I dunno, I mean if you were talking about, for example, the speed of light then you could say you were travelling at 0.99999999 etc. the speed of light but would never actually reach it so in this context it would not be quite the same thing.
Yes, clearly, yes, but my point is that the number 1 or any other integer* can* be represented perfectly without any need to use rational numbers although, as you have correctly pointed out, you have the ability to express it as such.
edit: Should read ‘irrational’.
It occurs to me you might reasonably ask yourself an inverse question about numbers. What are all the numbers? Imagine you drop a stone from a height of one unit. How many numbers are there that represent the heights it falls though on its way to height zero? Imagine this in an ideal universe, one where there is no noise and no quantisation effects. Mathematicians call all the numbers the Real numbers. There is no definition of how to create real numbers, not like Natural number, Integers, Rational numbers or Algebraic numbers. It is simply the entire set of all possible numbers on the number line. (Which means they still obey all the properties we expect.) So the problem isn’t about what sorts of numbers can represent what values. It is just about working out various subsets of numbers. And in the end we discover that the non-algebraic numbers are almost all the Real numbers. The other subsets are infinitely smaller. Even those irrational numbers that form part of the set of Algebraic numbers are a vanishingly small component of the Real numbers.
Representation of numbers is mostly just a matter of notation. Some infinite sums can be condensed to a notation that is a repeating decimal. It is just a notation, just as any non-repeating decimal is a shorthand for a ratio(nal) representation. Representations don’t convey much more than that. 2/2 = 1/1 = 1 just means that the Integers are a subset of the Rationals. Which is important. If they were not, we would end up violating the closure property we require of our system of numbers.
What is probably another core point to understand is that the modern understanding of how we get to these numbers is based upon the properties we require them to have. We don’t get the properties for free (usually) but rather we are confining the various possible forms of number to obey important properties that make them useful. I wrote of closure. We really do want a system where the application of the operations we need don’t spit out things that are not numbers. Sadly we have to wear things like division by zero. There is a whole area of Pure mathematics devoted to the study of systems with these sorts of properties (Group Theory). This shows that it is impossible to get to a system that is both useful and utterly “perfect”. Which is again a slightly surprising result. It isn’t just a result of our peculiar choice of numbers, but a result of the requirements we place on numbers.
Another important property we place on the Natural numbers is that application of the successor operation (you can think of this as “the next number after this one”) is unique. We express this (somewhat loosely) as saying: "There does not exist a number z such that succ(x) = z, and succ(y) = z, and x != y, and x and y are numbers. If we let this happen we would not be able to count, let alone add. And so it goes.
The way the properties of numbers pops out has been a surprise. As we discussed, the Greeks thought that rationals were the be all and end all. Irrationals in the form of algebraic numbers (ie the solution to x[sup]2[/sup] = 2) caused them enough mental turmoil, let alone having them confront the transcendentals. Yet they knew about pi. Just not enough.
Francis Vaughan, I know you are writing for good paedogogical effect, but why say something misleading like that there is no definition of the real numbers? There are and several have been given in this very thread.
The Greeks gave no such definition, of course, and yet they knew about quantities like √2, but these were regarded as magnitudes or incommensurable ratios and not as numbers.
I guess you have to give the OP credit for struggling with Zeno’s paradoxes and irrational numbers since he or she is following a fine tradition. And if he or she is really interested, the solution is to study some modern real analysis to learn how these questions have been resolved.
Yes, numbers are very versatile and my obsession with integers should not blind me to the usefulness of other kinds, as you have mentioned. Imaginary numbers even deal with the sqrt of -1 (I think) and so on it goes… I liked your analogy of dropping a stone from a height of 1 unit and passing through all the real numbers with which to described the stone’s height at any point because it provides a good visualization of the number system. A very thought-provoking post, thank you. 
No, guilty. Written in haste to try to make a point (badly.) :o
I have a query posed as two pairs of question-and-secondary questions, now that this thread is wrapping up, based on SD experience.
The first of each pair less important or even answerable as a question of how one particular OP ponders mathermatics, but concerns how some humans deal with cognitive challenges–that is, in the sense of “what good is algebra” drift in a recent long thread–what concepts people carry along in their mind.
The second of each paired question is about how mathematicians introduce/teach topics for those laymen to chew over first, here and in and in the infamous unending (heh) SDGQ thread “does .999…=1”.
Why did the poster (and many other laymen, like me) remember “.9999…!=1” as memorable, and worth fighting for (not by me) even after being explained for 100 pages of SDGQ?
