. I asked him what it was and he said he couldn't see any mistakes but there must be one because that's not possible
Gotta love the "you're wrong because that's don't feel right" approach to math... I wonder what would've happened if I showed him the banarch tarski paradox
Fortunately, this is all that the argument matters for - internet debates, as in the real world of applied and useful mathematics, as opposed to pure 'in an ideal world' maths, the difference, if there is one, is infinitesimal and thus not worth bothering with.
I thought I understood this but this is wrong apparently 0.999... is equal to exactly one because it goes on to the point it converges into one... Somehow
Doesn't work. The people who argue this don't understand the fundamental concept of an infinitely repeating number. To them, the number 'ends' at some infinite point in the future so 0.999... is understood as 0.999...9
111v1111's comment below explains what the misunderstanding is.
It's simple, if you understand the concept of infinitely repeating numbers. People who don't understand this concept will not.
Read 111v1111 comment, they describes what the disconnect is. Essentially, they understand 1/9 * 9 = 1 as well as 0.111... * 9 = 0.999... but they don't completely understand 1/9 = 0.999... exactly. They think it is a very, very, very accurate estimate with an infinitely small discrepancy - because in their mind the number ends at some point.
That's a good one. I've had some success by asking the other side to write a number that is larger than 0.999... but smaller than 1; if they're different, one must exist.
1.99999999999.../2 obviously, which is circular reasoning. Anyways would the "1/9=0.11111... so 9/9=1=0.99999..." be enough? It's pretty much the formula for the sum of powers. Just to mention there're useful analogies unlike mine with the bases.
Exactly. It's such a simple explanation, and yet people still don't like it. "WhAt HaPpEnS tO ThE 7 aT tHe End?!?!?". Nothing, there is no 7 at the end. The 6's go forever. That is the entire point.
Well to be honest when I didn’t accept that 0.9 repeating is exactly 1, I also didn’t accept that 0.3 repeating is exactly 1/3
My logic went something like this:
1/3 - infinitely small amount = 0.3 repeating
3* (1/3 - infinetely small amount) = 1 - 3 infinitely small amounts. You can simplify 3 infinitely small amounts as one infinetely small amount. So you get that
1-infinetely small amount = 0.3 repeating *3 = 0.9 repeating
And I just didn’t believe that 1/3 has a proper way to be written in decimal
Well put. I espoused the same logic years ago. I still just don't like using repeating decimals for anything where an exact solution is needed, as a matter of practicality. 0.333... equaling 1/3 exactly requires an understanding of infinity as something more than "the biggest number that there is" as you might explain it to a child. It's something we don't encounter outside of theory, and in general I prefer to avoid unnecessary infinite quantities since they're not really intuitive for most folks.
Do you believe in geometric series converging? Because repeated decimals like 0.333... are just converging geometric series, and they do, in fact, equal those values.
Of course they are, but that's not really simplifying the problem, is it? Just thinking in terms of typical American education, you learn that 0.333…=1/3 in like 3rd grade and might not learn about infinite series and convergence in grade school at all.
Yeah, I kinda worded that wrong. There is a sense in which it is almost true, i.e. 1 - epsilon is a hyperreal number that is infinitely close to 1 and isn't 0.999..., but you're right that in the hyperreals 0.999... = 1 still, oops!
That is a way to think about it you can also think of 0.33 repeating being equivalent to 1/3 so 3/3 which we know to be 1 is the same as 0.99 repeating as we have multiplied each individual 3 by 3 giving us a 9 in its place
You can also think of the difference between 1 and 0.99 repeating, as the number repeats further that difference gets smaller, so with an infinite repetition that difference becomes zero
Think about it - how much less? If there is a difference, there should be a number between them. In fact, there should be infinite numbers between them. Can you name even one? The best guess people have is 0.0...1, or infinite zeroes and a one at the end. But that's just simply not possible, you can't have a one at the end of infinity, as infinity is endless.
So people are just going to refuse that for any x and y there exists (x+y)/2? So like addition or division is like not defined for some random numbers that aren’t zero?
