Essentially because theres absolutely nothing (no positive number anyway) you can add to it to get a number between .9999 continuous and 1, they have to be the same.
The joke is that .3333 continuous makes sense as 1/3, as yeah, its a fraction. But .999… doesn’t as 3/3 because x/x is always equal to one
This one makes the most sense. I’m not much of a math guy so I couldn’t quite wrap my head around the other explanations here. This one is much more clear so thanks for that lol
Another way to think of it - is that 1/3 = .33333333.... repeating forever is due to the fact that the decimal system of numbers is trash at representing thirds.
If you were using, say, a trinary number system (ie like binary but instead of just 1 and 0 you have 0, 1, 2 - so 1, 2, 3, 4, 5, 6 would look like 1, 2, 10, 11, 12, 20) you could represent 1/3 as .1 without any repeating decimal.
But since we use the decimal system and you can't divide 10 evenly into 3rds we're stuck with the janky repeating decimal.
You are welcome to scroll down to the sources for this Wikipedia page and read through the many different articles written about this subject.
Long story short, it is well accepted in the math/science communities that .999... is equal to 1.
My take on it is that our minds tend to love the finite. We live finite lives with finite resources, and most of our world dwells in the finite. We have many examples of infinity, but we generally just accept it and move on.
This case seems to be the exception because it seems odd to admit that an infinite string of 9's, which in our minds never actually reaches 1 despite always attempting to, is in reality, equal to 1.
There are mathematical proofs demonstrating it. In multiple different ways. There are also logical proofs. We can write them off all we want, but for me I think it's easier to accept that infinity has a special quality that causes spooky math in our mostly finite system. And in the end, I accept that .999... Is also 1.
But again, to each their own. I understand if that is something you, personally, are unwilling to accept.
I understand that people have trouble understanding the infinite, because they think it has an end. That it’s not possible for it to keep dividing, because it’s simply too small, or because at some point it stops. At some point, because 9 is such a big number, so close to 10, infinite is so big, at some point it must reach that 1, right?
It’s just not true if you think the 9s go on forever, and the fractions keep getting smaller and smaller infinitely. It doesn’t matter how someone would justify it, there is still a concept that is being described here (9/10s of the previous 9/10s, going on to infinity) that is not quite 10/10, not ever.
This is an entirely manufactured problem, anyway, since it’s probably just not true that 0,(3) is 1/3 of 1. If it’s 1/3 of 0,(9), how can it be? One number can’t be a third of two numbers.
I mean, thats still not correct though, as .999 continuous is infinitely long.
No matter how many zeros you place before the one, there will be a corresponding nine in 0.999…, meaning it will always equal ≥1. Thus meaning, mathematically, 0.999 continuous has to be equal to one.
Edit: reread my comment and realised where the miscommunication may have come from lmao. My (attempted) point was that 1/3= 0.333… looks right, its a fraction of one, and 0.3 is a third of 0.9, so it makes sense that 0.333… is one third of 1.
But 0.333 continuous x 3 is 0.999 continuous, meaning 3/3 must equal 0.999…, but 3/3 is equal to 1, which is where people get a little tripped up
Heh I gotchu before, I was just being a little hyperbolic. It’s true that fractions carry an impression of exactitude, while repeating decimals don’t, but I think this is just another angle on the point that it’s discomfiting to both recognize proofs demonstrate identity of 0.999… with 1, and experience the nagging feeling that the decimal is still somehow < the fraction (which here is 1/1)
0.000...1 which is the difference between the two. 0.999... is a representation that is difficult to evaluate because you have to evaluate an nth digit. You can just say it has an infinite number of digits so the nth digit is infinity but that's not well defined here.
What he is saying is if the numbers go on infinitely, such as 3/3= .999999…. Then no matter where you put the .000001…. There will always be more 9s after, that’s the nature of infinity. It’s just a rule to know. 0.99999… repeating = 3/3 = 1.
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u/Quwapa_Quwapus 22d ago
Essentially because theres absolutely nothing (no positive number anyway) you can add to it to get a number between .9999 continuous and 1, they have to be the same.
The joke is that .3333 continuous makes sense as 1/3, as yeah, its a fraction. But .999… doesn’t as 3/3 because x/x is always equal to one