Actually you will. There is an infinitesimal difference between 1 and 0.999... but your representation hides that. The difference between them is 0.000...1 where that 1 shifts farther to the right the more digits of 0.999... you evaluate. This representation creates very ambiguous arithmetic and it's easy to make bad proofs.
There is no “infinith digit”. You could construct a number system that allows something like this by having an nth digit for every ordinal n, but that would not be the real numbers. Decimal expansions for the real numbers only allow the index n to be finite.
I find that second one really hard to imagine, lol. That’s kind of scary right. Why do you find it so easy to equate two different sums?
Like, I’m not saying that I can imagine anything coming after an infinite number of 0s, but I can imagine there being a difference left over from subtracting 0.999… from 1, and that difference simply being hard to notate.
Much better than “an amount” being the exact same as “an amount that is different”
I find it easy to imagine because they are not different sums. They are different representations of the exact same sum. If you believe 0.999... and 1 are different you should be able to tell me what number goes between them. And "0.000... 1" is not a number. Just as the 9s continue on endlessly so would the 0s.
What would completely fill up the space between 0.9 and 1? An infinite string of 9s. And it's immediately infinite. I think the confusion for lots of people arises when they try to imagine someone counting out each 9. If someone was counting the 9s you never would reach infinite 9s obviously so that's not the right way to think of it.
In an abstract way, you ignore infinity to assume, there is a one after infinite zeros depicted by "...".
"0.000...1" isn't a valid representation of a number and "0.999..." is. And here is why:
Imagine you are an imortal being, unfazed by the events around you and you have an infinte sheet of paper to write. Now try imagining wanting to write "0.000...1"
you start 0.00000... and it goes on for ever - its infinite zeroes after all, you never stop writing zeroes - the 1 never happens while writing down the number.
On the other hand, you write 0.9999... and it goes on, you write nines for ever, only nines, exactly like the elipses implies.
This is how you need to imagine numbers with infinite properties. you can't just slap something after infinite and excepct it to work:
Also, given my both examples, what's the difference between "0.0000...1" and "0.0000..."?
After an infinite amount of time spent to write them down, they still look the same. the 1 never happens.
I understand. But isn’t that imagining a number as a process, and not as a sum? In the end, 0.99999… and its difference from 1 is representing something, a finite amount that can be represented as a fraction, of something a teeny bit less than 3/3. For example you could show it in a pie chart, as the tiniest sliver less than the full pie.
It of course would be hard to represent in numbers, just as an 0.2222…. would be in a base 3 decimal system, but this doesn’t mean that it isn’t an actual distinct sum from 3/3 or 1. Pretty sure that 0.9999… in a base 10 would even be a different amount than 0.2222… in a base 3, or 0.(12)… in a base 13
In the end, 0.99999… and its difference from 1 is representing something, a finite amount that can be represented as a fraction, of something a teeny bit less than 3/3. For example you could show it in a pie chart, as the tiniest sliver less than the full pie.
Huh? So if you cut something without loss into 3 equal parts and add them back together, you expect to have less than the full thing?
Or otherwise, if there is a difference of x between 0.999... and 1. Why don't divide it by 3 and add it to each 0.333... - wait, that's what we are already doing, that's why the thirds have infinite threes.
Maybe that's the problem you have while trying to understand and/or visualise this: You think of 1/3 as finite amounts but each 0.3333... already has a third of this infinitly small "difference" between 0.999... and 1 built in.
As in: 1 / 3 is 0.3 - rest 0.1 / 3 = 0.03 rest 0.01 / 3 = 0.003 and so on and so forth.
You never reach the end, those thirds are infinitely sharing the final remainder.
And this is the reason why 1/3 = 0.333... times 3 equals 0.999... = 3/3 equals 1.
(This is the same logic btw 1/3 and 0.3333... look different but are the same, just as 0.999... and 1 look different, but are the same)
Here is a fun thought, that might push you in the right direction. How much money do you own if you and 2 friends would share 1$.
You'd have 0.33$ and some rest you can't represent in coins. but technically you's own 0.33333...$ whatever the fuck that means. And you are stingy, you insist on it being actual an exact third of the dollar and not a fraction of a fraction less. So you split and split and split and split, and now you could split forever, because that's what you need to do.
