Let me explain further my question by what i mean 'operation'. An 'operation' can be on a single number, example sqrt(2). An 'operation' can be 'binary' between two numbers, example 1+2. An 'operation' can be performed on more than two elements, example placing vectors on a grid and finding 'sum' by 'tail to head' method. I know that my last example can be reduced to performing 'binary' operations on vectors, however, if you place THREE vectors on a grid tail to head and you connect the tail of the first vector to the head of the third vector, you now have a new vector which is the 'sum' of the 3 vectors you started with. This can be done for any amount of vectors. To me, this does not LOOK like a 'binary' operation when done this way
Hi agentredlum! I'm really not sure what you're asking or why you are asking it. But, judging from what you want to hear. There are uncountably many "operations". We can take the n'th root of any number x: [tex]\sqrt[n]{x}[/tex] and we can do this for any real (nonzero) number n. Thus there are uncountably many operations...
Well, then that means there are more DISTINCT 'operations' that you can perform on the set of integers than there are integers themselves... FANTASTIC! I was worried that my 'notion' of 'operation' was a little too loose but i am happy that there is someone out there that does not find it too offensive. Taking roots is one FORM of operation, and i agree with you (happily) that there is an uncountable number of ways to take roots if you allow TRANSCENDENTAL exponents. Would you say there might be an uncountable number of FORMS? Example trgonometric functions may be considered a different form than taking roots. I am worried because given ANY real number you can write a power series expansion for it and that only uses the basic 4 'binary' operations (A COLOSSALL ACHIEVEMENT) but that doesn't mean there aren't any other operations out there we are not aware of. I'm not writing a paper, I'm not in school, just curious and fascinated by INFINITY. Thanks so much for your time.
Here is what i think i know so far. If any of this is wrong, please, somebody correct me. 1. The set of real numbers is uncountable 2. The set of rational numbers and the set of irrational numbers are DISJOINT, union of both gives the set of real numbers 3. The set of rational numbers is countable, the set of irrational numbers is uncountable 4. The set of irrational numbers includes algebraic irrational and transcendental irrational, these 2 sets are DISJOINT 5. The set of algebraic irrational is countable, the set of transcendental irrational is uncountable Conclusion:The uncountability of the Real Numbers is a consequence of the uncountability of the transcendentals Question:are there irrational numbers that are not algebraic or transcendental?
Hmmm...let me reword my question more carefully since it does not capture the 'spirit' of my inquiry. Are there irrational numbers that are not algebraic or ONLY transcendental. Let me explain by way of example between irrational and transcendental irrational. 1. sqrt(2) is irrational but algebraic 2. e is irrational, NOT algebraic so there exist irrational numbers that have a distinct property that is not shared by 'proper' irrationals (such as sqrt(2)) These irrationals by definition called transcendental Question:are there transcendental numbers that have a distinct property that is not shared by other 'proper' transcendentals such as pi, e? Why do i ask? To say sqrt(2)) is irrational conveys all information about its properties. To say e is irrational is true but does NOT convey all information about its properties since the transcendental nature of e is absent from the statement. Can a statement be made, "X is irrational, not algebraic, transcendental AND ALSO ________" fill in the blank so in this sense X differs from normal transcendentals like pi, e. I am aware that the union of algebraic and transcendental gives Reals, also aware that the union of rational and irrational gives Reals, but the latter does not provide any information that there exists a very important class of numbers within the irrationals.
I'll try to dissect this as best as I can. I can't make sense of this question. There are irrational numbers that are algebraic. (sqrt(2)). There are also irrational numbers that are transcendental (pi, e). Since transcendental is the dual of algebraic, it's either one or the other. What is your definition of a 'proper' irrational? From first glance, it's an algebraic irrational. But sure, irrationals that are not 'proper' you can define as transcendental. I'm confused. I thought proper meant an algebraic irrational number? Ignoring that, that kind of properties are you talking about? The only properties that you've listed thus far are algebraic and transcendental. Again, what properties are you talking about? The fact that it's algebraic? If so, I'll agree. It's very easy to show that sqrt(2) is algebraic. This is true. It's not an easy task to show an arbitrary irrational number is transcendental. Again, what kind of property are you looking for? I can easily fill in the blank and say greater than 0, less than 0, divisible by pi, etc.. Some other properties could be much more difficult to prove. This is true. Some properties of the irrational numbers are easy to verify ( > 0, < 0). Some are not (transcendental vs. algebraic). It just depends on the property that you're dealing with.
