gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 3x(-6)=(-6)+(-6)+(-6)2x(-6)=(-6)+(-6)1x(-6)=(-6)0x(-6)=0 (no -6's at all)So you get from one number of -6's to the previous number of -6's by eliminating them. When you get into the negative numbers domain, you have to start substracting them, because there were none added to dispose of.(-1)x(-6)=-(-6)(-2)x(-6)=-(-6)-(-6)(-3)x(-6)=-(-6)-(-6)-(-6) 1 Quote Link to comment Share on other sites More sharing options...
manudude03 Posted February 10, 2011 Report Share Posted February 10, 2011 A lot of the understanding in Mathematics as others have said is that you need to go on gradually. I remember in my final year of school, I was in a class of 4 doing Further Mathematics and we were doing De Moivre's Theorem. My teacher also had to cover a class of 4th years so we were all in the same room. That class would have been lucky if they had even heard of complex numbers (not taught until final year), let alone some of the theory behind it. About 15 minutes into the class, one of the fourth years asked "Sir, what on earth is all this on the board?". Rather than try to explain it, the teacher just replied "Only the hardest stuff you'll learn in school". Of course, that is just another way of saying "If you don't understand it, don't worry about it :)" One thing I have always been critical of is that schools generally teach you how to pass exams instead of how to think. That's probably why my final year was 10 times better than any of my other years since we would continuously be asked to prove stuff without the teacher's help, and their job was basically to teach us any new theories and address any other problems we had. Myself and 2 of my classmates tried for the British Maths Olympiad in our final year, all of us failing miserably. It was probably my favourite "exam" ever since the conditions were so relaxed, we were allowed to chat to each other (but not about the questions :P), have food+drink etc. The first question, which was supposed to be a warmup was: "Find 4 prime numbers less than 100 which are factors of 3^32 - 2^32." If you know how to do this, it takes less than 5 minutes to get all 4 numbers. One of my classmates hadn't realised the significance of 32 and ended up doing 3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3.........., might have been 9x9x9x9x9.... to make it faster, can't remember. 2 hours later (no calculator allowed), we just said to him "you do know you're doing this the hard way, right?" 30 mins after that, it clicked and he found 3 of the numbers. If you really care, 3^32 - 2^32 = 1,853,015,893,884,545. A few weeks later, that problem was given to the 4th year class mentioned above. To make it "easier" we let them use calculators and let them discuss how to go about doing it. Collectively they didn't get close, they guessed the lowest. I specifically told them it was just a topic they would have done already in disguise, that didn't help. I really only the latter story because it is a pretty basic problem in disguise, yet unless you recognise it and know how to do it, you will be struggling. The fact that the three of us were the best Maths students in the school and I was the only one to get all 4 numbers (one of them I believe is doing a postgraduate Maths degree at Cambridge) just shows how difficult the easy stuff can be. A couple of off-topic points: 1. Would anyone laugh if a first grade (or first year of primary) when being taught how to count asked their teacher to prove that is the order? 2. I was thinking about statistics earlier and have come to the conclusion that statistically statistics are statistically irrelevent. Quote Link to comment Share on other sites More sharing options...
gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 (edited) is it like 3**32-2**32=(3**16+2**16)(3**8+2**8)(3**4+2**4)(3**2+2**2)(3+2)=(big number)(big number)*97*13*5 I suppose the second big number is not so big. 2**8= 2563**8=6561---------2**8+3**8=6817 aha 6817=17*401! so 5, 13, 17, 97! Edited February 10, 2011 by gwnn Quote Link to comment Share on other sites More sharing options...
manudude03 Posted February 10, 2011 Report Share Posted February 10, 2011 Yep, that's the proper way (there is some other theorem by Fermat to get 17 too) Quote Link to comment Share on other sites More sharing options...
gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 Yes so the more rules and theorems you know, the easier maths problems become. Fair enough and true as well, but it seems to be the opposite of winston's point :) applying new and nifty formulas is not my idea of thinking for yourself. Quote Link to comment Share on other sites More sharing options...
gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 I think 1, 2, 3, 4, etc each are just defined to be the successor of the number one less than them, therefore you don't have to prove they are in this order. So yes, I would LOL right in their face! Quote Link to comment Share on other sites More sharing options...
manudude03 Posted February 10, 2011 Report Share Posted February 10, 2011 New and nifty formulas? It's just basic algebra ( x^2-y^2=(x+y)(x-y) ). When I was doing it, I just did the 3^8+2^8 myself, only learnt of the other theorem afterwards, and still haven't found another use for it :P Quote Link to comment Share on other sites More sharing options...
gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 I meant to say that it's way too advanced for winstonm :) Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 3x(-6)=(-6)+(-6)+(-6)2x(-6)=(-6)+(-6)1x(-6)=(-6)0x(-6)=0 (no -6's at all)So you get from one number of -6's to the previous number of -6's by eliminating them. When you get into the negative numbers domain, you have to start substracting them, because there were none added to dispose of.(-1)x(-6)=-(-6)(-2)x(-6)=-(-6)-(-6)(-3)x(-6)=-(-6)-(-6)-(-6) Finally, someone who talks sense. :) Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 New and nifty formulas? It's just basic algebra ( x^2-y^2=(x+y)(x-y) ). When I was doing it, I just did the 3^8+2^8 myself, only learnt of the other theorem afterwards, and still haven't found another use for it :P To some of us, basic algebra is an oxymoron. ;) Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 I meant to say that it's way too advanced for winstonm :) No, I knew the order of numbers was defined. :P Quote Link to comment Share on other sites More sharing options...
helene_t Posted February 10, 2011 Report Share Posted February 10, 2011 Let's try to prove that 2+2 = 4 2+2 = 2+(1+1) because 2 is defined as 1+1= (2+1)+1 we have to assume by axiom that we are allowed to move brackets in sums. Makes sense, if I have two apples and get one more and later get one more again, it doesn't matter if the two latter apples arrive as one batch or not= 3+1 because 3 is defined as 2+1= 4 because 4 is defined as 3+1 Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works... Instead, math was always presented as "do it this way because the me and the book say to do it this way". For me, at least, having an understanding of the why seems almost an intergral part of my ability to learn. Quote Link to comment Share on other sites More sharing options...
wyman Posted February 10, 2011 Report Share Posted February 10, 2011 You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works... Instead, math was always presented as "do it this way because the me and the book say to do it this way". For me, at least, having an understanding of the why seems almost an intergral part of my ability to learn. This is a weird pedagogical thing. One would like to give children the building blocks so that they have full understanding; however, two problems: 1) sometimes it's easier (and often more effective -- people in general do poorly with abstraction) to give them something half-built, not talk about the internals, and allow them to continue building, swap out parts, mimic what others have built, ask questions about why -- with these pieces -- we can't build X, Y, or Z; and 2) sometimes teachers don't understand the building blocks well enough to teach them. Certainly I don't trust the vast majority of elementary educators to do any justice to mathematics. It is striking to me that when I ask what you learned in math between, say, 3rd grade (when people usually get into times-tables and basic multiplication) and 8th grade (possibly 7th), where you start (pre-)algebra, most people can only come up with "long division" or "we multiplied 3 digit numbers by 2 digit numbers. Then we multiplied 3 digit numbers by 3 digit numbers." This has to be a function of how kids are learning at those ages; retention just seems to be really bad, so we have to pound it in there by rote. Part of me says "there has to be a better way," but part of me says "they're kids, and I think in elementary school, it's more important to teach them to be good people and let them develop socially than it is to teach them long division." Squeezes seem so easy in comparison. Quote Link to comment Share on other sites More sharing options...
helene_t Posted February 10, 2011 Report Share Posted February 10, 2011 You may find it hard to believe, but nowhere along my educational route did anyone ever simply stop and say, these are axioms. An axiom is.... We use axioms because...This is how it works... Instead, math was always presented as "do it this way because the me and the book say to do it this way". For me, at least, having an understanding of the why seems almost an integral part of my ability to learn.I didn't come across axioms in junior hi either. There were a couple of proofs or pseudo-proofs but whether the fundaments on which they were based were axioms, lemmas or just common sense wasn't clear. It doesn't really matter, either. In my "proof" that 2+2=4, you may wonder whether the "axiom" I invoke, namely that I am allowed to move brackets, is really part of the axiomatic basis for number theory as it is taught in universities, or if it was something I made up for the purpose of this "proof". But it doesn't matter, since I am not going to provide a coherent account of number theory here anyway. Not did your teachers at high school. If you take maths to a serious level (I mean, taking the graduate classes made for students who specialize in maths for its own sake, rather than as a tool to use in subjects), at some point you will run into assertions that are not so obviously true. And then it may be of interest to know (and maybe even to be able to prove!) that the axiom of choice, for example, is something that can't be proven so although it is intuitively obvious (to most of us), it is something we just have to assume by axiom, and we could also assume its negation, and it may be of interest to explore how maths would look like if based of the negation of the axiom of choice. It relies a lot of intuition, and this intuition is used in two ways: to decide which axioms we chose (we chose the axiom of choice rather than its negation because it is more intuitive), and to make short-cuts in the proofs (we don't bother to prove everything, most of the time we accept something as proven as soon as it is intuitively obvious that it can be proven). But for everyday purposes, maths is about what just has to be true by sheer logical necessity. Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 I didn't come across axioms in junior hi either. There were a couple of proofs or pseudo-proofs but whether the fundaments on which they were based were axioms, lemmas or just common sense wasn't clear. It doesn't really matter, either. In my "proof" that 2+2=4, you may wonder whether the "axiom" I invoke, namely that I am allowed to move brackets, is really part of the axiomatic basis for number theory as it is taught in universities, or if it was something I made up for the purpose of this "proof". But it doesn't matter, since I am not going to provide a coherent account of number theory here anyway. Not did your teachers at high school. If you take maths to a serious level (I mean, taking the graduate classes made for students who specialize in maths for its own sake, rather than as a tool to use in subjects), at some point you will run into assertions that are not so obviously true. And then it may be of interest to know (and maybe even to be able to prove!) that the axiom of choice, for example, is something that can't be proven so although it is intuitively obvious (to most of us), it is something we just have to assume by axiom, and we could also assume its negation, and it may be of interest to explore how maths would look like if based of the negation of the axiom of choice. It relies a lot of intuition, and this intuition is used in two ways: to decide which axioms we chose (we chose the axiom of choice rather than its negation because it is more intuitive), and to make short-cuts in the proofs (we don't bother to prove everything, most of the time we accept something as proven as soon as it is intuitively obvious that it can be proven). But for everyday purposes, maths is about what just has to be true by sheer logical necessity. It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to. The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line. I could be wrong about all this, though. Quote Link to comment Share on other sites More sharing options...