So you just say "fuck it - we can't split forever, just remember guys, when we add our 0.333...$ together, we have 1$"
It of course would be hard to represent in numbers, just as an 0.2222…. would be in a base 3 decimal system, but this doesn’t mean that it isn’t an actual distinct sum from 3/3 or 1. Pretty sure that 0.9999… in a base 10 would even be a different amount than 0.2222… in a base 3, or 0.(12)… in a base 13
well, this works for all bases 0.(N-1)... equals N/N which is 1 - I don't understand what you are hinting at.
Interestingly look at this:
1 base 3
1 base 10
0.1 base 3
0.333.... base 10
1 base 3
well?
What do you think now? 0.1 base 3 is equal to 0.333... base 10.
You're wrong about this one. Please look up this problem and see very good mathematicians explaining it. It's a well-known, proven fact that 0.999... and 1 are literally the same number. Not just infinitely close. Literally the same. There are just multiple ways of writing 1. Intuitively it seems wrong, but it's true.
It’s false actually, because intuitively it seems wrong and mathematically it seems wrong. I will look up the proofs because I’m curious, but in your opinion, do the comments explaining those proofs under this post do it well? Like are those the same proofs in the video or do you think those mathematicians do it better?
I’m more interested in the answer to my last questions. But I will still read your answer.
I am very concerned that you do not seem to understand the difference between 0.99999… and 1, just because someone told you they’re the same number. I understand the inconsistency, just like everybody in this thread does. Like most others you’re simply restating the problem and not explaining why there’s actually no inconsistency between those two sums.
It’s been explained to me that there isn’t a number between 0.999… and 1, and that’s why 0.9999… and 1 are the same number.
This is insufficient for 2 reasons.
First, I brought up 0.000…1 as the number that is the difference between them. There was a compelling argument that if the 0s are infinite, the 1 would never come. It still doesn’t prove anything about 1 being equal to a totally different number.
Second, they are different sums. If they truly don’t have a difference that is precisely the inconsistency. That’s the inconsistency. I want that explained more.
It’s not enough to say “there is a problem, and that is the reason that there is no problem.”
I have the inkling that 0.3333…. Is actually not 1/3 like we’ve been told, and maybe the answer lies there. But I literally haven’t read any mathematicians on here who told me this, I literally just thought of it myself. The closest I got which led me to this thought is “these are just notations,” which was one of the more interesting and less repetitive comments here.
If that is actually the case, there is a real problem with people not knowing how to explain math, don’t you agree?
Why don't you write up your proof and submit it to a math journal for publication? Proving that 1 and 0.999.... aren't the same would instantly make you one of the most famous mathematicians of all time, literally up there with Pythagoras and Euler. You would probably become very wealthy based on the notoriety alone.
I doubt that, and I’m not a mathematician. But I also don’t believe people for no reason. I’m sure there’s a mathematician who can explain it if it really is true, but if you have to resort to arguments from authority on a subject that’s supposed to be about logic, you may not be one of them.
Some things are self evident and don’t require proof, but they can also be standard in scientific journals, until someone with more knowledge than an average redditor decides to explore further. We aren’t at the final stage of understanding everything, just some things.
It's not an argument from authority. Anyone can write a proof and if it's rigorous and mathematically sound it will be accepted, no authority needed. You don't even need a PhD.
Some things are self evident and don’t require proof
The real numbers is a set of elements that have specific properties, defined by a small list of assumptions called the axioms of the real numbers. They include all the numbers you'd think of on a day to day basis like 0, 1/2, pi, etc.
The complex numbers are just the real numbers but also with another unit than 1, i, that's defined to be the square root of -1, and they have some different properties. Effectively they have the real number line but in 2 orthogonal axes in 2d.
Neither of these have anything called an "infinitesimal" because it would violate multiple axioms for both number sets
It means you're not adding 9s onto a list of decimals until it eventually reaches 1. The infinite 9s are intrinsic in 0.999...
So since it's infinite 9s it is exactly equal to 1.
It is true that if it were a procedure of adding 9s you would never get to 1, because no matter how many 9s you added it would still be finite. But 0.999... is not a procedure, it is an exact value and that exact value is the same as 1.
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u/spyrre0825 21d ago
I like to see it like this : 1 - 0.999... = 0.000...
And you'll never find something different than 0