mm-2089 === Subject: Re: [[Urgent question]a probability question > In an experiment, two cards are drawn from a deck of well-shuffled > pack > find the probability for the event that > (a)the two cards have a sum of 3 > (b)the two cards have a sum of 3 if one of the cards is an ace of > heart. There are C(52, 2) = 1362 possible two card hands of which 16 add up to 3. There are 51 one card hands excluding AH. Four of those, together with AH, will add up to 3. Alternately, P(add to 3 | one is AH) is P(add to 3 and one is AH / P(one is AH) = (4/1352)/(51/1362). === Subject: Re: discrete math > How do i prove that the square root of 2 is irrational using the > unique factorization theorum? Suppose sgrt(2) is rational. Let sqrt(2) = p/q with gcd(p, q) = 1. Then 2 = p^2/q^2 and so 2*q^2 = p^2. Here is where UF gets into it: 2 is a prime factor of p^2 and hence of p since only prime factors of p are factors of p^2. So 2^2 is a factor of p^2 and consequently 2 is a factor of q^2 and, using UF again, 2 is a factor of q. But that gives 2 as a common factor of p and q. Consequently, sqrt(2) cannot be rational. === Subject: Re: converting rectangular, spherical and cylindrical... cordinates > Looking for any help I can get on these problems. > http://www.public.iastate.edu/~gbhatt/HW.pdf If you're referring to problem 1(1), all I see is an integral of a horrible function of x from x=a to x=a. This is trivially integrated (zero). I'm going to assume that the lower limit should read x=-a. > right now I am stuck converting the limit; sqrt (a^2-x^2) and the > negative, to spherical cordinates. You have to look at the region specified by the limits of all three integrals and find some way of expressing it in terms of spherical coordinates. So, eg, z=a is a plane; z = a + sqrt(a^2 - x^2 - y^2) is a sphere centered at (0,0,a); the limits of the x and y integrals tell you that you want the hemisphere above the plane z=a. Your best bet is to use the usual spherical polar expressions for x and y but set z-a = r*cos(theta). Limits for r, theta, phi should now be obvious. P.A.C. Smith 'If the Apocalypse comes, beep me.' <*> http://www.srcf.ucam.org/~pas51 === Subject: Re: how many 2x2 matrix is identical to its inverse > In Z/26Z > I am assuming that this notation indicates that we are talking about arithmetic > module 26... > I want to know how many 2x2 matrices and their inverses are > identical? i.e. > [a b] [a b] [1 0] > x = > [c d] [c d] [0 1] > Please give me some hints. > Well, the above expands to: > a^2 + bc = d^2 + bc = 1, (a + d) b = (a + d) c = 0 a+d, > b, and c. Only some of them also satisfy the first part. I get four cases, three > are easy and have three solutions, and the fourth case is not so easy and has > more solutions, and I don't see an obvious way to solve it short of writing out > a 13x13 multiplication table, but I am too lazy to do that now :-) > meeroh > -- > If this message helped you, consider buying an item > from my wish list: === Subject: Re: need help >2. f(x)=x^3-4x^2+x+6 > Not sure the easiest way to do this by hand. You need the roots If a polynomial over integers has an integer root x, then x divides the free coefficient. Therefore, any integer roots are among +-1, +-2, and +-3. If you try out all six, you find that -1, 2, and 3 are the roots. (This often works for homework problems because they usually have pretty solutions.) meeroh If this message helped you, consider buying an item from my wish list: === Subject: Re: converting rectangular, spherical and clindrical coordinates by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9Q4D2Q22360; >Looking for any help I can get on these problems. >http://www.public.iastate.edu/~gbhatt/HW.pdf >right now I am stuck converting the limit; sqrt (a^2-x^2) and the >negative, to spherical cordinates. >Any help is greatly appreciated. >Troy Well, you CAN'T convert just one integral, you have to convert the entire system! If you are trying to do each integral separately, that may be your problem. By the way, the problem (1) you give has both lower and upper limits of the integral with respect to x equal to a. If that were actually the case, the problem would be trivial: the integral is equal to 0! I'm going to assume that it is actually from -a to a. Okay, start by drawing a coordinate system and marking x= -a, x= a (strictly speaking, you should draw the PLANES x= -a, x= a). The region to be integrated over must lie between them. The integral over y has lower limit y= -sqrt(a2- x) and upper limit y= sqrt(a2- x). Squaring both sides of those gives y2= a2- x or x2+ y2: a circle, centered at the origin with radius a. The region to be integrated over must lie over that circle. The integral over z has lower limit z= a and upper limit z= a+ sqrt(a2- x2- y2). If we subtract a from both sides and then square we get (z-a)2= a2- x2- y2 or x2+ y2+ (z-a)2= a2, the equation of a sphere, centered at (0, 0, a) with radius a. Since the lower limit is a, the region to be integrated over is the upper hemispher of that. Putting this into cylindrical coordinates is very easy precisely because the region lies above a circle of radius a centered at (0,0): r must vary from 0 to a and theta from 0 to 2pi in order to cover that. z must vary from a to a+ sqrt(a2- x2- y2) just as above but now we need to replace x and y by r and theta. Of course, x2+ y2= r2 so the z integral is from a to a+ sqrt(a2- r2). Of course, you have to replace x in the integrand by r cos(theta) and the differential of volume in cylindrical coordinates is rdz dr dtheta. Integral (1) in cylindrical coordinates is integral(r= 0 to a)integral(theta= 0 to 2pi)integral(z= a to a+ sqrt(a2- r2)(r cos(theta))r dz dr dtheta. Spherical coordinates is a bit harder. Of course, theta still measure the angle around so the theta integral is still from 0 to 2 pi. phi measures the angle from the z axis down. We need to calculate the angle the lower plane, z= a makes. Draw a line from (0,0,0) up to (0,0,a) and then another line from there to a point on the circumference of the circle and then a third back to (0,0,0). That gives us a right triangle with both near and opposite sides of length a. That means that the base angle, phi, is pi/4. The phi integral is from 0 to pi/4. rho is the hard part. The lower limit is the length of the line from (0,0,0) to a point