Thank you for your reply, talking about this helps me a lot. Yes, I would call an algebraic irrational a 'proper' irrational but the word 'proper' i would not confine to mean only algebraic. i would also use it to describe transcendentals. Are all transcendentals 'proper'?? I am asking if there are transcendental numbers that can be put in a class with a property that is not shared by ANY other transcendental numbers. Example:sqrt(2) is irrational, pi is irrational but pi 'enjoys' the property of being transcendental and this property is not shared by sqrt(2)) so pi is not a 'proper' irrational. so if you put sqrt(2)) and pi in the same class somehow you are holding back information. They really don't 'belong' together because pi is a different kind of irrational. (My opinion) Replace irrational with transcendental in the above example Example:pi is transcendental, X is transcendental but X 'enjoys' the property of being ________ and this property is not shared by pi or any OTHER 'proper' transcendentals similar in properties to pi. Again, 'proper' here being used simply as an english word and NOT as a mathematical definition. I think historically it went something like this..from the integers you went to the rationals, then irrationals, then reals. At some point you had to go back because a class of numbers was discovered within the irrationals that had a PROFOUND property that separated them from other irrationals. So my question, by way of analogy, is there a class of numbers within the transcendentals, that has a PROFOUND property which separates them from other transcendentals? I am not talking about 'trivial properties' like "6 is the only number that gives 3 when divided by 2"
No, there are too many of them. If by "property" you mean something you could write down as a formula or algorithm, then there are only countably many of them. If a formula or algorithm is a finite string, and your alphabet is countable, then there are only countably many strings of length 1; countably many strings of length 2; dot dot dot; and taking the union over all n, there are only countably many possible finite-length strings. But there are uncountably many reals. So most reals can't be named, described, computed, or characterized by any finite string of symbols, even if you have countably many symbols. The vast majority ("almost all" is the technical phrase) of the real numbers are an amorphous, unnamable, indescribable blur.
I agree with everything you say except the first sentence in your reply. Why do you say no? Are you saying that ALL PROFOUND properties that distinguish one class from another become trivial because there is an UNCOUNTABLE infinitude of them? The question still remains...is there an UNCOUNTABLE subset of transcendentals that has a property not shared by any other UNCOUNTABLE subset of transcendentals? My interest is not in programming so i would allow infinite series, after all, no one can write down ALL polynomials, no one can write down a SINGLE infinite polynomial, no one can write down the COMPLETE diagonalization in Cantor's argument, yet these ideas are used 'freely' when describing properties of real numbers.
No, I didn't say that. What I'm pointing out is that there are only countably many finite-length strings, but uncountably many real numbers. So most real numbers can't possibly be named or characterized by a formula or algorithm. There's a mathematical theory of what numbers can be defined. http://en.wikipedia.org/wiki/Definable_real_number There are other ways to get at this. For example the definable numbers are a little different than the computable numbers. But regardless, only countably many numbers can have names or descriptions. So we have no way to say "this real number is different from that real number because of such and so property," except for a countable set of reals. The rest have no names or descriptions. Well sure, the uncountable set of transcendentals greater than or equal to pi is different than every other uncountable set of transcendentals. But note that you could only do this trick with countably many transcendentals, because we can have at most countably many different names for numbers. So if you have two real numbers that can't be named, how can you distinguish between them? Yes, we can describe properties of the real numbers. For example in set theory we can prove: "There are uncountably many real numbers." But we can name only countably many of them. That's a bit of a philosophical mystery. You might be interested in constructive mathematics, where a mathematical object is not said to exist unless we have a specific construction for it. http://plato.stanford.edu/entries/mathematics-constructive/
WOW! How come i didn't think of that? Very nice trick sir, the "or equal to" is what makes it work, it makes it BRILLIANT!! Also accept your note very seriously and heed your warning about this trick. Hopefully the trick can be extended and used elsewhere. I came close...i considered uncountable transcendental subsets GREATER THAN any given transcendental but i did not come up with "or equal to" THANX! Now having said that i must also say that i am a little disappointed...is this the best that we can do? Can't we find a more PROFOUND property? For instance the property that distinguishes transcendental from algebraic is PROFOUND (AMAZING) and not just a simple 'order relation' Thanx again because i have learned something very important today, the links you provided were also very helpfull. I've got some thinking to do, have a GREAT day!! PLEASE POST MORE TRICKS!!!
Maybe you'll like the following two kinds of numbers: Computable numbers: http://en.wikipedia.org/wiki/Computable_number Definable numbers: http://en.wikipedia.org/wiki/Definable_number In general we have Rational --> Algebraic --> Computable --> Definable --> Real So these might be the classes you're looking for...
