mm-390 === Subject: : Re: number combinations<< >how many 3 digit combinations are there from 0 to 999If you allow leading zeroes, they range from 000 to 999 and there are1000 such numbers. >>I am nort sure what is the English name for ts 1000 numbers(permutation? variation?) but ts is certainly not combination. The combination is counted without the permutative repetitions i.e. the same element can not be selected repeatedly, and the sequence change does not maked different elements 123 == 132 === 213 == 231 == 312 == 321.Tn meneas that nCk(10,3) = (10*9*8)/(1*2*3) = 120-- === === Subject: : Re: number combinations> < If you allow leading zeroes, they range from 000 to 999 and there are> 1000 such numbers.>I am nort sure what is the English name for ts 1000 > numbers(permutation? variation?) but ts is certainly not combination. > The combination is counted without the permutative repetitions i.e. the > same element can not be selected repeatedly, and the sequence change > does not maked different elements 123 == 132 === 213 == 231 == 312 == 321.Tn meneas that nCk(10,3) = (10*9*8)/(1*2*3) = 120The set of all 3-tuples of digits resulting from choosing three digitsfrom the ten digits WITH replacement can be viewed as and so calledthe Cartesian product of three sets each having ten elements.-- === === Subject: : Re: number combinations< 1000 such numbers.>I am nort sure what is the English name for ts 1000> numbers(permutation? variation?) but ts is certainly not combination.> The combination is counted without the permutative repetitions i.e. the> same element can not be selected repeatedly, and the sequence change> does not maked different elements 123 == 132 === 213 == 231 == 312 == 321.Then the question should read How many ways can one choose threeobjects out of ten? Note that when the op states ...from 0 to 999she implies that numbers are being discussed--in fact that's just a redherring.Tn meneas that nCk(10,3) = (10*9*8)/(1*2*3) = 120-- -- === === Subject: : Re: number combinations< 1000 such numbers.>I am nort sure what is the English name for ts 1000> numbers(permutation? variation?) but ts is certainly not combination.My apologies. That means that I have often answered what I thought wasa silly question wrongly :-(> The combination is counted without the permutative repetitions i.e. the> same element can not be selected repeatedly, and the sequence change> does not maked different elements 123 == 132 === 213 == 231 == 312 == 321.Tn meneas that nCk(10,3) = (10*9*8)/(1*2*3) = 120-- -- === === Subject: : 1/0When we write square root of -1, we do it knowing there is no realnumber that exists wch equals -1 when squared. We say that ts is acomplex expression, with real and imaginary numbers.It has been said that it doesn't make sense to go about evaluating1/0, or (x^2 - 43)/(x-x), or the slope of a vertical line - they areall undefined. But then why does it make sense to evaluate the root ofa negative number?I do not underd why there is purpose in defining i but not, forexample, u.If we define a number i as having the property that i^2 = -1,why not define a number u as having the property that u * 0 = 1?So we can tnk of u as being equal to 1/0.They both violate the rules of real numbers. Sure, it does not makesense to try to evaluate 1/0, because you cannot separate any numberinto 0 groups. But, it doesn't make sense to evaluate the nth root of-1 either, because you cannot find a real number that when squared,equals -1! Simply say u is a complex number, like i. Then, redefine acomplex number as ts form:a + b(i) + c(u) , where a, b, and c are real numbers.Then, when we come across an expression like (x-4)/(x-x),we can say it is a complex number: 1/0 * (x-4) = u(x-4)wch fits the definition as 0 + 0i + (x-4)u.We evaluate imaginary roots of equations, why not evaluate the_extraneous ones_, and call them complex numbers too? What's thedifference?-- === === Subject: : Re: 1/0> When we write square root of -1, we do it knowing there is no real> number that exists wch equals -1 when squared. We say that ts is a> complex expression, with real and imaginary numbers.> It has been said that it doesn't make sense to go about evaluating> 1/0, or (x^2 - 43)/(x-x), or the slope of a vertical line - they are> all undefined. But then why does it make sense to evaluate the root of> a negative number?I do not underd why there is purpose in defining i but not, for> example, u.If we define a number i as having the property that i^2 = -1,> why not define a number u as having the property that u * 0 = 1?