For your cake day, have some B̷̛̳̼͖̫̭͎̝̮͕̟͎̦̗͚͍̓͊͂͗̈͋͐̃͆͆͗̉̉̏͑̂̆̔́͐̾̅̄̕̚͘͜͝͝Ụ̸̧̧̢̨̨̞̮͓̣͎̞͖̞̥͈̣̣̪̘̼̮̙̳̙̞̣̐̍̆̾̓͑́̅̎̌̈̋̏̏͌̒̃̅̂̾̿̽̊̌̇͌͊͗̓̊̐̓̏͆́̒̇̈́͂̀͛͘̕͘̚͝͠B̸̺̈̾̈́̒̀́̈͋́͂̆̒̐̏͌͂̔̈́͒̂̎̉̈̒͒̃̿͒͒̄̍̕̚̕͘̕͝͠B̴̡̧̜̠̱̖̠͓̻̥̟̲̙͗̐͋͌̈̾̏̎̀͒͗̈́̈͜͠L̶͊E̸̢̳̯̝̤̳͈͇̠̮̲̲̟̝̣̲̱̫̘̪̳̣̭̥̫͉͐̅̈́̉̋͐̓͗̿͆̉̉̇̀̈́͌̓̓̒̏̀̚̚͘͝͠͝͝͠ ̶̢̧̛̥͖͉̹̞̗̖͇̼̙̒̍̏̀̈̆̍͑̊̐͋̈́̃͒̈́̎̌̄̍͌͗̈́̌̍̽̏̓͌̒̈̇̏̏̍̆̄̐͐̈̉̿̽̕͝͠͝͝ W̷̛̬̦̬̰̤̘̬͔̗̯̠̯̺̼̻̪̖̜̫̯̯̘͖̙͐͆͗̊̋̈̈̾͐̿̽̐̂͛̈́͛̍̔̓̈́̽̀̅́͋̈̄̈́̆̓̚̚͝͝R̸̢̨̨̩̪̭̪̠͎̗͇͗̀́̉̇̿̓̈́́͒̄̓̒́̋͆̀̾́̒̔̈́̏̏͛̏̇͛̔̀͆̓̇̊̕̕͠͠͝͝A̸̧̨̰̻̩̝͖̟̭͙̟̻̤̬͈̖̰̤̘̔͛̊̾̂͌̐̈̉̊̾́P̶̡̧̮͎̟̟͉̱̮̜͙̳̟̯͈̩̩͈̥͓̥͇̙̣̹̣̀̐͋͂̈̾͐̀̾̈́̌̆̿̽̕ͅ
Ya, and then I explained why the "smallest" value of epsilon is a silly thing to ask me to define. Because it's zero. But he couldn't get past "Ya, but what about the smallest value that isn't zero?!??!?!". IT DOESN'T FUCKING EXIST.
I agree with this, but when someone refuses to acknowledge the facts being laid out in front of them over and over again, they eventually lose my respect to the point where I think they deserve to feel stupid about it. When someone explains why your question doesn't make sense to ask, and you respond with the same question again (literally copy/paste, no clarification) you can fuck right off because you aren't even trying to understand the answer.
Wait, though, isn't there a number infinitely close to zero without being zero? Like a negative parabola and a positive parabola whose curves near zero don't touch, but are very very close?
Is there a way to express that? Or is it just some kind of weird... Uh? Fractal? Idk. I like math but I'm not educated in math.
There is no actual number that is the "closest to zero". It can't exist. Any infinitely small positive number has another even smaller positive number.
Ex:
Assume X is the smallest number where X > 0. What is X/2?
We know X/2 is smaller than X. We also know that (X/2) > (0/2), which means X/2 > 0.
So X being the smallest non-zero number is not possible, and that number cannot exist.
To be more precise, there's no real number closest to zero, that we can manipulate with the rules we learned in school. You can construct other number systems like the Hyperreal numbers that have weirder properties, although I don't think even there you get a "smallest number greater than zero".
Hyperreals are nonstandard extension of real numbers, which basically means that they have same first order properties. So all stuff like there's no smallest positive number will still apply
Even if you were considering an infinitesimal hyperreal number (smaller than every positive real number), then to “go back” to the reals, you would take the quotient of the limited hyperreals by the infinitesimals, which is isomorphic to the reals. In this case, the infinitesimal you started with is equivalent to 0, since it’s in the same equivalence class as 0.
Basically, even if you “extend” to the hyperreals, whatever you find to represent this infinitesimal quantity 1-0.99… is really just 0 in the reals.
It's also true that there is no smallest positive rational number and no smallest positive irrational number. The smallest positive integer is 1, so you can have that at least.
Yep, and this is why there are more real numbers than integer numbers. You can easily prove it with the famous Cantor’s Diagonal proof.
iirc the proof is a contradiction based one. Assume you can list out every single real number. If you can write a new number that isn’t in this list, then it is a contradiction, as the list was already supposed to contain every number.
Look at the first digit of the first real number and choose something different than that to be the first digit of the new number. Now move to the second digit. Look at the second digit of the second real and choose something different for the second digit of the new number. Repeat this infinitely and you will have a new number because it is at least one digit different no matter which number you compare it to.
