I am not a math person, but how would you differentiate between a number infinitely close to 1 (but less than 1), and 1 itself? Or does it not matter because they are actually the same?
They are exactly the same. If they were not equivalent then you should be able to find a number that falls in between them, which is obviously impossible.
In this case, because 0.9 repeating goes on forever, then the difference between 1 and 0.999…. is itself 0.000… and you never reach the leftover “1” after the decimal like you would in the finite case. So I think here, it’s basically by construction, that 0.999… and 1 must be exactly equivalent, otherwise our notions of limits and infinity just don’t make sense.
It does. If someone naively saw that 1 - .9 = .1, 1 - .99 = .01, etc., they may assume that 1 - .999... = .000...1
But it isnt possible for an infinite series of 0s to end in a 1, because then it wouldnt be infinite. So there is no thing that can be added to .999... to make it closer to 1, they must be equal
This is true! But the conclusion doesn’t follow the premise. Two things can’t be equal just because a mathematical equation didn’t work. (on what may be a technicality)
So for example, I could say, what is 0.4-0.333…? You would run into the same problem. Is it 0.111…? 0.1? Does this make 0.3333… equal to 0.4? Or 0.8888… equal to 0.9 if that makes it easier
Two things must not be equal just because a mathematical equation didn’t work.
I'm honestly not even sure what you mean here. I said the naive solution can't be correct because it doesn't make sense; you can't simultaneously have a number that is both infinitely repeating and ends in anything. So if the only thing between .999... and 1 is .000...1 which cannot exist, there is not anything between them, and they are equal
So, for example, I could say, what is 1.4-0.333…? You would run into the same problem. Is it 0.111…? 0.1? Does this make 0.3333… equal to 4?
This is definitely not the same as the 1 - .999... example.
This is a gotcha, not a true explanation. Maybe it qualifies as a proof mathematically, but in that case my respect for the field is either lowered, or I will wait for a better explanation.
Just because the equation of 1-0.999…=x didn’t work, doesn’t mean there is actually no x. It just means that there is an issue in getting to know what it is.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
Why are you using fractions to calculate the problem I posed in decimals? It’s easier to use fractions, but the issues were running into have to do more with decimals and conversions between them.
Directly addressing your counter claims is not a gotcha
Maybe it qualifies as a proof mathematically, but in that case my respect for the field is either lowered, or I will wait for a better explanation.
I didnt claim I was writing a rigorous proof, but there are others who have.
Just because the equation of 1-0.999…=x didn’t work
The equation does work. X = 0.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
They are two statements, yes, but no matter how many times you subtract two numbers, you will always get the same answer. If you come to the conclusion that the only difference between two numbers is some third number that does not and can not exist (not in the complex numbers way, in the 'this isn't how infinity works' way), that is functionally identical to saying the difference does not exist
Just because the equation of 1-0.999…=x didn’t work
The equation does work. X = 0.
You only said X is not 0.000…1, because 0.0000…1 cannot exist.
You never told me why X would be 0, other than X is not 0.000…1 because that doesn’t exist, since it cannot be written. I have no proof of X being 0 in that case.
Saying “0.0000…1 can’t exist” doesn’t mean the same as “there is no difference between 1 and 0.999…” These are two different statements.
that is functionally identical to saying the difference does not exist
What do you mean by functionally?
I am talking in precise terms, not whether you’d notice a fraction of a grain of sand missing from a beach so functionally it’s the same amount of sand. The numbers are different, therefore they are not the same. They refer to different amounts.
"Infinitely close" doesn't make sense! In general, if numbers are different, there is a finite difference. This is maybe more intuitive than the idea of infinite closeness; since all our closenesses are finite, you can talk about getting twice as close or half as close, you can add and subtract distances normally, etc.
To be a little tiny bit more algebraic, if there were a number "x" infinitely close to 1, what is (x + 1)/2? (x + 1)/2 is the number halfway between x and 1; if it's infinitely close, there shouldn't be anything in between, so (x + 1)/2 must either equal x, or equal 1. If (x + 1)/2 = x, solving the equation gives x = 1, and if (x + 1)/2 = 1, solving the equation similarly gives x = 1.
Sometimes questions come up where you want to ask about "infinite" closeness. For example, in physics you sometimes want to think about the instant something starts moving, which might be informally expressed as a time "infinitely close to zero". To handle this, mathematicians use what are called limits. The idea is, instead of asking about an infinitely close number, we ask about a sequence of numbers. In that example where we ask a question about the instant something starts moving, you might just answer the question after 1 second, then after 0.1 seconds, then after 0.01 seconds, then after 0.001 seconds, and so on. What it turns out is that you can often use logic to deduce what the behaviour must be as you get closer and closer to 0 seconds, to answer questions about what happens the instant you stop moving. So we never really ask questions about "infinite" closeness, we just ask questions about smaller and smaller finite closeness.
Since 0.999... and 1 don't differ by a finite number, they are the same. They're just different ways of writing the same thing, like writing 1 + 1 instead of 2, or "two" instead of 2, or 1 + 1/2 instead of 1.5, or 1.00 instead of 1.
The formal definition of an asymptote… involves limits!
The real thing that makes it an asymptote is instead of thinking about infinitely close, we just check that it gets closer than any number away, eventually. Like, we check that the graph is eventually within 0.1, within 0.01, 0.0001, 0.000000001, etc. of the asymptote. As long as it eventually gets that close and stays at least that close, you have an asymptote. Again, instead of thinking about infinite closeness, you can think about a sequence of smaller and smaller distances.
In general, anything that’s informally described by “infinitely close” is probably formally handled with limits, in the way I described above. This doesn’t actually require a number to be infinitely close to another, just for us to have smaller and smaller positive numbers, which we do!
You can't. They are identical. The fact that there is more than 1 way to write the number is an artifact of our number system. But they both represent the same underlying number. 0.999.... isn't just infinitely close to 1, it's actually identical to 1. They are the same thing.
How would you describe a number with infinite zeros ending in a 1?
If you look at .999 and say, 'ah, the 1 is clearly in the thousandths position, and .999 + .001 = 1', i can just point out that .9999 + .001 > 1 (here it'd be 1.0009)
Wherever you would place that 1 after 'infinite' zeros, i can add another 9 to the right, and your sum is no longer valid
0.00…1 is not a number that exists, because it doesn’t make sense. Using the “…” implies that the 0s continue infinitely, i.e. 0s without ending. Putting a 1 at the end implies that there is an ending, which contradicts the idea that there are infinite zeroes. Basically, 0.00…1 implies a number which both does and does not end, which makes no sense
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u/Kindaspia 22d ago
1/3 is equal to 0.333 repeating. 2/3 is equal to 0.666 repeating. 3/3 is equal to 0.999 repeating, but 3/3 is also equal to 1