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^ this isn't understandable if you never taken a real analysis class. I mean, you kind of have to know what rational numbers and cauchy sequences are.
Okay so basically I really dislike the decimal expansion...I think the concept of infinity makes things a bit wonky at times; from my linear algebra class I know this is not the first time mathematicians have had some issues with using infinity in arguments. But maybe I just need to revise my strict understanding of what a real number can and can't be; though it seems strange to me that you can equate a number to an "object" like 0.333... which does not even exist theoretically
Why doesn't it ''exist'' theoretically?
http://en.wikipedia.org/wiki/0.999...
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On June 19 2014 03:23 sOda~ wrote:Show nested quote +On June 19 2014 00:42 radscorpion9 wrote:On June 18 2014 06:39 PassionFruit wrote:On June 18 2014 06:08 wingpawn wrote:Not. That. Cool. The world in which 0.9999999999... equals 1 doesn't even make sense. But at least, we have this: + Show Spoiler + Just think of the decimal as a representation of a fraction. .3 repeating is 1/3. .9 repeating is 3 x 1/3. Ergo, one. I think if you were being consistent, you would have to deal with the same problem in asserting that 0.33333... equals 1/3. Since 0.333... is an infinitely long number, I think the problem is that you can't actually imagine the "complete" number, so you can never really say at any point that it is equal to 1/3, because the sequence by definition will never terminate. Infinity after all is *not* a number, so I would expect that an infinite series of 3's after the decimal would not be a number either. So equating that to a real number like 1/3 is problematic. edit: I think it might be best to consider 0.333... as being an equivalent way of writing (in shorthand) the limit of some function like f(x) = 1/x as x --> 3; something which is never reached by the definition of limit. edit 2: Actually I think it makes things confusing again, because of course the limit has to exist, and if its an infinitely long series of numbers, then technically there is no limit *that you can write in decimal form*. The only limit that exists is the precise fractional form. Okay so basically I really dislike the decimal expansion...I think the concept of infinity makes things a bit wonky at times; from my linear algebra class I know this is not the first time mathematicians have had some issues with using infinity in arguments. But maybe I just need to revise my strict understanding of what a real number can and can't be; though it seems strange to me that you can equate a number to an "object" like 0.333... which does not even exist theoretically. But anyway; to the OP, I also like math and I kind of wish I was completely devoted to it but I worry that my brain is not up to the challenge  . I hope it turns out to be your passion over the long term Formally the reals are the completion of QQ with respect to the "usual" absolute value (the arch. one). Thus when you talk of a real number you are really talking of is a cauchy sequence of rational numbers modulo the equivalence: (a_n) ~ (b_n) iff (a_n-b_n) converges to zero. Rarely do people write numbers like this; just dealing with the notation is pretty horrible. The decimal representation of a real number is a throwback to this definition; the real number with decimal expansion a_0.a_1a_2 a_3 .... corresponds to the sequence of rational numbers a_0, (a_0a_1) / 10, (a_0a_1a_2) / 100, .... . As mentioned this representation is not unique but its a reasonably good (mostly) canonical way writing down expressing a concrete real number.
Okay I spent a long time reading about this on Wikipedia, and going back to what you said. There are some technical difficulties I'm having, but as long as we consider any real number as actually representing a sequence of rational numbers, then that's okay, because then 0.999... isn't actually a number but an infinite sequence which makes more sense, and if 0.999... = 1 is reinterpreted to mean that the two Cauchy sequences of (0, 0.9, 0.99, ...) and (1,1,1,...) are equivalent, where equivalent means the difference between the sequences tends to zero, then that's fine.
But I want to understand something clearly. Does this mean there is no such thing as a single 'number' in R? It sounds like the whole thing is quite deceiving, and rather than dealing with numbers we are in fact dealing with sets of sequences that (either explicitly or not) converge to those values, with the operations between sets defined on the wiki page entitled "construction of the real numbers".
