No, but in reality you, any machine or natural process can never reach infinity because it would take an infinite amount of time. The mathematical representation of 'infinite' is just a way around that so the concept can be used in mathematics. I didn't present it as a way to define infinity, more like a way to view it as a concept.
But in reality problems run further than that. A computer cannot accurately compute/store/manipulate the decimal expansion of any rational number a/b where b has factors other than 2 or 5, since such rational numbers do not have terminating decimal representations in Base 10. Even if it were programmed to store the rational number as a and b, it cannot exactly manipulate any irrational numbers, so any dealing with things like pi, sqrt(2), e etc are beyond it's ability to exactly handle. Then you get into the entire realm of how to handle such approximations in terms of precisions etc. So if you define your mathematics by physical limitations, you find that the vast majority of otherwise well defined concepts become blurred and vague. Hence, it's better to gain a firm grasp of definitions within analysis on more understandable things like integers, rationals, algebraics and irrationals and then move onto the less familiar like infinities and infinitesimals because then you have a well defined framework to put them in.
That's very true and I know all that. I guess what I'm trying to say is that infinity only exists as an abstract Please Register or Log in to view the hidden image!
Actually, this is not quite true. Software packages such as Maple, Mathematica (and certain extensions to Matlab, even?) are able to deal with certain irrationals in certain contexts.
There's a package in C++ STL called "rational", but I think you have to start with rational numbers to end up with rational numbers. The same is true for mathematica, I think, unless you set some sort of tolerance or something.
Mathematica actually favors symbolic computation whenever possible, and you have to force it to give numeric results with N[] when you want the decimal expansion. So Alphanumeric's point remains since it seeks to avoid working with the decimal representation. (Sqrt[2]/3) * Sqrt[3] * Sqrt[6] / 4 is exactly 1/2 in Mathematica, for example, but is .49999999999999999999 in my better-than-average floating point tool of choice. About the only tool that I feel is strong for FP->symbolic is Plouffe's Inverse Symbolic Calculator http://pi.lacim.uqam.ca/ Which works wonders on .51166335397324424423
Sure, but you have to start with rational number in order to get rational numbers out. So, for example, if I type: \(\sqrt{0.5}\) Mathematica spits out a decimal approximation to the number. But if I type \(\sqrt{\frac{1}{2}}\) I may get \(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\). So, in other words, mathematica has some line of code which gives rationals if you start with rationals, and decimals if you start with decimals. That link is cool, thanks!
\(\infty - \infty \not= 0\), period. German mathematician Georg Cantor proved that infinity can have different sizes, and it could be easily demonstrated using limits. \( \lim_{n \rightarrow \infty} n = \infty\) now, obviously so does \( \lim_{n \rightarrow \infty} n^2 = \infty\) but also obviously, n^2 grows faster than n, so even though they both equal infinity, if these two limits were subtracted, it would not equal zero because one infinity is bigger than the other.
That is not what Cantor was talking about, at all. Part of the problem in this thread, and part of the problem with understanding the concept of infinity in general, is that the term is in a sense overloaded. Cantor's multiplicity of infinities relates to the cardinality of a set. The set (a,b,c,d,e) has a finite number of elements. So does the set (1,2,3,4,5). The two sets can be placed in a one-to-one correspondence with one another; they have the same cardinality. The set of all natural numbers is an infinitely large set, as are the set all rational numbers, the set of all real numbers, and the set of all possible curves in a plane. The set of all rational numbers can be placed in a one-to-one correspondence with the set of all natural numbers. Just like (a,b,c,d,e) and (1,2,3,4,5) have the same cardinality, so do the sets of all natural numbers and the set of all rational numbers. This is the first non-finite cardinal number, \(\aleph_0\). Any infinite set that can be placed in a one-to-one correspondence with the natural numbers is called a countable set. Cantor proved fairly early in his career that the set of all real numbers cannot be placed in a one-to-one correspondence with the natural numbers by a rather convoluted argument. His early proof was convoluted; much later he developed a very simple and very elegant proof called the diagonalization argument. The set of real numbers is larger than natural numbers. It is uncountably infinite. The set of all curves in a plane is in turn "bigger" than the set of all real numbers. Cantor also showed that the the set of all countably infinite sets (the power set of the countable infinite sets) is also uncountable. He also showed that there are no infinite sets with cardinality intermediate between that of the natural numbers and that of this power set. Since this is the next largest cardinal set, it is called \(\aleph_1\). How big are the reals compared to \(\aleph_1\)? That they are the same is the subject of the continuum hypothesis. This problem was listed as problem number one of the twenty three most vexing problems to mathematicians at the onset of the 20th century. Kurt Gödel showed that the continuum hypothesis was consistent with the standard axioms of mathematics. The second of Hilbert's twenty three problems was to prove that mathematics is consistent. Gödel showed that this goal was unachievable. Some mathematical questions are inherently unanswerable. Twenty some years later, Paul Cohen proved that the contradiction of the continuum hypothesis was consistent with the standard axioms of mathematics plus the axiom of choice. Camilus is talking about a different concept of infinity, the use of infinity in limits. This concept of infinity is not about the cardinality of a set. It is much closer aligned to the geometrical concept of the point at infinity.