on the plane z= a. By symmetry, that won't depend on theta but it will depend on phi. Again, we have a right triangle with near side length a, now with general angle phi. Since cos(phi)= near side/hypotenuse= a/r, we have r= a/cos(phi) or r= a sec(phi). The upper limit is the length of the line from (0,0,0) to the sphere. That is NOT a constant since we are not measuring from the center of the sphere. I think the simplest way to do that is to multiply out the (z-a)2 in the formula for the sphere to get x2+ y2+ z2- 2az+ a2= x2+ y2+ z2= rho2 so we get rho2- 2az= 0. In polar coordinates, z= rho cos(phi) so this is the same as: rho2- 2a rho cos(phi)= 0 or rho= 2a cos(phi). Of course, x= rho cos(theta)sin(phi) and the differential of volume in spherical coordinates is rho2sin(phi)dtheta dphi drho. The integral, in polar coordinates is integral(theta= 0 to 2pi)integral(phi= 0 to pi/4)integral(rho= a sec(phi) to 2a cos(phi)(rho3cos(theta)sin2(phi)dtheta dphi drho. === Subject: Re: Where is my exam? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h9Q4D0M22333; Owen [alt.undergrad.math 24 Oct 03 07:21:08 -0400 (EDT)] http://mathforum.org/epigone/alt.math.undergrad/nerdclelmoa > I was wondering if anyone knew where the exam for Calc 101 > is today? I've emailed several people in the department and > left several phone messages and no ones responded where my > exam is. Did you look on table in the back of the tutoring room? You might also look in room 325 for it, on the table by the bulletin board. Dave L. Renfro === Subject: Explaining math definition problem I'm an independent researcher, which means that I use my *own* funding, and my *own* direction to go out and see what knowledge I can obtain. Some of my research has been in the area of mathematics. Getting important research findings is one thing, and getting them noticed, is another. At least here on Usenet I can talk freely to people around the world. What I'd like to explain is my disturbing and to me fascinating finding of a problem with a math definition that's over a hundred years old. In looking over various replies to my previous posts on this subject, I've seen assertions that definitions can't cause problems, which is something that I can address quickly at the start. Over a hundred years ago, the great German mathematicians Karl Gauss played with numbers of the form a+bi, where 'a' and 'b' are integers. In his honor they were later called gaussian integers, though a number like 1+2i is not an integer. The gaussian up-front is important. Later mathematicians came up with other numbers they called algebraic integers, which include gaussian integers. They thought they'd found THE set, or superset you might call it, which includes all numbers with certain special properties of integrality. The most important property to point out is the ability to have primeness between numbers. For instance, with integers, 2 and 3 are coprime, that is, they don't share non-unit factors, that is, factors other than 1 or -1, with each other. Just be clear here, factors of 1, are called units or unit factors. But notice that with rationals, you have 2(3/2) = 3, so 2 and 3 *do* share a factor and are not coprime in that ring, which is typically called a field *because* every element except 0, has a multiplicative inverse. What Gauss had started considering, which other mathematicians extended, was the idea of sets of numbers where you kept interesting properties of the set of integers, like being able to say two numbers were coprime. What I've found is a problem with their set of algebraic integers, as unfortunately, despite what many mathematicians think, it's too small. That's it. The definition they use is too small to do what they think it does, which is include all these interesting numbers with special properties. But because they *think* it's big enough, mathematicians have an error in their discipline based on their false assumption, as they've come up with more arguments based on that assumption, which then aren't actually proven. It's like when the Greeks with their word atom thought they had the smallest thing, and later our civilization used it, and broke atoms apart, though part of the definition is that they are *indivisible*, as people can define things, and later *refine* their definitions. Now my research finding isn't hard to show quickly in broad strokes. On of my important analysis tools is a simple technique to factor polynomials into non-polynomial factors. For instance, with the polynomial P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 that technique gives you P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 so I can factor to get P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). where the a's are the roots of the cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). Now despite the complexity, you can rely on *simple* ideas still, by noticing that setting x=0, pulls out constant terms, as P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's with x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3. You may not realize it, but what you just saw is revolutionary, both in the special techniques, and most importantly with the consequences that quickly follow. That's because P(x) has another special feature as P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) where that 49 is just begging to be divided off, which gives, of course, P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. But remember, my three factors with the a's from before had *constant* terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, 1, and 22, which is the result that is so earth shattering. Here the principle is like if you have S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) in that setting x=0 gives you *constant* terms within the expression, which you can conveniently, also look at to see how it works. S(0) = (7(0) + 7)(0 + 1) = 7(1). The point is that the 7 is constant, so x's value means nothing to it. So from before with P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) I know that dividing through by 49, it must go like P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) and as the constant terms are *independent* of the value of x, it MUST be that in general P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). The problem now though is that