Then I spent an hour trying to understand this statement from the same wiki page:
For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...)
Equivalence is only defined between Cauchy sequences, so I don't even understand how they can cavalierly interchange a sequence with a single number and continue on talking as if its not a big deal. If I apply your explanation and state that the decimal expansion of pi is in fact represented by a Cauchy sequence, then that makes sense; but here they seem to be saying that "representation" and "equivalence" are the same thing!
On June 19 2014 04:32 zksa wrote:^ this isn't understandable if you never taken a real analysis class. I mean, you kind of have to know what rational numbers and cauchy sequences are. Show nested quote +Okay so basically I really dislike the decimal expansion...I think the concept of infinity makes things a bit wonky at times; from my linear algebra class I know this is not the first time mathematicians have had some issues with using infinity in arguments. But maybe I just need to revise my strict understanding of what a real number can and can't be; though it seems strange to me that you can equate a number to an "object" like 0.333... which does not even exist theoretically Why doesn't it ''exist'' theoretically? http://en.wikipedia.org/wiki/0.999...
I actually have taken a class called "Analysis 1", which was a full year course geared towards math specialists in my first year so I do understand enough to figure out what he means with supplementary education from the wiki page. I do know what Cauchy sequences are pretty well by now .
I treat the 0.999... part the same way mathematicians treat infinity. This is all theoretical anyway so its probably superfluous to say that, but anyway, we know that infinity is not a real number, and in the same way I can not accept an infinite decimal expansion of 9's as being a real number. To me it just seems seems like saying "infinitely large" and "infinitely close to 1" express a similar description of an object that can not be 'pinned down' to an exact value; whatever we imagine the value as being we would be constantly wrong as it is always growing larger or closer to 1. It just exists as a concept, or as Soda wrote as being represented by an infinite sequence of rationals, though in some cases that sequence can converge to the number in question if the limit *exists*. 0.999... does not "exist" so in this case I treat it purely as representing a Cauchy sequence.
The key point here being I never really understood what a "real" number means, I always operated off of intuitive definitions I learned through grade school; our class in university never explicitly went over how the real numbers were rigorously defined.
As long as they are defined properly then we can get around these difficulties, and I think for the most part my complaints were addressed by Soda. After reading the wiki page I can go back and understand what he's saying, though some things on the wiki page confuse me as I mentioned above
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I study math too. Complex Analysis will blow your mind. its like seeing through the matrix or like reading gods blueprint!
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I really don't like set theory. I wish people wouldn't insist on using it for every freaking thing. That's why I shy away from the more abstract maths, though I really enjoy applied mathematics.
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On June 19 2014 07:01 fmod wrote:
I really don't like set theory. I wish people wouldn't insist on using it for every freaking thing. That's why I shy away from the more abstract maths, though I really enjoy applied mathematics.
The world of holomorphy is the only one where I'm okay with doing analysis, cause everything works well.
I was like you OP, then I took a course about algebric geometry where we where up to the Nullstellensatz after 5 hours, and I never understood what the fuck schemes work  I've kinda lost my crush for the subject now...
Edit : also fuck categories, especially because doing a course on them doesn't occure to any teacher, but using it is no problem.
