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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.
My "easier" way to think of this one:
1 - 0.999... = 0That is, if you subtract 0.999... from 1, what is the result? It's an infinitely small value, which can only logically be expressed by 0.
Is it not 0 . ...1 ?
That's exactly what it is--but when the "..." is infinitely long, you never get to the "1". There is no "1" at all.
Actually infinity is easy to understand.
If you were to walk in a straight line, you would never get to the end of the earth - it is infinite.
Its finitism that is impossible to understand.
3*(1/3) = 3/3 = 1 3*(1/3) = 3*(0.333...) = 0.999... 0.999... = 1
This a problem of the number base you're using, not infinity. One third is a finite number which cannot be expressed in base 10.
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
No, because that "some point" will never happen. There is no last nine to round up, because if there were a last nine, they wouldn't be infinitely many.
There are many different proofs of this online, more or less rigorous.
Because it isn't 0.9; it's 0.999... with the ellipsis saying "repeat this to the infinite" being part of the number. And you don't need to round it up to get 0.999... = 1, since the 9 keeps going on and on, so their difference is infinitesimally small = zero.
Another thing showing that they're the same number is that there is no number between them. For example:
One way to tell if two numbers are equal is to show there’s no real number between them. Try to formulate a number that’s between 0.999… and 1. You can’t do that.
But between 0.999 and 1 is 0.9999.
If something comes ever increasingly close to, but never physically touches something else, would you say it’s touching it?
0.999… means infinitely repeating 9s. There’s no more 9 to add that hasn’t already been added. If you can add another 9, then it’s not infinitely repeating.
So it never ends, and it stays 0.9… infinitely?
Still not a 1.
It's an infinite number of nines after the decimal.
Or think of it another way. What number would you subtract from 1 to get 0.999... ? The answer is 0.
let x = 0.999...
so 10x = 9.999...
subtract first line from second:
9x = 9
divide by 9
x = 1
an asymptote 😎
In the real world when you see .9 you often should round it. You rarely have as much precision as presenting - .5 should generally be seen as 1 unless you have reason to believe the measurement is that precise.
.333... Is a third. That's just a quirk of base 10. If you go to a different number system you won't run into that particular issue.
The most common other base people know of is binary. Base 2. So in binary the fraction would be 1/11 and then 1/11(binary)=1/3(base 10).
I remember talk back in the day that base 12 is good for most common human problems. Some people were interested in trying to get people to switch to that.
1/3 of 12 is 4.
So 4/12=1/3=3.33333.../10
.333... Is just the cursive way of writing 1/3.
I still don't "grasp" infinity. I'd recon you'd need an infinite mind to grasp infinity.
1/3 of 10 is 3
3 x 3 is 9
Yet
1/3 of 1 is .3
.3 x 3 is 1?
Just does not compute for me.
1/3 of 10 is 3.333...
1/3 of 1 is .333...
It's like when people come to America and are surprised when tax isn't included in sale prices. The .0333... you forgot to add on will get you in trouble with the universes math IRS.
0.999… means the value of the limit of a sequence {0.9, 0.99, 0.999, …} as number of 9s (or length of a sequence) goes to infinity, and the limit is very clearly 1 in my mind.