conclusion can be used to show that unequivocally beyond any reasonable doubt the definition of algebraic numbers is TOO SMALL, as at times 5a_1/7 and 5a_2/7 are not included. You see, they get left out, which is a problem because from the *assumption* of mathematicians, they should be included, if the ring of algebraic integers is the ring that mathematicians thought it was. Some of you may find yourselves fearful of using your own mathematical understanding, if you realize I'm right, and then realize that mathematicians are disputing the result, especially if you see posters tossing out far more complicated math in reply to my post, but remember, math isn't magic. Logic rules mathematics, so look for what makes sense. And remember that you can't assume that posters are on your side. I don't want you to assume that I'm on your side either. You see, I don't need you to assume anything, as what I need you to do is check. While some mathematicians may erroneously believe now that it's in their interest to hide the problem I've revealed, that mistake in thinking does not help the rest of the world. After all, what good does it do everyone else for mathematicians to hide their definition problem? What's in it for you? James Harris === Subject: Re: Explaining math definition problem What I'd like to explain is a disturbing and fascinating finding of a problem with a math definition that's over a hundred years old. Over a hundred years ago, the German mathematician Karl Gauss played with numbers of the form a+bi, where a and b are integers. In his honor they were later called gaussian integers, though a number like 1+2i i s not an integer. The gaussian up-front is important. Later mathematicians came up with other numbers they called algebraic integers, which include gaussian integers. They thought they'd found THE set, or superset you might call it, which includes all numbers with certain special properties of integrality. The most important property to point out is the ability to have primeness between numbers. For instance, with integers, 2 and 3 are coprime, that is, they don't share non-unit factors, that is, factors other than 1 or -1, with each other. Just be clear here, factors of 1, are called units or unit factors. But notice that with rationals, you have 2(3/2) = 3, so 2 and 3 do share a non-unit factor and are not coprime in that ring, which is typically called a field because every element except 0, has a multiplicative inverse. What Gauss had started considering, which other mathematicians extended, was the idea of sets of numbers where you kept interesting properties of the set of integers, like being able to say two numbers were coprime. What I've found is a problem with their set of algebraic integers, as unfortunately, despite what many mathematicians think, it's too small. That's it. The definition they use is too small to do what they think it does, which is include all these interesting numbers with special properties. But because they think it's big enough, mathematicians have an error in their discipline based on their false assumption, as they've come up with more arguments based on that assumption, which then aren't actually proven. It's like when the Greeks with their word atom thought they had the smallest thing, and later our civilization used it, and broke atoms apart, though part of the original Greek definition is that they are indivisible, as people can define things, and later refine their definitions. Now my research finding isn't hard to show quickly in broad strokes. On of my important analysis tools is a simple technique to factor polynomials into non-polynomial factors. For instance, with the polynomial P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 that technique gives you P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 so I can factor to get P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). where the a's are the roots of the cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). Now despite the complexity, you can rely on simple ideas still, by noticing that setting x=0, pulls out constant terms, as P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's with x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3. You may not realize it, but what you just saw is revolutionary, both in the special techniques, and most importantly with the consequences that quickly follow. That's because P(x) has another special feature as P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) where that 49 is just begging to be divided off, which gives, of course, P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. But remember, my three factors with the a's from before had constant terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, 1, and 22, which is the result that is so earth shattering. Of course, the distributive property is important here. Distributive Property: a(b+c) = ab + ac The point is that the 7 is constant, so x's value means nothing to it. So from before with P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) I know that dividing through by 49, it must go like P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) and as the constant terms are independent of the value of x, it MUST be that in general P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). The problem now though is that conclusion can be used to show that unequivocally beyond any reasonable doubt the definition of algebraic numbers is TOO SMALL, as at times 5a_1/7 and 5a_2/7 are not included. You see, they get left out, which is a problem because from the assumptions of mathematicians, they should be included, if the ring of algebraic integers is the ring that mathematicians thought it was. Some of you may find yourselves fearful of using your own mathematical understanding, if you realize I'm right, and then realize that mathematicians are disputing the result, but remember, math isn't magic. Logic rules mathematics, so look for what makes sense. You see, I don't need you to assume anything, as what I need you to do is check. While some mathematicians may erroneously believe now that it's in their interest to hide the problem I've revealed, that mistake in thinking does not help the rest of the world. After all, what good does it do everyone else for mathematicians to hide their definition problem? === Subject: Re: Explaining math definition problem > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > a^3 - 3a^2, which has roots, 0, 0 and 3. > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. > S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) > S(0) = (7(0) + 7)(0 + 1) = 7(1). > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) > P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). You're getting better. I almost understood this. === Subject: Re: Explaining math definition problem > I'm an independent researcher, which means that I use my *own* > funding, and my *own* direction to go out and see what knowledge I can > obtain. Some of my research has been in the area of mathematics. > Getting important research findings is one thing, and getting them > noticed, is another. > At least here on Usenet I can talk freely to people around the world. > What I'd like to explain is my disturbing and to me fascinating > finding of a problem with a math definition that's over a hundred > years old. In looking over various replies to my previous posts on > this subject, I've seen assertions that definitions can't cause > problems, which is something that I can address quickly at the start. > Over a hundred years ago, the great German mathematicians Karl Gauss > played with numbers of the form a+bi, where 'a' and 'b' are integers. > In his honor they were later called gaussian integers, though a number > like 1+2i is not an integer. The gaussian up-front is important. > Later mathematicians came up with other numbers they called algebraic > integers, which include gaussian integers. > They thought they'd found THE set, or superset you might call it, > which includes all numbers with certain special properties of > integrality. > The most important property to point out is the ability to have > primeness between numbers. > For instance, with integers, 2 and 3 are coprime, that is, they don't > share non-unit factors, that is, factors other than 1 or -1, with each > other. > Just be clear here, factors of 1, are called units or unit factors. > But notice that with rationals, you have 2(3/2) = 3, so 2 and 3 *do* > share a factor and are not coprime in that ring, which is typically > called a field *because* every element except 0, has a multiplicative > inverse. > What Gauss had started considering, which other mathematicians > extended, was the idea of sets of numbers where you kept interesting > properties of the set of integers, like being able to say two numbers > were coprime. > What I've found is a problem with their set of algebraic integers, as > unfortunately, despite what many mathematicians think, it's too small. > That's it. The definition they use is too small to do what they think > it does, which is include all these interesting numbers with special > properties. > But because they *think* it's big enough, mathematicians have an error > in their discipline based on their false assumption, as they've come > up with more arguments based on that assumption, which then aren't > actually proven. > It's like when the Greeks with their word atom thought they had the > smallest thing, and later our civilization used it, and broke atoms > apart, though part of the definition is that they are *indivisible*, > as people can define things, and later *refine* their definitions. > Now my research finding isn't hard to show quickly in broad strokes. > On of my important analysis tools is a simple technique to factor > polynomials into non-polynomial factors. > For instance, with the polynomial > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > that technique gives you > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > so I can factor to get > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > where the a's are the roots of the cubic > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's with x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3. > You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. > That's because P(x) has another special feature as > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > where that 49 is just begging to be divided off, which gives, of > course, > P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. > But remember, my three factors with the a's from before had *constant* > terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, > 1, and 22, which is the result that is so earth shattering. > Here the principle is like if you have > S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) > in that setting x=0 gives you *constant* terms within the expression, > which you can conveniently, also look at to see how it works. > S(0) = (7(0) + 7)(0 + 1) = 7(1). > The point is that the 7 is constant, so x's value means nothing to it. > So from before with > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > I know that dividing through by 49, it must go like > P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) > and as the constant terms are *independent* of the value of x, it MUST > be that in general > P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). > The problem now though is that conclusion can be used to show that > unequivocally beyond any reasonable doubt the definition of algebraic > numbers is TOO SMALL, as at times 5a_1/7 and 5a_2/7 are not included. > You see, they get left out, which is a problem because from the > *assumption* of mathematicians, they should be included, if the ring > of algebraic integers is the ring that mathematicians thought it was. > Some of you may find yourselves fearful of using your own mathematical > understanding, if you realize I'm right, and then realize that > mathematicians are disputing the result, especially if you see posters > tossing out far more complicated math in reply to my post, but > remember, math isn't magic. > Logic rules mathematics, so look for what makes sense. And remember > that you can't assume that posters are on your side. I don't want you > to assume that I'm on your side either. > You see, I don't need you to assume anything, as what I need you to do > is check. > While some mathematicians may erroneously believe now that it's in > their interest to hide the problem I've revealed, that mistake in > thinking does not help the rest of the world. After all, what good > does it do everyone else for mathematicians