Thanx for the links, i think i have a rudimentary understanding but i want to learn more. It occured to me, after my previous post, to start a thread on MATH TRICKS. Tricks are awsome! (trix are for kidz) I hope you can participate. It is in Number Theory category but the trick could be about anything you find interesting, even physics. I hope you post, 4000 posts you must have a few tricks under your thinking cap? I hope everyone posts a lot so many can benefit.
Rational --> Algebraic --> Computable --> Definable --> Real Hmmmm......is there any class between Definable and Real? Let me give an 'IMPORTANT' example of each class, so you can get an indication of how much i understand, hopefully i won't make a mistake. Rational, 1/3 I will send $10 to the person who figures out why i picked 1/3 Algebraic, sqrt(2) sorry, no prize here...LOL Computable, e Definable, Chaitins omega Real, all numbers (division by zero not allowed)
I'm a little concerned that I haven't been clear enough to explain this. There is no difference between using greater-than or greater-than-or-equal. Either way, you get an uncountable set of transcendentals that differs from any other uncountable set of transcendentals. But you can only do this with a countable number of transcendentals. You can't distinguish among the rest of them using any finite string of symbols. There is no "trick" in using [itex]\lt[/itex] versus [itex]\leq[/itex]. If you can explain why you think these are different or that one gives a great insight as opposed to the other one, then perhaps I'll be able to explain what I'm talking about more clearly. Perhaps you could defined what you mean by "property." I'm taking it to mean something you can describe in a finite number of symbols. What do you mean by property?
What do you mean by "distinct property"? Every number has a "distinct property" different from every other number- that of being itself. And the fact that sqrt(2) is algebraic is missing from the statement that it is irrational. I still don't see what point you are trying to make. Yes, of course, in any set of numbers, you can find some properties that distinguish some of them from others. But what, exactly, are you looking for?
Well sure, the uncountable set of transcendentals greater than or equal to pi is different than every other uncountable set of transcendentals. But note that you could only do this trick with countably many transcendentals, because we can have at most countably many different names for numbers. So if you have two real numbers that can't be named, how can you distinguish between them? If you do not include 'or equal to' in your trick then i choose 2 DIFFERENT sets that can satisfy 'greater than pi property' 1st set... 'The set of all transcendentals greater than or equal to 2e' ...2nd set... 'The set of all transcendentals greater than or equal to 3e' ...these 2 sets are different yet they 'enjoy' the same mentioned property, namely they are both greater than pi. Many more can be picked, an uncountable number of them and they can all satisfy greater than pi. So this property ALONE is not enough. Do you see now why 'or equal to' is so brilliant? Greater than or equal to pi defines only ONE set...UNIQUE!! Just like you said in your post...read it again, YOU SAID IT!!!!! So you have found a property that makes an uncountable set of transcendentals (without omissions of course) UNIQUE What do i mean by without omissions? It is not allowed to say...Take the set of all transcendental greater than or equal to pi TWICE. Remove 2e from the second set. Now you have 2 different sets sharing the same property and UNIQUENESS COLLAPSES. What i mean by property is like 'greater than or equal to' personally, i don't find this property interesting, unless it turns out to be the only one possible in my search, then it will be VERY interesting to me. Finite length and not too complicated to state. Another property i find more interesting is 'not the root of any polynomial equation with integer coefficients' This is what i mean by property, nothing too fancy, although if you have a fancy complicated idea i would like to hear it. Thanx again man. Let me know if i got my point across.
I'm looking at your markup (by hitting Reply but not writing a reply to the earlier post) and you have an open QUOTE and a close QUOTE around your entire post. If you just surround the parts I say with QUOTE tags and leave your own text outside of QUOTE tags it all works. In general I use the Preview Post button till I have the markup right. It usually takes me much longer to do the markup than it does to just write down the post itself. I appreciate your crediting me with brilliance but there is no substantive difference between less-than and less-than-or-equal, leading me to think I'm not communicating the meaning of this example to you. If [itex]\{x_1, x_2, ...\}[/itex] is a countable set of transcendentals, then the sets [itex]A_{x_n} = \{x \in R : x[/itex] is transcendental and [itex] x_n \lt x \}[/itex] are each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals. The sets [itex]A_{x_n} = \{x \in R : x[/itex] is transcendental and [itex] x_n \leq x \}[/itex] are also each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals. It only differs from the other set because it includes [itex]x_n[/itex]. I posted that example in response to your request to show some property that can be used to distinguish one uncountable set of transcendentals from another. But you can only form these sets for a countable number of transcendentals [itex]x_n[/itex], because you have no way of naming more than countably many transcendentals. This point is valid no matter whether you illustrate it using [itex]\lt[/itex] or [itex]\leq[/itex].