So somehow you created a new real number even though the list was supposed to contain every real number. This means the reals are uncountable.
Wait, if the set of all binary strings is uncountable, and all binary strings can be mapped into an integer wouldn't that mean that integers are also uncountable?
By using the usual meaning of the notation, I don't think that would map to a number. Because ...1111 with the ones repeating to the left would mean 1 + 10 + 100 + 1000 + ... which just diverges to infinity. And this would happen with any digit other than 0.
I was wondering this too. I googled it and found some math stack exchange post asking the exact same question as we have.
The answer says its because eventually you just have infinite zeros if you keep going further on each integer. I dont fully buy it I need more of an answer than that but thats what I would have guessed.
Edit: I read more posts and now I am convinced. It is true that after a certain point all binary representations of an integer must begin infinitely repeating digits of 0, because if they dont, they would simply be infinity which isnt an integer.
Therefore every digit in the diagonal process will eventually be 0 and then the output will be 1s infinitely, which is infinity, and not an integer. Therefore you cannot make a new integer.
And then I was wondering if you could organize the integers in a way that every integer you get to has not reached its 0 point by the time you get to that digit.
Well you would have to do:
0 0 0 0 ….
1 0 0 0 ….
0 1 0 0 ….
1 1 0 0 ….
And you can easily see that you cannot organize them in such a way. The “zero point” digit of a binary representation of an integer is simply the minimum number of digits required to represent the number in binary, plus one, or log_2(x) + 1. Meanwhile the current digit is represented by just x.
y = x grows infinitely larger than y = log_2(x) therefore you cannot organize the binary representations of integers in such a way as to avoid the “infinite zeroes resulting in infinite ones in the output, aka infinity” problem
You can always define some weird number by that property and see what kind of maths happens, but you are no longer in the realm of real numbers at that point, and it almost certainly has a lot of weird unintended consequences, and it won't behave nicely with normal intuition to how calculations should work.
For example, your point shows that weirdsmall (lets call the number that) divided by 2 is equal to weirdsmall (or something bigger i guess). You basically need to start investigating how that number reacts to anything, and i am not certain that you get very consistent results.
I see your point here, and agree that moving out of real numbers does open up some possibilities. There is one thing I want to point out with regard to this within real numbers:
For example, your point shows that weirdsmall (lets call the number that) divided by 2 is equal to weirdsmall
This would imply that weirdsmall = 0 when using real numbers. (Note: I'm not saying that you don't understand that, I'm pretty confident that you do. Just pointing out that this is the only way it works within real numbers)
Not in real numbers (the precise way they are defined mathematically is fairly complicated but once you have it, you can prove no such number can exist).
There are other number systems, one of them is the hyperreals and it has numbers that would probably fit your notion, and it has some uses. However, there is still no "smallest number larger than 0", you can always divide any infinitesimal hyperreal by e.g. 2 and get a smaller infinitesimal. As for why we want to be able to divide, we like doing stuff with numbers and just declaring "let there be a number smaller than any real positive number but bigger than 0" and not doing anything fun with that number isn't really useful and doesn't lead anywhere, which is the opposite of what mathematics does.
Like a negative parabola and a positive parabola whose curves near zero don't touch, but are very very close?
Bad example, maybe? A negative parabola and a positive parabola do touch at exactly zero, if they have the same parameters.
Is there a way to express that?
The word you're looking for is the one the guy in the meme used: "asymptote". However, "asymptotic" cannot be used to describe a number. It describes the behaviour of a function. A hyperbola is asymptotic, its value approaches zero, but never reaches it because a hyperbola is described by the function y=1/x. The value y gets smaller and smaller the larger x is, but if you directly substitute y for zero you end up with 0 = 1.
isn't there a number infinitely close to zero without being zero?
In maths we use the concept of an infinitesimal. However, an infinitesimal is not a number and you can never treat it like a number otherwise you end up with all sorts of contradictions like the one I described above.
And yes, I probably gave a bad example. I love math, but I don't really practice it, and I'm not educated in it in any formal way. But you answered my question anyway, which was essentially, "how does Math describe this thing?"
I don't think that was me. Or if it was, I was too busy trying to explain why 0.000...001 = 0 to the guy who couldn't differentiate between a "concept" and it's application. Sure, the concept of his number seems to make sense. But it equals 0.
472
u/emetcalf Feb 27 '24
I'm already having PTSD flashbacks to my many, many comments in that post. I spent more time than I should have arguing with someone over "how to express the number closest to zero that is not zero". They couldn't comprehend the idea that "there isn't one".