I think you misunderstood me. I just said Cantor proved that infinity can have different magnitudes, and that it could be easily demonstrated that they have different magnitudes using limits to show that \(\infty - \infty \not= 0\)., or even infinite series. Im not saying Cantor was saying that, its just how my calc instructor made the idea more concrete. for example, would you agree that \(\sum_{n=1}^{\infty} n = 1 + 2+3+4+\dots\) diverges to \(\infty\) So will \(\sum_{n=1}^{\infty} n^2 = 1+4+9+16+\dots\) now since both equal infinity, if we were to subtract the second by the first, we would be subtracting \(\infty - \infty\). But its easy to see that the \(\sum_{n=1}^{\infty} n^2\) grows faster than \(\sum_{n=1}^{\infty} n\) so we would not get \(\sum_{n=1}^{\infty} n^2 - \sum_{n=1}^{\infty} n = \infty - \infty \not= 0\).
Your calc instructor should know better than to conflate Cantor's concept of cardinality with Weierstrass' concept of limits. They are very different things.
this had nothing to do with Cantor's set theory. You dont study that in multivariable calculus. It was concerning "what does infinity minus infinity equal?" I just said Cantor, using his one-to-one correlation was able to show that some infinities are larger than others. With that established, THEN, you can use limits and infinite series to show why \(\infty - \infty \not= 0\).
Logic and refutation: 1)"Any value" presumes an actual value which is indefinite, but real. 2)No, it doesn't. "Larger than any real number" is similar to the well defined analysis concept of "|x|<d for all d<0". This means that |x|=0. Larger than any real number means that infinity is not a member of the reals. If it were then infinity+1 would be a member of the reals, since the reals are a field, and you'd have a real larger than infinity. 1)Infinity "is not a member" of a set of logical numbers, called "the reals". As anyone who knows what a goddam number is should also know. 2)You continue to think of the mathematicians use of 'real' and 'imaginary' as the same as the everyday use of those words. 2)No, you continue to think that my use of the phrase "real number", means I think numbers are real. You prefer this interpretation, because you then imagine you have room to pour scorn and contempt. I don't equate the mathematical meaning with "everyday use" at all; I'm actually pointing out that mathematics is logic - logic is real because it's a real process. But you can't make something out of logic, or numbers. 2)They aren't. If a number is real then it means that it belongs to the set which is defined as the Cauchy completion of the rationals. If a number is imaginary then it means it's a member of the set produced by multiplying each and every element of the set of reals by i, where i*i = -1. The complex numbers are not 'complicated' but the set of numbers formed by linear combinations of elements from the set of reals and the set of imaginaries. 1)If a number is "real", it belongs in a certain logically defined set. Numbers are not physically real, for god's sake. 2) When I say "5 is real" I don't mean there's a planet somewhere which is populated by 5's, I mean that it satisfies the property that it's the limit of a Cauchy convergent sequence of rational numbers. It's the same as saying "3 is odd". I don't mean 3 is a bit weird and I don't want to hang around with it but I mean it can be represented in the form 2n+1 for an integer n. You are making the common mistake of thinking that because you've never opened a book on analysis in your life and you keep reading people using words you think you know the meaning of then your ignorant assumptions about end results are true. Or put another way, you're being a crank. 1)And you keep using "real" and "imaginary" to imagine complete bumpf into existence. In much the same way you also use logic, in that caustic manner you generally default to. Numbers are imaginary, logically. Let's see you invent a real number. When you've achieved that, let's see you invent its imaginary double, then make something real with them.
Nice straw men. Where did I make such claims? I can easily come up with a real number noone else has ever considered. \(\sqrt{2948278583}+10^{\pi\sqrt{5}-\ln(4\phi)}-\sqrt[3049]{395823848289482834}\) What exactly is its 'imaginary double'? A concept of your own concoction I think.
Uhh.. actually that one was considered by Ramanujan for transcendental phi when he was about 2 months old. As for imaginary double.. *chuckle* I think you are wrong. --- As for this 'discussion' on what the 'real' in real number means... :spank: :itold:
I never heard of imaginary doubles either lol. As for infinity minus infinity, show me how Im wrong, I see it as very apparent using these methods.
