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Infinity. We're simply too dumb to grasp it. Example:
That "..." means "it continues to the infinite". And yet when you show this reasoning to people, they keep "looking" for the last 9, to claim that 0.999... is not the exact same as 1.
And that applies to all humans. You might counter it rationally, you might train yourself to recognise "it's infinite, so theoretically it'll behave in a certain way", but you don't grasp it. I don't, either.
Why would it be the same as 1, wouldn’t it always be 0.9 unless you round up at some point.
There's lots of proofs for this but this is the simplest one.
.333... = 1/3
.333... • 3 = .999...
1/3 • 3 = 1
Therefore .999... = 1
Why is .333 being treated the same as a third?
You could have .3 of 2.7 and that wouldn’t be a third. So I don’t see why .3 times 3 would be anything other than 0.9?
In this case you literally divide 1 by 3. And that's 0.3333 . And if you multiply 1/3 by 3 you get 1 and if you multiply 0.3333 by 3 you get 0.9999. So these two are the same.
.333... Not .333
The "..." Here represents an infinitely repeating number.
In this context 1/3 = .333...
Just pretend I added dots. But that still doesn’t change anything?
Imagine a pizza, I can divide that pizza into halves, thirds, quarters, etc. because conceptually they represent splitting a defined thing into chunks that are the sum of its whole. 1/3 can exist in this world of finites.
0.333… is unending. I can’t have 0.333… of a pizza, because 0.333… is a number and that makes as much sense as saying I’ll have 2.8 pizza. Do I mean 2.8 times a pizza, 2.8% of one? Etc.
1/3 being equal to .333... Is incredibly basic fractional math.
Think about it this way. What is the value of 1 split into thirds expressed as a decimal?
It can't be .3 because 3 of those is only equal to .9
It also can't be .34 because three of those would be equal to 1.2
This is actually an artifact of using a base 10 number system. For instance if we instead tried representing the fraction 1/3 using base 12 we actually get 1/3=4 (subscript 12 which I can't do on my phone)
Now there are proofs you can find relating to 1/3 being equal to .333... But generally the more simplistic the problem, the more complex the proof is. You might have trouble understand them if you haven't done some advanced work in number theory.
Is there a number system that's not base 10 that would be a "more perfect" representation or that would be better able/more inherently able to capture infinities? Is my question complete nonsense?
Different bases would have different things they cannot represent as a decimal, but no matter what base you can find something that isn't there.
For real world use base 12 is much nicer than base 10. However it isn't perfect. Circles are 360 degrees because base 360 is even nicer yet, but probably too hard to teach multiplication tables.
I get its basic shit that’s over my head. I’m just trying to understands
If the only reason is because 1/3 of 1 = 0.9, than id say the problem is with the question not the answer? Seems like 1 cannot be divided without some magical remainder amount existing
If I have 100 dogs, and I split them into thirds I’ve got 3 lots of 33 dogs and 1 dog left over. So the issue is with my original idea of splitting the dogs into thirds, because clearly I haven’t got 100% in 3 lots because 1 of them is by itself.
Likewise would 0.888… be .9? If we assume that magical remainder number ticks you up the next number wouldn’t that also hold true here as well?
And if 0.8 is the same as 0.888888888…, than why wouldn’t we say 0.7 equals 0.9, etc?
It's over the head of everyone. That's why I shared it here.
No, but 0.899... = 0.9. This only applies to the repeating sequences of the last digit of your base. We're using base 10 so it got to be 9.
Then you split the leftover dog into 10 parts. Why 10? Because you use base 10. Three of those parts go to each lot of dogs... and you still have 1/10 dog left.
Then you do it again. And you have 1/100 dog left. And again, and again, infinitely.
If you take that "infinitely" into account, then you can say that each lot of dogs has exactly one third of the original amount.
0.333... represents 0.3 repeating, which has an infinite number of 3s and is exactly equal to 1/3.
I don't agree that they are the same.
It's just that the difference is infinitely small
The difference is zero, so they’re equal.
Well, you state that as a fact, but I’m going to say that the difference is infinitely small, so they are equal