1A)
Why did the focus of that thread, IIRC, by the mathematics teachers here, settle on the idea of limits?
Why isn’t statement " ‘1/2 != .5’ but ‘1/2 = .500…’ " (i.e., this thread) more well-known as a puzzler and, for many, worth fighting against?
2A)
Why, at least in this thread, did the pedagogy swiftly tend toward number theory and representation?
The focus didn’t tend towards number theory in this thread. I don’t think anyone has even brought up number theory. The focus has been on representation, because we pedagogues are pretty sure that representation is at the core of the OP’s misunderstandings.
And another problem with Francis Vaughan’s post is calling the real numbers “all the numbers”. If you want to look at “all the numbers”, then you have to go at least to the complex numbers, and there are many, many ways you can extend the concept of “number” even beyond that.
I think that is a bit unjust because I feel he was using that term for my benefit and my limited understanding of mathematics. I would not have thought that Francis Vaughan was unaware of what ‘all the numbers’ really meant. It isn’t easy to adapt your phraseology, when you are an expert, to cater for less able individuals.
Then what have we been discussing these last several days?
Philosophy of mathematics and real analysis.
If I understand correctly “Number Theory” is the term for the field that discusses the properties of integers. Fermat’s Last Theorem and Goldbach’s conjecture are number theory - what we’ve been talking about is (I suppose) analysis - the field that deals with limits, infinite series, etc.
For many people, a number is its decimal representation. It’s then understandable that lack of a finite decimal representation is seen as a serious flaw.
I try to ask leading questions, to see where the problem lies. For example, 0.99999… has a 2nd flaw beside its infinitude — its number has TWO different decimal representations. I asked the guy a year ago whether 0.3333… was just as bad as 0.9999… He agreed, briefly, that it was, but went back to repeating “I move my pencil from .99 to .999” :smack:
I tried to ask OP whether infinite repeating decimals had the same flaw, in his view, as irrationals:
… but got no answer.
I warned you. :rolleyes:
abashed. Is -1 an integer? Can you show me -1 balls in a bag? Can you subtract 5 from 3? What is x in 5x + 12 = 3?
Start from there and solve that. Then start working on irrationals.
Great cite. Thank you.
I am totally stealing this line: “It’s not just incorrect. It’s an entirely new category of stupid!” Yes, I’m well aware of the irony that the character shouting it was later proven wrong.
“0.999 <> 1” is obvious. “After all they’re written differently” is all the analysis needed to raise the question. Conversely “1/2 <> 0.5” requires more knowledge; the idea of precision. After all, “everybody knows zeros to the right are immaterial.” Something “everybody knows” is obvious. Turns out however, that it’s wrong or at least not fully right.
As discussed in the great “push vs pull” thread, most puzzlers come from people who were taught a simplified reality back in grade school and who absorbed the “truth” that “Things *are this way." Rather than the truer truth the teacher should have taught them: "Things are roughly *this way and that’s close enough for your needs today; But there’s lots more details and exceptions you’ll learn later if you pursue more math / physics / history / political science / religion / medicine / IT / whatever.”
2A) As said by others above above “number theory” is a particular branch of math that’s not really involved here. Industry jargon trips another newbie.
As to representation, the OP pretty well telegraphed that his bedrock confusion was his grade school teacher’s lesson that “A number *is * the thing we write with a string of base-10 digits.” That was good enough then, but isn’t when faced with irrationals. He was just now coming to grips with rooting out that bit of defective foundation and replacing it with something more generalized and robust.
Bleh. Left out a minus sign. 5x + 12 = -3. Or make it 5x +18 = 3 if you prefer.
Yeah, I know. I thought about that, but since we were talking about the number line I left it at “all”. Which isn’t strictly correct, but seemed reasonable for the purposes here.
“All” is pretty difficult to define. After complex numbers you still have quaternions and octonions. And that is just in forms that you can actually consider directly useful outside of pure mathematics. They are interesting in that octonions require you to give up commutivity and associativity. Then again many consider that something that makes them less interesting. Given that all of these are tuples of reals, for the purposes of representation issues I felt it was reasonable to stop at reals.
Similarly, in some sense, a matrix is a number. Yes, we can also think of a matrix as an ordered set of numbers, but the same can be said for complex numbers, and quaternions, etc.
And if you go Gödel, even equations are numbers.
Mathematics is beautiful and weird.
Its numbers all the way down… 