to hide their definition > problem? > What's in it for you? > James Harris James, Again, the problem is not with the definition; It is with the subsets of numbers. If I create a ring with all even integers [2,4,6...] and want to introduce 9 into the ring, I can't because 9 is odd. If the numbers you are describing do not belong to the ring, then they must be in another ring. I think you need to write an argument without all this crap about how mathematicians are evil and are hiding an error. That is beside the point and I don't think you're educated enough in mathematics to make that call. David Moran === Subject: Re: Explaining math definition problem > On of my important analysis tools is a simple technique to factor > polynomials into non-polynomial factors. > For instance, with the polynomial > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > that technique gives you > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > so I can factor to get > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > where the a's are the roots of the cubic > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's with x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3. > You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. > That's because P(x) has another special feature as > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > where that 49 is just begging to be divided off, which gives, of > course, > P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. > But remember, my three factors with the a's from before had *constant* > terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, > 1, and 22, which is the result that is so earth shattering. > Here the principle is like if you have > S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) > in that setting x=0 gives you *constant* terms within the expression, > which you can conveniently, also look at to see how it works. > S(0) = (7(0) + 7)(0 + 1) = 7(1). > The point is that the 7 is constant, so x's value means nothing to it. > So from before with > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > I know that dividing through by 49, it must go like > P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) > and as the constant terms are *independent* of the value of x, it MUST > be that in general > P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). Your recent discoveries have come to the attention of the Bureau of Advanced Supercomputing, Crytography Unit, Washington DC. Several researchers here feel that your achievement may well have major implications in developing methods of linearizing non-linear systems, and possibly weather forecasting, and chaos interpretation. On behalf of the BAS, I would like to extend to you this invitation to host a colloquium on your work for NOAA at Woods Hole Oceanographic Institute this fall. Your travel per diem and lodging are available per your expense account provided. We are also asking at this time that you keep confidential any discoveries which you may have made, as these could have implications to national security or other defense related matters of global importance. On behalf of BAS and NOAA, we hope that you will accept this request to join Respectfully, Richard Hertz, PhD, Prof Emeritus, UW-Madison co-Chair, NSA/BS algorithms Washington DC dickhertz@nsa.mil.gov === Subject: Re: Explaining math definition problem Never trust a guy called critter. Anyhow, isn't it unlawful to impersonate someone on the web? MB > Your recent discoveries have come to the attention of the Bureau of Advanced > Supercomputing, Crytography Unit, Washington DC. Several researchers here > feel that your achievement may well have major implications in developing > methods of linearizing non-linear systems, and possibly weather forecasting, > and chaos interpretation. > On behalf of the BAS, I would like to extend to you this invitation to host > a colloquium on your work for NOAA at Woods Hole Oceanographic Institute > this fall. Your travel per diem and lodging are available per your expense > account provided. > We are also asking at this time that you keep confidential any discoveries > which you may have made, as these could have implications to national > security or other defense related matters of global importance. > On behalf of BAS and NOAA, we hope that you will accept this request to join === Subject: Re: Explaining math definition problem > Anyhow, isn't it unlawful to impersonate someone on the web? > MB IFF during division by additive identities, hence, suppose not, ergo, a contradiction. I'd rather not be made to argue this point for the next n+1 years....... Rich Hertz === Subject: [JSH] Re: Explaining math definition problem Adjunct Assistant Professor at the University of Montana. >I'm an independent researcher, which means that I use my *own* >funding, and my *own* direction to go out and see what knowledge I can >obtain. Some of my research has been in the area of mathematics. >Getting important research findings is one thing, and getting them >noticed, is another. >At least here on Usenet I can talk freely to people around the world. >What I'd like to explain is my disturbing and to me fascinating >finding of a problem with a math definition that's over a hundred >years old. In looking over various replies to my previous posts on >this subject, I've seen assertions that definitions can't cause >problems, which is something that I can address quickly at the start. >Over a hundred years ago, the great German mathematicians Karl Gauss >played with numbers of the form a+bi, where 'a' and 'b' are integers. >In his honor they were later called gaussian integers, though a number >like 1+2i is not an integer. The gaussian up-front is important. >Later mathematicians came up with other numbers they called algebraic >integers, which include gaussian integers. >They thought they'd found THE set, or superset you might call it, >which includes all numbers with certain special properties of >integrality. >The most important property to point out is the ability to have >primeness between numbers. James, if you DO NOT KNOW the history or the reasons behind something, then DON'T TRY TO GUESS WHAT THE REASONS ARE! You are just plainly, simply, and completely wrong about how and why algebraic integers came about. It had very little to do with