Ok, sorry - you never quite know who you’re talking to. It seemed from what you wrote that you thought calculators don’t round unless you tell them to. Anyway, never mind - I just don’t get any obvious insight from that at all into the question at hand, but maybe that’s just me.
This is great fun when I teach concepts like improper integrals and series convergence to my high school students. Is the series annoying during its early terms? Who cares, lop off the first hundred trillion terms if it makes your life easier. Integration from x=3 to infinity is troublesome? Change the bound and integrate from x=Graham’s number to infinity if it’s more convenient. If we’re playing in infinite waters, staggeringly obscenely horrifically large numbers might as well be 3.
Not all calculators are created equal.
Indeed they can be quite different.
I may be old fashioned but I was taught the order of operations. 2+3x4=14 to me.
Point being whatever is considered “right” today is not necessarily how your calculator will calculate it (in my experience most calculators do the problem in order of input so would say the answer to the problem above is 20).
This is quickly becoming “How many high minded and pretentious arguments can we fit in one thread?” I wanted to call bullshit on your answer because that was not what I was taught in grad school but it appears that you are correct according to the current consensus among mathematicians.
It makes no intuitive sense whatsoever to say that all infinities are the same size but they seem to know what they are talking about according to Scientific American.
““There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities…. I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.”
The sentiment is lovely but mathematically inaccurate. One of the most mind-blowing facts a young mathematician learns is that, in a specific, rigorous way, there are exactly as many numbers between 0 and 1 as there are between 0 and 2, 0 and a million, or even in the entire set of real numbers! Don’t worry, it’s natural to feel dubious about that. It seems impossible that a set could be the “same size” as a set that contains it plus some other stuff! But that’s one of the marvelous mysteries of infinity.”
No one will drive us from the paradise which Cantor created for us.
Cantor’s ideas were controversial in 1891, but not when you went to grad school.
This is a thread about infinities, and Cantor had already been alluded to. It’s analogous to bringing General Relativity into a discussion of gravity and the solar system. It may not be required, but it’s hardly “high minded and pretentious”, and some things are not going to make sense without it.
Since I feel the OP was answered I have to say I cannot get my head around this one.
I cannot imagine how the set of infinite numbers between 0 and 1 is the same size as the set between 0 and 2 or 1,000,000.
Unless you want to say infinity is not a number therefore there can be no "bigger"or “smaller”.
Seems dubious to me though. There definitely seems to be a concept of bigger and smaller infinities.
This is where precision in language helps. The cardinality of the set of numbers between 0 and 1 is the same as the cardinality of the set of numbers between 0 and 2, because we can match them up in a one-to-one correspondence: x in (0,1) corresponds to 2x in (0,2).
The measure of the set of numbers between 0 and 1 is, as you would expect, half of that between 0 and 2.
Whack-a-Mole, read this thread and get back to us when you’re finished:
A proper subset of a set is a subset of that set that is not the set itself. When dealing with finite sets it is clear they’re the same size if and only if they can be put into a one-to-one correspondence with each other. It’s clear that no finite set exists that can be put into a one-to-one correspondence with a proper subset of itself.
The definition though of an infinite set is precisely a set that can be put into one-to-one correspondence with one of its proper subsets. We can easily show that such sets and such correspondences exist in familiar sets, for example take the natural numbers = {1,2,3…}, clearly the even numbers = {2,4,6…} are a proper subset, however the function from the natural numbers to the even numbers, f(n) = 2n, is a one-to-one correspondence.
Now not all infinite sets can be put into a one-to-one correspondence with each other, for example the natural numbers are an infinite subset of the reals, but they can’t be put into a one-to-one correspondence with the reals, so this is one sense that there are bigger infinities.
I was partially kidding only because a whole lot of very complex topics got introduced very quickly and mixed into one thread. I have been here for a long time and I know that even bringing up one of them is sure to bring up controversy but this thread has brought out the all-star group of questions all in one. I know something about many of them them but I am not a mathematician or theoretical physicist. All I am trying to do is help outside readers learn something in ways that helped me understand some concepts at all because plain English translation is one of the things I do best but I admit that the idea that there are the range of numbers between 0 and 1 is exactly the same as those between 0 and 1 billion to start the whole bullshit cycle all over again.
Don’t get me wrong. I know that you and many others know much more than I do on that question but I cannot even begin to understand why. I am reasonably intelligent and well educated but I don’t respond well to mathematical proofs. Is there a way to put the explanation into fairly plain language?
In his defense, he is asking more questions than that. The 0.999… is just the start and that has almost been settled in record time but there are others including approaching the speed of light and whether some infinities are bigger than others.