wyman Posted February 10, 2011 Report Share Posted February 10, 2011 It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to. The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line. I could be wrong about all this, though. Well, if we had a scale that measured in apples (of course, we could use 1 lb weights), +apples would go on one side and -apples would go on the other. The sum is always the net reading on the scale. This is the number line with something concrete illustrating it. This probably doesn't help with -A x -B = A x B, but it should help with adding negatives to positives and negatives to negatives. This should show that taking one away from one side is the same as adding one to the other; armed with this knowledge we can also talk about what 3 - (-4) is. We start with 3 on the + side of the scale, and then we take away -4; that is, we take away 4 from the negative side. Taking from one is the same as adding to the other, so we lump 4 more on the + side for a net of +7. Idk, stream of consciousness Quote Link to comment Share on other sites More sharing options...
BunnyGo Posted February 10, 2011 Report Share Posted February 10, 2011 It appears to me that one of the basic difficulties in teaching mathematics is that we are dealing with abstractions rather than real objects, and as children we are used to objects. That to me explains why it is easier to get kids to understand that 2 apples + 2 apples = 4 apples. It is not so easy to grasp the concept of -2 apples + -2 apples = -4 apples when there are no minus appole objects to point to. The former we can illustrate - the latter we can only conceptualize with a substitution for actual minus objects, the number line. I could be wrong about all this, though. I agree, it's a very tricky problem. Part if it is that the teachers don't understand the abstraction usually. That said, shapes are pretty concrete and contain a lot of the abstractness (this also requires a teacher who really "gets it" so maybe it wouldn't be that applicable). But you could give a child a grid and 9 checkers and ask them to make a square. Count the sides of the square, etc. Then with 16 checkers. What about 17? Huh...can't do it? How about 12? Still no? Can you make a rectangle with 12 pieces? How long are it's edges. In the end, you can really develop numbers from geometry...but it takes a bit of work. Would be interesting to try though. Quote Link to comment Share on other sites More sharing options...
hrothgar Posted February 10, 2011 Report Share Posted February 10, 2011 That said, shapes are pretty concrete and contain a lot of the abstractness (this also requires a teacher who really "gets it" so maybe it wouldn't be that applicable). But you could give a child a grid and 9 checkers and ask them to make a square. Count the sides of the square, etc. Then with 16 checkers. What about 17? Huh...can't do it? How about 12? Still no? I can make a square with 12 checkers... Quote Link to comment Share on other sites More sharing options...
Winstonm Posted February 10, 2011 Author Report Share Posted February 10, 2011 I can make a square with 12 checkers...Sounds like the geometry version of "Name That Tune". Quote Link to comment Share on other sites More sharing options...
BunnyGo Posted February 10, 2011 Report Share Posted February 10, 2011 I can make a square with 12 checkers... On a grid with no leftovers? Quote Link to comment Share on other sites More sharing options...
hrothgar Posted February 10, 2011 Report Share Posted February 10, 2011 On a grid with no leftovers? Yes, the square wouldn't be filled in, but it would still be a square, and even have an equal number of checkers on each side Quote Link to comment Share on other sites More sharing options...
gwnn Posted February 10, 2011 Report Share Posted February 10, 2011 I think he means a 5x5 square which has 12 white and 13 black squares? Quote Link to comment Share on other sites More sharing options...
PassedOut Posted February 10, 2011 Report Share Posted February 10, 2011 I can make one with 17 checkers, and I even have a checker left for the center. Quote Link to comment Share on other sites More sharing options...
PassedOut Posted February 10, 2011 Report Share Posted February 10, 2011 I think he means a 5x5 square which has 12 white and 13 black squares?Yes, otherwise the center checker would not be on the grid. :ph34r: Quote Link to comment Share on other sites More sharing options...
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