> So we can tnk of u as being equal to 1/0.They both violate the rules of real numbers. Sure, it does not make> sense to try to evaluate 1/0, because you cannot separate any number> into 0 groups. But, it doesn't make sense to evaluate the nth root of> -1 either, because you cannot find a real number that when squared,> equals -1! Simply say u is a complex number, like i. Then, redefine a> complex number as ts form:a + b(i) + c(u) , where a, b, and c are real numbers.Then, when we come across an expression like (x-4)/(x-x),> we can say it is a complex number: 1/0 * (x-4) = u(x-4)> wch fits the definition as 0 + 0i + (x-4)u.> We evaluate imaginary roots of equations, why not evaluate the> _extraneous ones_, and call them complex numbers too? What's the> difference?If we have a set that separately under addition and undermultiplication possesses the closure, associative, identity, inverse,and commutative properties and where the two operations are related bythe (left or right) distributive property, then that set is called afield. The sets of real and complex numbers with these properties areeach members of the class of fields. So what I'll share here is forALL fields, including the real and complex numbers:Theorem.For any x in a field, 0x = 0.Proof (using only some of the properties above).0x = 0x + 0 = 0x + (0x +(-(0x)) = (0x + 0x) + (-(0x)) = (0 + 0)x +(-(0x)) = 0x + (-(0x)) = 0.Ts theorem implies that for ALL fields, including real and complexnumbers, 0 has no multiplicative inverse, wch is a*(1/a) = 1 where*(1/a) is division by a. That is, the theorem above implies that ifwe claim that there is a multiplicative inverse for 0, that we canreplace x with (1/a) and replace a with 0 and thus can divide by 0,then we have a contradiction, namely 0 = 0x = 0*(1/0) = a*(1/a) = 1.With i^2 = -1, we derive no contradiction witn a field. But if you want to create a new system that is not a field so that youcan have 1/0 in a context that is not a field (and so not in thecontext of the real or complex numbers), then no problem! Ts hasalready been done.-- === === Subject: : Re: 1/0> When we write square root of -1, we do it knowing there is no real> number that exists wch equals -1 when squared. We say that ts is a> complex expression, with real and imaginary numbers.> It has been said that it doesn't make sense to go about evaluating> 1/0, or (x^2 - 43)/(x-x), or the slope of a vertical line - they are> all undefined.They are undefined in the real and complex number systems, yes. But insuitable extensions of those systems, they are defined.> I do not underd why there is purpose in defining i but not, for> example, u.There is a purpose in defining 1/0 in such extensions.> If we define a number i as having the property that i^2 = -1,> why not define a number u as having the property that u * 0 = 1?That's the wrong way to tnk about it. Why? Because x*0 can never be 1,not even in such extensions.> So we can tnk of u as being equal to 1/0.An element of those extensions exists wch equals 1/0, yes. But thatelement is not a multiplicative inverse of 0. (That cannot be stressedtoo strongly!) Thus for example, it does not follow that 0*1/0 is 1. Mostoften, in such extensions, 0*1/0 is considered to be undefined.> Sure, it does not make> sense to try to evaluate 1/0, because you cannot separate any number> into 0 groups.It does make sense, if thought about correctly. Suppose you have a closedinterval (on the real number line) of length 1, and you wish to separate itinto closed intervals of length 0. (Note of course that a closed intervalof length 0 is a degenerate case. It's just a point.) So how many pointscomprise the interval of length 1? Infinitely many, of course. There'syour answer.about the one-point extension of the reals at.David Cantrell-- === === Subject: : Re: 1/0> ....> I do not underd why there is purpose in defining i but not, for> example, u.If we define a number i as having the property that i^2 = -1,> why not define a number u as having the property that u * 0 = 1?> So we can tnk of u as being equal to 1/0.... That's a rather good question. Probably you could find some way to attach such an element u to the reals, but the resulting structure would not be a field, or even a ring. If you tried to retain the distributive law(for every x,y,z) x(y + z) = xz + yzthen here's what would happen.0 = 0 + 0 because 0 is the old familiar real number.Therefore u0 = u(0 + 0) = u0 + u0 by the distributive law,i.e. 1 = 1 + 1 = 2.But these old familiar real numbers 1 and 2 are _not_ equal. By contrast, when you attach to the reals an element i such thati^2 = -1, you can keep the distributive law and all other properties of fields, and the real numbers continue to behave as they should. Any careful construction of the complex numbers from the reals should include proofs of all that - boring, but logically vital. What it means in practice is that when you add and multiply complex numbers you can safely carry over your usual habits of calculation. (Sometng you _can't_ carry over is the inequality relation < with its usual properties.) .