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On June 19 2014 05:55 radscorpion9 wrote:Show nested quote +On June 19 2014 03:23 sOda~ wrote:On June 19 2014 00:42 radscorpion9 wrote:On June 18 2014 06:39 PassionFruit wrote:On June 18 2014 06:08 wingpawn wrote:Not. That. Cool. The world in which 0.9999999999... equals 1 doesn't even make sense. But at least, we have this: + Show Spoiler +http://www.youtube.com/watch?v=5Yt9moC-peM Just think of the decimal as a representation of a fraction. .3 repeating is 1/3. .9 repeating is 3 x 1/3. Ergo, one. I think if you were being consistent, you would have to deal with the same problem in asserting that 0.33333... equals 1/3. Since 0.333... is an infinitely long number, I think the problem is that you can't actually imagine the "complete" number, so you can never really say at any point that it is equal to 1/3, because the sequence by definition will never terminate. Infinity after all is *not* a number, so I would expect that an infinite series of 3's after the decimal would not be a number either. So equating that to a real number like 1/3 is problematic. edit: I think it might be best to consider 0.333... as being an equivalent way of writing (in shorthand) the limit of some function like f(x) = 1/x as x --> 3; something which is never reached by the definition of limit. edit 2: Actually I think it makes things confusing again, because of course the limit has to exist, and if its an infinitely long series of numbers, then technically there is no limit *that you can write in decimal form*. The only limit that exists is the precise fractional form. Okay so basically I really dislike the decimal expansion...I think the concept of infinity makes things a bit wonky at times; from my linear algebra class I know this is not the first time mathematicians have had some issues with using infinity in arguments. But maybe I just need to revise my strict understanding of what a real number can and can't be; though it seems strange to me that you can equate a number to an "object" like 0.333... which does not even exist theoretically. But anyway; to the OP, I also like math and I kind of wish I was completely devoted to it but I worry that my brain is not up to the challenge . I hope it turns out to be your passion over the long term Formally the reals are the completion of QQ with respect to the "usual" absolute value (the arch. one). Thus when you talk of a real number you are really talking of is a cauchy sequence of rational numbers modulo the equivalence: (a_n) ~ (b_n) iff (a_n-b_n) converges to zero. Rarely do people write numbers like this; just dealing with the notation is pretty horrible. The decimal representation of a real number is a throwback to this definition; the real number with decimal expansion a_0.a_1a_2 a_3 .... corresponds to the sequence of rational numbers a_0, (a_0a_1) / 10, (a_0a_1a_2) / 100, .... . As mentioned this representation is not unique but its a reasonably good (mostly) canonical way writing down expressing a concrete real number. But I want to understand something clearly. Does this mean there is no such thing as a single 'number' in R? It sounds like the whole thing is quite deceiving, and rather than dealing with numbers we are in fact dealing with sets of sequences that (either explicitly or not) converge to those values, with the operations between sets defined on the wiki page entitled "construction of the real numbers". Then I spent an hour trying to understand this statement from the same wiki page: Show nested quote +For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...) Equivalence is only defined between Cauchy sequences, so I don't even understand how they can cavalierly interchange a sequence with a single number and continue on talking as if its not a big deal. If I apply your explanation and state that the decimal expansion of pi is in fact represented by a Cauchy sequence, then that makes sense; but here they seem to be saying that "representation" and "equivalence" are the same thing!
So if C is the collection of all cauchy sequences of rational numbers and A = (a_n) is one particular cauchy sequence in C then you can consider the set of all cauchy sequences which are equivalent (as we both described it above) to A; write C(A) for this set.
Then a real number X is a subset of C of the form C(A) for some sequence A. In this case A is the sequence "representing" our X
If B=(b_n) is another cauchy sequence which, lets say, is equivalent to A (so B is in C(A)). Then C(A) = C(B); this is because if C=(c_n) is a cauchy sequence equivalent to B then C is also equivalent to A, and vice versa. In this sense you can change the representative but you can be sure we are still talking about the same real number.
What this allows us to say is that X, as a subset of C, exists independently of of the choice of representative.
Thus (in a certain sense) care needs to be taken when moving between a representative sequence A and the corresponding real number C(A). If for instance we try to define a map f: R-> R which sends a real number represented by a decimal expansion a_0.a_1 a_2..... onto the integer a_0 then you can ask what is f(1)? Since 1=C(1,1,1,...) is represented by 1.000... we have f(1)=1. Yet we also see that 1=C(0.9,0.99,0.999...) so f(1) = 0; our map f doesn't make sense. You need to be sure when the defining talking of a real number that what you are saying is independent of the choice of representative.