the ability have primeness between numbers, which is obviously your bastardization of being able to say two elements are coprime. It had to do with extending the notions of unique factorization, and the ->main<- reason for their creation in the first place (through the work of Kummer) was to extend a result of Gauss, conjectured by Euler, called Quadratic Reciprocity. [.snip.] >What I've found is a problem with their set of algebraic integers, as >unfortunately, despite what many mathematicians think, it's too small. >That's it. The definition they use is too small to do what they think >it does, which is include all these interesting numbers with special >properties. >But because they *think* it's big enough, mathematicians have an error >in their discipline based on their false assumption, as they've come >up with more arguments based on that assumption, which then aren't >actually proven. >It's like when the Greeks with their word atom thought they had the >smallest thing, and later our civilization used it, and broke atoms >apart, though part of the definition is that they are *indivisible*, >as people can define things, and later *refine* their definitions. >Now my research finding isn't hard to show quickly in broad strokes. >On of my important analysis tools is a simple technique to factor >polynomials into non-polynomial factors. >For instance, with the polynomial >P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 >that technique gives you >P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 >so I can factor to get >P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). >where the a's are the roots of the cubic >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). >Now despite the complexity, you can rely on *simple* ideas still, by >noticing that setting x=0, pulls out constant terms, as >P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) >as the cubic defining the a's with x=0 is >a^3 - 3a^2, which has roots, 0, 0 and 3. >You may not realize it, but what you just saw is revolutionary, both >in the special techniques, and most importantly with the consequences >that quickly follow. >That's because P(x) has another special feature as >P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) >where that 49 is just begging to be divided off, which gives, of >course, >P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. >But remember, my three factors with the a's from before had *constant* >terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, >1, and 22, which is the result that is so earth shattering. >Here the principle is like if you have >S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) This example is with a reducible polynomial and constant coefficients. It is worthless and a red herring. >in that setting x=0 gives you *constant* terms within the expression, >which you can conveniently, also look at to see how it works. >S(0) = (7(0) + 7)(0 + 1) = 7(1). >The point is that the 7 is constant, so x's value means nothing to it. >So from before with >P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) >I know that dividing through by 49, it must go like >P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) >and as the constant terms are *independent* of the value of x, it MUST >be that in general >P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). No, this calculation is incorrect. Here are the correct calculations: nonconstant term constant term P(m)/49 = [ (5a_1(m)/w_1(m) + (7/w_1(m)) -1) + 1 ] * [ (5a_2(m)/w_2(m) + (7/w_2(m)) -1) + 1 ] * [ ( ( (h_3(m)+22)/w_3(m) ) -22 ) + 22 ] where w_1(m) is the common factor of 7 and 5a_1(m) + 7 w_2(m) is the common factor of 7 and 5a_2(m) + 7 w_3(m) is the common factor of 7 and 5a_3(m) + 7=(5a_3(m)-15)+22. IT IS NOT TRUE THAT THE CONSTANT TERM NECESSARILY CONSISTS OF EVERY SUMMAND WHICH DOES NOT CHANGE AS m CHANGES. IT IS NOT TRUE THAT IN THE NON-CONSTANT TERM EVERY SUMMAND MUST BE MULTIPLIED BY SOMETHING THAT CHANGES AS m CHANGES. Your error is in thinking that the independent term is always equal to 7/w_i(m). It is not. You never bothered to do the actual calculations explicitly, and you screwed up. [.rest deleted.] Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to this extent, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Explaining math definition problem > I'm an independent researcher, which means that I use my *own* > funding, and my *own* direction to go out and see what knowledge I can > obtain. Some of my research has been in the area of mathematics. > Getting important research findings is one thing, and getting them > noticed, is another. > At least here on Usenet I can talk freely to people around the world. > What I'd like to explain is my disturbing and to me fascinating > finding of a problem with a math definition that's over a hundred > years old. In looking over various replies to my previous posts on > this subject, I've seen assertions that definitions can't cause > problems, which is something that I can address quickly at the start. > Over a hundred years ago, the great German mathematicians Karl Gauss > played with numbers of the form a+bi, where 'a' and 'b' are integers. > In his honor they were later called gaussian integers, though a number > like 1+2i is not an integer. The gaussian up-front is important. > Later mathematicians came up with other numbers they called algebraic > integers, which include gaussian integers. > They thought they'd found THE set, or superset you might call it, > which includes all numbers with certain special properties of > integrality. > The most important property to point out is the ability to have > primeness between numbers. Well, that's the problem with the defintion. Idiots Number Theorists need to be reminded at least once a generation that primeness is NOT a property of integers. It is a property of NON-NEGATIVE integers. It is necessary that you say that in ALL CAPS. Otherwise the meta-meta-dorks in logic can't hear you. === Subject: Re: Explaining math definition problem [snip historically and technically inaccurate preample] > Now my research finding isn't hard to show quickly in broad strokes. > On of my important analysis tools is a simple technique to factor > polynomials into non-polynomial factors. > For instance, with the polynomial > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > that technique gives you > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > so I can factor to get > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > where the a's are the roots of the cubic > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's with x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3. > You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. What you just presented and others may have seen was completely wrong! Your original polynomial: P(x) = 14706125x^3 - 900375x^2 + 157640x + 1708 can, indeed be factored as P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). But the next logical step, which you did not take, is to determine a_1, a_2, and a_3 so that the equality holds for *all* 'x'. Otherwise, calling both expressions, P(x) is false and misleading. Since you did not post the values of a_1, a_2, and a_3 which satisfy the stated equality, I have taken the liberty of posting them for you. They are: a_1 = -(7/5)+(1/5)(1078+175640x-900375x^2+14706125x^3)^(1/3) a_2 = -(7/5)-(1/10)(1-3^(1/2)I)(1078+175640x-900375x^2+14706125x^3)^(1/3) a_3 = -(7/5)-(1/10)(1+3^(1/2)I)(1078+175640x-900375x^2+14706125x^3)^(1/3) For these values of a_1, a_2 and a_3 (and only these values), does P(x) = 14706125x^3 - 900375x^2 + 175640x + 1078 = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). In fact, the following specific cases can be easily confirmed: x=0, P(0) = 1078 x=1, P(1) = 13982468 x=2, P(2) = 114399858 which holds for either the original representation of P(x) or the other representation. Since you did not provide the expressions for a_1, a_2, and a_3, but merely asserted *incorrectly* that two of them go to zero when x = 0, there is no way to determine the equivalence of the two representations for P(x) for any other values of x from your post. Note, however, that *none* of the correct a_1, a_2 or a_3 go to zero when x = 0! You have arbitrarily picked numbers that produce the constant term you are looking for and then claim that this results from setting x = 0. This is false. In fact, for the case x= 0: a_1 = (1/5)(-7+7^(2/3) 22^(1/3) = 0.650704... a_2 = (1/10)(-14+I7^(2/3) 22^(1/3)(I+3*(1/2))) = -2.42535...+1.77596...I a_3 = (1/10)(-14-7^(2/3) 22^(1/3)(1+3^(1/2)I)) = -2.42535...-1.77596...I None of these values is zero at x=0, but the value of P(0) is correctly given as 1078. You also failed to state that the constant term in the original expression for P(x) is also 'pulled out' by setting x = (105 + (129487)^(1/2) I)/3430 or setting x = (105 - (129487)^(1/2) I)/3430. These values work for both of the representations for P(x) as long as the values I have given above for a_1, a_2, and a_3 are used in the second representational form. The rest of your post is based on the serious gap which occurred when you jumped to the conclusion, without any proof, that exactly two of the 'a's go to zero when x = 0. They do not. Before you repost, please correct the problem by using the correct values of a_1, a_2 and a_3, instead of just waving your hands and declaring that two of the 'a's go to 0 when x = 0. Wacky, isn't it. But, hey it's just basic math. Yup, yup, yup! -- A man with integrity identifies, acknowledges and corrects his errors. One with it ignores, denies or defends them. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Explaining math definition problem >[...] >A man with integrity identifies, acknowledges and corrects his errors. One >with it ignores, denies or defends them. I believe there's an error there... >Democracy: The triumph of popularity over principle. ************************ David C. Ullrich === Subject: Re: Explaining math definition problem >[...] >-- >A man with integrity identifies, acknowledges and corrects his errors. One >with it ignores, denies or defends them. > I believe there's an error there... math passes scrutiny, however.) >-- >Democracy: The triumph of popularity over principle. > ************************ > David C. Ullrich === Subject: Re: Explaining math definition problem >I'm an independent researcher, which means that I use my *own* >funding, and my *own* direction to go out and see what knowledge I can >obtain. Some of my research has been in the area of mathematics. >Getting important research findings is one thing, and getting them >noticed, is another. ??? The editors of the journals you've sent papers to have certainly noticed your results. >At least here on Usenet I can talk freely to people around the world. >What I'd like to explain is my disturbing and to me fascinating >finding of a problem with a math definition that's over a hundred >years old. In looking over various replies to my previous posts on >this subject, I've seen assertions that definitions can't cause >problems, which is something that I can address quickly at the start. >Over a hundred years ago, the great German mathematicians Karl Gauss >played with numbers of the form a+bi, where 'a' and 'b' are integers. >In his honor they were later called gaussian integers, though a number >like 1+2i is not an integer. The gaussian up-front is important. >Later mathematicians came up with other numbers they called algebraic >integers, which include gaussian integers. >They thought they'd found THE set, or superset you might call it, >which includes all numbers with certain special properties of >integrality. Guffaw. Show us a reference where a mathematician states that the algebraic integers include all numbers with certain special properies of intergrality. ************************ === Subject: Re: Explaining math definition problem > With the polynomial > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > that technique gives you > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > so I can factor to get > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > where the a's are the roots of the cubic > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > as the cubic defining the a's with x=0 is > a^3 - 3a^2, which has roots, 0, 0 and 3. > You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. > That's because P(x) has another special feature as > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > where that 49 is just begging to be divided off, which gives, of > course, > P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. > But remember, my three factors with the a's from