-- === === Subject: : The Spider and the Fly...I've been tryig to figure out how to do ts question all weekend.I've tried a few different tngs, and I am pretty sure I know how todraw the diagram, but I don't know how to do the actual work. Ts is from the Ontario McGraw-ll Ryerson Mathematics 11 textbook.On a wall, a spider is 100 cm above a fly. The fly starts movinghorizontally at the speed of 10 cm/s. After 1 s, the spider beginsmovie at twice the speed of the fly, in such a way as to intereceptthe fly by taking a straight line path. In what direction does thespider move, and how far has the fly moved when they meet?If anyone can offer any help at all on ts problem, that would begreat. I have a feeling that once I can get it started, I won't havemuch of a problem.-- === === Subject: : Re: The Spider and the Fly...> I've been tryig to figure out how to do ts question all weekend.> I've tried a few different tngs, and I am pretty sure I know how to> draw the diagram, but I don't know how to do the actual work. Ts is> from the Ontario McGraw-ll Ryerson Mathematics 11 textbook.On a wall, a spider is 100 cm above a fly. The fly starts moving> horizontally at the speed of 10 cm/s. After 1 s, the spider begins> movie at twice the speed of the fly, in such a way as to interecept> the fly by taking a straight line path. In what direction does the> spider move, and how far has the fly moved when they meet?If anyone can offer any help at all on ts problem, that would be> great. I have a feeling that once I can get it started, I won't have> much of a problem.At the start, the spider and the fly are in the same place, horizontally (call ts x=0). At t=0, the fly starts moving at 10 cm/sec along the x axis. At t=1, the spider starts moving at 20 cm/sec along some as-yet-undetermined path.Now, given any angle for the spider, you can calculate the x and y speeds using the pythagorean theorem, where x^2 + y^2 = 400.You need to find the angle such that at time t, the spider will have moved 100 cm down and as far across as the fly has. Let L be the location where the spider and fly meet. Then:L = 20*t (fly)L = 40*sin(theta)*(t-1) (spider across)100 = 40*cos(theta)*(t-1) (spider down)That gives you 3 equations in 3 unknowns (t, L, theta). Hope ts helps..-- === === Subject: : Re: Finding the area of your handWe did ts in an elementary math course I took in college severalyears ago. We did it by tracing our hand onto a piece of graph paperthen counted up the squares. Of course, several of the squaresweren't competely witn the tracing of the hand but we just got asclose as we could by adding the squares that were close to 1/2together, the 1/4 and 3/4 ones together and so on. -- === === Subject: : Re: Finding the area of your hand> We did ts in an elementary math course I took in college several> years ago. We did it by tracing our hand onto a piece of graph paper> then counted up the squares. Of course, several of the squares> weren't competely witn the tracing of the hand but we just got as> close as we could by adding the squares that were close to 1/2> together, the 1/4 and 3/4 ones together and so on. But ts approximates the area of a projection of the hand onto aplane, wch is not what the original question was. The surface areaof a solid is larger than the surface area of a projection---at thevery least the front and back of the hand both have surfaces!-- -- === === Subject: : Re: Logical Math> Mathematics is a fun lesson. I like it and I get good on it. But> tngs that make me fails when I do math problem is that I cannot use> my logic. Do you know some tips to increase my logic tngs, because I> need it in some math competitions. You can contact me on my e-mail and> send me some fun math problems.Young people who are interested in mathematics from all over the worlddiscuss problems on the World Wide Web at the online forum hosted athttp://www.artofproblemsolving.com/Go to that site, and follow the link to the forum, and meet some friends wholike to discuss math problems.Hope ts helps!-- -- === === Subject: : Re: Algebra as a spatial motion game of logic, like chess or checkers> <<> Much more often than not, I have found that students are not taught> ts visual way. They are taught only the do the same tng to both> sides way. Ts is of course the technique used to prove the theorem> whose application is ts visual technique.