In terms of using the real numbers though all of this is, most of the time, unimportant; as you mentioned you did a whole course in analysis (as did I) without worrying about this kind of stuff. If you are doing analysis the key facts about the real numbers are that Q sits inside them, that the real numbers come with an absolute value, and the cauchy sequences of real numbers converge to real numbers (i.e., the reals are complete). This is mostly all you need to know.
On June 19 2014 07:27 corumjhaelen wrote:
Edit : also fuck categories, especially because doing a course on them doesn't occure to any teacher, but using it is no problem.
thats because a course on category theory would have anyone in the audience clawing out their eyes; i've never found anything quite as boring.
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On June 19 2014 08:06 sOda~ wrote:Show nested quote +On June 19 2014 07:27 corumjhaelen wrote:
Edit : also fuck categories, especially because doing a course on them doesn't occure to any teacher, but using it is no problem. thats because a course on category theory would have anyone in the audience clawing out their eyes; i've never found anything quite as boring.
Obviously, but taking more than 30min to introduce the vocabulary might be good.
In the same spirit my algebric geometry teacher said that specialists of category theory were doing intellectual masturbation. I guess we're always that for someone :p
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Do things like set theory or category theory have any real world applications?
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On June 18 2014 02:04 GeneralStan wrote:
Interesting to me that you're Real Analysis and Abstract Algebra in your first year. At my University you generally don't take that level of math until third year.
Also as an engineering/physics type, I"m going to go ahead and pass on Math classes aimed at Math majors.
One thing I've noticed about the mathematician's classes is that they have to prove everything. They also focus on edge cases and boundaries and try to break their own theorems. That's all well and good, and somebody has to do it.
But I view mathematics as a tool. Give me the formulae, tell me they work, and I'm more than happy to believe you
ahhh spoken like a true engineer
On June 18 2014 06:08 wingpawn wrote:Not. That. Cool. The world in which 0.9999999999... equals 1 doesn't even make sense. But at least, we have this: + Show Spoiler +http://www.youtube.com/watch?v=5Yt9moC-peM
hahahaha hated my maths units in 1st and 2nd year (get so damned sleepy everytime I attend class) but it not so much anymore because it made me get that joke hhahaha. I didn't even know what what I learnt was called Real Analysis.
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United States10328 Posts
On June 19 2014 08:08 corumjhaelen wrote:Show nested quote +On June 19 2014 08:06 sOda~ wrote:On June 19 2014 07:27 corumjhaelen wrote:
Edit : also fuck categories, especially because doing a course on them doesn't occure to any teacher, but using it is no problem. thats because a course on category theory would have anyone in the audience clawing out their eyes; i've never found anything quite as boring. Obviously, but taking more than 30min to introduce the vocabulary might be good. In the same spirit my algebric geometry teacher said that specialists of category theory were doing intellectual masturbation. I guess we're always that for someone :p
lol, it does seem to be a problem that it's necessary to learn a bunch of super dry notational stuff before being able to do algebraic geometry; for example I never took it because I could never fit commutative algebra into my schedule
My algebraic topology class did, very helpfully, take a week to talk about category theory at the beginning; I agree it's not terribly fun to learn "oh this is a colimit" without motivation, but many of the "toy examples" already have a lot of other theory behind them :/ so it's kind of a circular problem: you can't study X without understanding seemingly unmotivated concept Y, but the motivations for Y are X and some completely unrelated thing Z that you might've never encountered before. Then you end up really lost for a while until one day it magically all makes sense. It's like trying to build something where the pieces are all interdependent, and the only way you'll build it is to somehow put everything together simultaneously.
It's very helpful to work together with people on this kind of thing, and perhaps even better to have a friend or professor who has already put everything together who can give you perspective. On the other hand, you have to be very careful to still do your own thinking instead of just riding on other people's coattails... I suspect I coasted entirely too much in my math classes, and ended up with lots of surface comprehension and little true understanding.