This thread has the potential to settle the meaning of life, the universe and everything once and for all.
This article (in four parts) is the clearest online “beginner’s explanation” I know of for some of these infinity-related topics.
It’s a long while since I’ve done any of it, but Cantor’s basic stuff is not difficult so much as weird and unintuitive at first.
His idea of “size” in infinite sets (called cardinality, or aleph number) depends on whether you can specify a one-to-one mapping between elements. Since infinite sets never “run out” of elements, it’s initially strange what things are deemed the same size by this definition.
For example, you’d think that there are “obviously” twice as many integers {1,2,3,4…} as there are even numbers {2,4,6,8…}. But not so, because you can map n<->2n such that every element of one set has a unique partner in the other set. With this mapping, if you specify any member of one set, I can tell you the corresponding member of the other set, with no “spares”.
So, when are infinities different in “size”? I will defer to the Wiki for this, because it’s not easy to write out in text, but Cantor’s diagonal argument shows that the reals are uncountable, i.e. you can’t form a one-to-one mapping between the natural numbers and the reals. It’s the first part of this article:
I also found this, which I glanced through and found fairly accessible. It’s written more technically, but Cantor’s stuff doesn’t really require prior knowledge of other branches of math, just careful logical thought. I’m going to go through this myself later for a refresher.
Jeez, I just noticed that Asympotically Fat already put up a post on the prior page with identical content on basic Cantor. Ah well, sorry for the duplication.
Stop right there. Infinity is not a number.
Think about that. And then stop trying to apply anything you know about numbers to infinity, because it’s something else.
Specifically, infinity is a set. Cantor actually invented set theory in order to rigorously define infinity and make it usable in math.
Until then, everybody, generally speaking, had the same problem you are having. They understood that 0.999… equally 1 and exactly one. But no thing and no process could ever be infinite, because infinite means unending. If you have something that never ends, how does it suddenly equal a definite thing?
Cantor answered that question. Make it a set that can be equal to other sets. So take the set of all points from 0 to 1. How large is it? We can’t count that high, but we can use logic and see that if we put those points into a one-to-one correspondence to the set of all points between 0 and 1,000,000,000,000,000,000,000 we can pair them all up in a logical way that leaves no points unaccounted for. Therefore all of these infinities are exactly the same size.
Are their other infinities? Yes. If you try to put all the points between 0 and 1 into a one-to-one correspondence with the infinite numbers of even numbers, then you find an infinite number of points left over. This is the difference between the countable infinite of the number line and the uncountable infinity of the real numbers. (As Riemann below said somewhat more formally.)
Are there still other infinities? Yes. Raise an infinity to a higher infinity and the result is yet another infinity. Which means there are an infinity of infinities, although those rest don’t match up with any concepts we can point to.
All of this means you have to stop thinking of infinity as a number. That’s the hardest leap to make. Numbers are arithmetic; infinities are mathematics. The difference is huge.
That’s Riemann above. That is, when you read the thread his post is above mine. Only in reply mode is it below. Time for bed.
Miscellaneous remarks.
(1) OP asks “where do mathematicians draw the line?” That’s the confusion right there: sometimes, including this case, it just doesn’t make sense to “draw a line.”
(2) I think the
1/3 = .33333…
3/3 = .99999…
intuition is perfectly fine. It has nothing to do with the finiteness of calculater precision. Multiply the infinite string of 3’s by 3 and get an infinite string of 9’s.
(3) Although the 1 = .99999… fact has little to do with cardinal numbers directly, there is an interesting connection. When trying to prove a fact like P(Aleph_null) = C you want to construct a precise bijection between the power-set and the set of reals. If you approach this by treating the reals as the set of decimal renderings, the .9999… = 1.0000… facts are problems that require jigglings. I became aware of this upon reading an interesting letter from Cantor to Dedekind. He commented that he was having trouble completing one of his proofs but correctly regarded the detail as just a nuisance.
(4) Last thread I dealt with the paradox by introducing the “Axiom of Archimedes”, but was booed down. But I think it’s good to remind us that the ancient Greeks had already dispensed with any difficulty. The Axiom (which was introduced to cope with some of Zeno’s paradoxes and implicit in Euclid’s work before Archimedes) insists that numbers aren’t different from each other unless they have a, well, difference! It may not be obvious that (1 - .999…) = 0, but the difference is smaller than any posititve number.
Infinity cannot be reached by finite incrementation of the finite. Infinity isnt a big number that you eventually reach by adding more digits; it’s a different thing.
Therefore, there is no line that can be thus crossed.