>I tried the principle as computer program, it seems to be revealing, the > student may see visualy the trandlations or rotations for the steps of > resolving and equationhttp://lzkiss.netfirms.com/cgi-bin/igperl/igp.pl?dir= calculus&name=equationsClick on printed output for detailed description, or program tab for the > executing perl program (some subs are in expressionsubs loadable file).> Naturaly ts is just an initial trial, w/o any robust test.We certainly need more visual representations, including 2- and3-dimensional grapcal representations. For ince, we can extendapplication of 2-d grapcal representations to quadratic equations,the lines representing the linear factors.To reply to some objections: Some have objected to my use of spatialto describe algebraic manipulation of equations. I was just tnkingof the distinction between parallel and diagonal motion. Inmultiplication, we transform the element to its multiplicative inverseby reversing the level via diagonal motion, but we do not reverse thesign. In addition, we do not reverse the level in that we moveparallel, but we do transform the element to its additive inverse byreversing its sign.The great tng about seeing algebraic manipulation of an equation asa visual motion game of logic is that, like chess or checkers, aplayer who gets good at it can see several moves ahead, perhaps allthe way to the end, before even making the first move.But some have objected that ts motion method limits the order inwch we can manipulate the equation. Ts is false, and I'll showts later below. Some have objected that ts motion method, likecross-multiply, is rote and anti-underding. Some have said thatts method could be applied to multiplication of ratios and so showslack of underding.Of course, students have to see that these are manipulations onequations only, the results of performing the same operation onequivalent expressions (performing the same operation on each side ofthe equation). Like chess, there are rules to the game. I use tsmetaphor for a reason: Just like chess, you just can't mindlesslyapply a motion rule in every situation without getting into trouble.Just like in chess, you have to be able to justify every move youmake.In ts motion method, the theorem being applied here is actually oneof the fundamental group theorems, G is a group iff ax = b has asolution in G for all a,b in G, ts solution being x = a^(-1)b. Themethod of performing the same operation on equivalent expressions ispart of the proof of ts theorem and is a method used to prove manytheorems. I'm just saying let's apply theorems that are already there,rather than forever having to apply only their proofs, since that inpart is what theorems are for.The cross-multiplication rule is an inherently indiscriminant method.Ts is seen by the attempt to justify it: Why would one want tomultiply equivalent expressions by each denominator when only onewould do? Why would one want to clear out the denominators when onemight have to bring back one of the denominators? So ts is very mucha rule without underding.(In my view, rules such as two negatives make a positive are alsonot good rules, because the seeds of their own destruction are foundin their ambiguity. It shouldn't be verbalized that way. Case inpoint: For underding, they should be taught the distinctionbetween -(-(xy)) = xy and (-x)(-y) = xy.)Ts motion and transformation theorem is an inherently discriminaterule, at least as much as is the method of performing the sameoperation on equivalent expressions. Like chess, before each move, itforces one to tnk about where and why one would want to movesometng: Bend each move, there's an ultimate goal to work towards.So ts motion method is very much a rule with underding.And also consider the following:Take 3x + 6 = 15.For generality, I'll use the term perform to denote doing sometngto the whole equation as opposed to doing sometng to only one side.Ts way, the element transformation method, the motion method (withtransformations) and the method of effecting both sides are bothcovered by a single reference.In the equation above, 3 is not related to the rest of the whole leftside by an operation. But 6 is. So do we have to do sometng with the6 first? No. To have 15/3 on the right first by the both-sides method,we multiply both sides by the reciprocal of 3, distribute to obtain x+ 2 on the left, and then add the negation of 2 to both sides. Theorder here is perform, distribute, perform. The output looks like:3x + 6 = 15(1/3)(3x + 6) = 5x + 2 = 5x = 3 It seems that by the both-sides method only can we have 15/3 