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God I hate math. I had to take real analysis and numerical analysis to apply for econ graduate programs, I almost passed out from boredom in class every day. It just seems all so meaningless to me unless you put them in a real context, the only useful thing I've learned from years and years of math is probably stochastic calculus and mathematical statistics.
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Cool to read, I'm starting university in september and I'll be pursuing a Math degree. I love everything about math but I also understand why most people hate it or find it boring.
It's hard to explain for me why I like math so much.
It's also the pleasure of proving whether something is true or not (proofs). Analysing situations with a logical method which ensures you that what you find out is right if you proceed correctly. That's not something you can do in real life because everything is subjective.
Another thing I love about math is fantasy. Most people think that since everything in math is "necessary", that means there is no imagination involved. Nothing more false than that. There are always multiple ways to find the solution of a problem (or a proof) and you can have fun looking for the more simple, "elegant" or different one.
Then the next step is applying math to reality which is also extremely interesting. Here though, you will always be making mistakes because reality is always different (even if slightly) from math. Nonetheless, it gives great satisfaction when you solve a practic problem with methods that other people would think they are "just useless abstract formulas".
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On June 18 2014 06:08 wingpawn wrote:Not. That. Cool. The world in which 0.9999999999... equals 1 doesn't even make sense. But at least, we have this: + Show Spoiler +http://www.youtube.com/watch?v=5Yt9moC-peM
This is the proof i would give.
0.99999999999999999 =
0.9 + 0.09 + 0.009 + 0.0009 + .... =
9( 1/10 + 1/100 + 1/1000 +... )
the sum between the brackets is a geometric progression with common ratio q = 1/10.
Now the formula for a sum of a geometric progression is:
a( q^n -1 ) / ( q - 1) where "n "is the number of terms, "a" is the first term.
Here n is infinite, while the first term is 1/10, so you get that the sum is:
1/10 * ( (1/10)^ (infinite) -1 ) / ( 1/10 - 1) = 1/10 * ( 0 -1 ) / (1/10 - 1) = 1/9.
Multiply this by 9 and you get your result: 1/9*9 = 1.
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Mathematics are cool.
I guess there's no love for English u_u
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On June 17 2014 23:59 zksa wrote:
To take a Math class aimed at Math majors. It's so bizarrely different than your calculus/linear algebra (for engineers..) courses. I think Algebra and/or Real analysis are the coolest. Go do it, you won't disappointed. There are like zero downsides.
Is it an absolute zero or just a tiny fraction after a 0.00000... infinite series featuring like an infinite number of zeros before "1"? Also, if it is zero, how can I feel divided in my mind about the decision?
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On June 21 2014 08:00 KingAlphard wrote:
Cool to read, I'm starting university in september and I'll be pursuing a Math degree. I love everything about math but I also understand why most people hate it or find it boring.
It's hard to explain for me why I like math so much.
It's also the pleasure of proving whether something is true or not (proofs). Analysing situations with a logical method which ensures you that what you find out is right if you proceed correctly. That's not something you can do in real life because everything is subjective.
Another thing I love about math is fantasy. Most people think that since everything in math is "necessary", that means there is no imagination involved. Nothing more false than that. There are always multiple ways to find the solution of a problem (or a proof) and you can have fun looking for the more simple, "elegant" or different one.
Then the next step is applying math to reality which is also extremely interesting. Here though, you will always be making mistakes because reality is always different (even if slightly) from math. Nonetheless, it gives great satisfaction when you solve a practic problem with methods that other people would think they are "just useless abstract formulas".
Reality isn't different from math, it just requires a complicated mathematical model that begins on the smallest level possible 
But I agree, math is so ubiquitous while being so theoretical. Great stuff
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On June 18 2014 00:47 obesechicken13 wrote:
Nice try maths professor!
nice one
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EPT