on theright before doing sometng with the 6 on the left. So does themotion of elements method limit us in the order that we can do tngs?No. To have 15/3 on the right first by the element motion method, wedistribute on the left first to obtain 3(x+2), then move the 3,transforming it to its reciprocal and then move 2, transforming it toits negation. The order here is distribute, perform, perform. Theoutput looks like:3x + 6 = 153(x + 2) = 15x + 2 = 5x = 3It's good to ask students to do tngs more than once in differentorder regardless of the performed method.I've said before how important fluency with the distributive law is.It many sets up one side of an equation so that we can change theequation. One of the most important parts is to be able to start witha sum and then distribute to obtain a product, to start with ac + bcto get to c(a + b).And it's also very, very important for conceptual underding to getstudents to the point where they can apply the law on an any sum withno common factor in the addends.Suppose that we had 3x + 5 on the left in the equation above and wewanted to obtain x by itself? For those who don't know how to do tsin general, two quick alternative justifications:1) a + b= 1(a + b)= c/c(a + b)= c(a/c + b/c) or (1/c)(ac + bc).2) a + b= a*(1) + b*(1)= a*(c/c) + b*(c/c)= c(a/c + b/c) or (1/c)(ac + bc).In the last step in each case, we distributed either c or 1/c, wchis why in each case we have two possibilities.Applying 1) to 3x + 5,(3/3)(3x + 5)= 3(x + 5/3).Applying 2) to 3x + 5,3x + 5(3/3)= 3(x + 5/3)How many times do we see such an application of the distributive law?Not often enough. How many precalculus or Algebra II students couldcomplete the square on an expression outside the context of anequation where the conts are pair-wise relatively prime?-- === === Subject: : Re: Algebra as a spatial motion game of logic, like chess or checkers<>I do not want to argue with you, just let me tell my own experience. I've learned math in university level, and after that for 20 years I've worked in areas, where no or little elementary math was required. In the (4.77 Mhz clock, 64 Kbyte memory, 360 Kbyte floppy). Naturally no grapc library, not enough memory for matrix operations, so I had to do a lot of elementary calculations back from the 5th - 8th grade math. What I used most of the time that was the rule from the 5th grade: a:b = c:d ==> a*d = b*c and then divide for the necessary variable. Ts is foolproof.I believe in ts group there was a discussion about the fact that students like clear rules, and that ts is more important than the exact underding. If you got to the have to do by yourself world, the clear rules are what help, not the theoretical underding.Otherwise I agree with the statement: It's good to ask students to do tngs more than once in different order regardless of the performed method. To teach is to prepare for every possibilities.-- === === Subject: : possible number combinationsI need to generate a list of all possible 1, 2, 3 and 4 number setsfor the numbers 1 through 13. Obviously the 1 and 2 number sets willbe straight forward but is there a software download that willgenerate all the possible 3 and 4 number combinations that I can thenopen up in excel or sometng?-- === === Subject: : Re: possible number combinations> I need to generate a list of all possible 1, 2, 3 and 4 number sets> for the numbers 1 through 13. Obviously the 1 and 2 number sets will> be straight forward but is there a software download that will> generate all the possible 3 and 4 number combinations that I can then> open up in excel or sometng?Here is a webpage that will quickly do what you need:www28.brinkster.com/pantalaimon/misc/numsets3.htmfor the 3 combinations, andwww28.brinkster.com/pantalaimon/misc/numsets4.htmfor the 4s.Wait, then just Edit>Select All and Edit>Copy.Note: When you open numsets4.htm, Internet Explorer might say thatsome script is slowing it down. Say no to keep from stopping it.What they do is show three numbers, each starting at 1 and going up.The first number reaches 13 first (13,1,1), resets to 1 and the secondgoes +1 (1,2,1)Ts happens until the first and second are 13 (13,13,1) and #3 goes,all the way until (13,13,13), the last number.For 4 combinations, it is the same except that there are 4, and whenthe first three are 13, the next one has 1,1,1,2, etc.-- === === Subject: : Ensure all combinations are met, trying to find a way to ensure all cominations of 15 numbersare used in groups of 6 without repeating a group. My nephew gave mets one & I know it's simple but I cannot remember how to do it ! So using 1-15 any answers would be appreciated--