before had *constant* > terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, > 1, and 22, which is the result that is so earth shattering. > Here the principle is like if you have > S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) > in that setting x=0 gives you *constant* terms within the expression, > which you can conveniently, also look at to see how it works. > S(0) = (7(0) + 7)(0 + 1) = 7(1). > The point is that the 7 is constant, so x's value means nothing to it. I don't understand what you mean by this. Your usages of 7 is confusing for me. Could you please restate this with the example S(x) = 7x^2 + 19x + 10 = (7x+5)(x+2) S(0) = (7(0)+5)(0+2) = 5(2) And tell me what is constant and thus independent of what? I just wish to understand what all this fuss is about. > So from before with > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > I know that dividing through by 49, it must go like > P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) > and as the constant terms are *independent* of the value of x, it MUST > be that in general > P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). Quaternion === Subject: Re: Explaining math definition problem > With the polynomial > > P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 > > that technique gives you > > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > > so I can factor to get > > P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). > > where the a's are the roots of the cubic > > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > > Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as > > P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) > > as the cubic defining the a's with x=0 is > > a^3 - 3a^2, which has roots, 0, 0 and 3. > > You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. > > That's because P(x) has another special feature as > > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > > where that 49 is just begging to be divided off, which gives, of > course, > > P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. > > But remember, my three factors with the a's from before had *constant* > terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, > 1, and 22, which is the result that is so earth shattering. > Here the principle is like if you have > > S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) > > in that setting x=0 gives you *constant* terms within the expression, > which you can conveniently, also look at to see how it works. > > S(0) = (7(0) + 7)(0 + 1) = 7(1). > > The point is that the 7 is constant, so x's value means nothing to it. > I don't understand what you mean by this. Your usages of 7 is confusing > for me. Could you please restate this with the example > S(x) = 7x^2 + 19x + 10 = (7x+5)(x+2) > S(0) = (7(0)+5)(0+2) = 5(2) > And tell me what is constant and thus independent of what? I just wish to > understand what all this fuss is about. Well you pointed them out, as 5 and 2 are constant in the factors of your S(x), while its constant term, of course, is 10. My examples have the further distinction that the polynomial has 7 as a factor in general, as, for instance, I have P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) which you'll notice has 49 as a factor. The math I can use is VERY simple, thankfully, as mathematicians are fighting this result, so I use very basic math to give them less room to obfuscate. Here what I'm using is the distributive property. Remember that one? Distributive Property: a(b+c) = ab + ac And believe me if you have to break it down to the distributive property then you have some SERIOUS denial on the part of mathematicians. Can you believe that people? Having to break it all the way down to the freaking *distributive property*? James Harris === Subject: Re: Explaining math definition problem > The math I can use is VERY simple, thankfully, as mathematicians are > fighting this result, so I use very basic math to give them less room > to obfuscate. > Here what I'm using is the distributive property. Remember that one? Um, no, what you're using is an intuitive believe that your non polynomial factors should have certain properties in common with polynomial factors. And they don't. You keep saying they MUST, but a number of people have actually worked out explicit expressions for those terms which makes it abundantly clear that they DON'T. Can you write down a1, a2, a3 explicitly for your polynomial as others have done (not another polynomial, YOUR polynomial) and show they have the behavior you want them to have? - Randy === Subject: Re: Explaining math definition problem > Here what I'm using is the distributive property. Remember that one? > Distributive Property: > a(b+c) = ab + ac > And believe me if you have to break it down to the distributive > property then you have some SERIOUS denial on the part of > mathematicians. > Can you believe that people? Having to break it all the way down to > the freaking *distributive property*? There is no reason to udnerestimate me like this..you are more naive than I think.. No I read the other comments and even *I* understand how you have to express the a_i for all x and not calculate them for x=0 and then determine, because they have constant factors there, that they are independent for all x.. You constructed it in a manner that causes, confusion, but nevertheless not enough to fool people here (and to be honest, they all got it the minute they read your post) Perhaps if you could explain yourself completely, once and for all, just completely, every point, write it all in a multi-page pdf, double-check it, then let others calmly correct points using little asterixes that point to the relevant sections, then you would be seen as a misunderstood person and not as a troll..or even worse, like what you are now, a toy. Quaternion === Subject: Re: Explaining math definition problem > Here what I'm using is the distributive property. Remember that one? > > Distributive Property: > > a(b+c) = ab + ac > > And believe me if you have to break it down to the distributive > property then you have some SERIOUS denial on the part of > mathematicians. > > Can you believe that people? Having to break it all the way down to > the freaking *distributive property*? > There is no reason to udnerestimate me like this..you are more naive than I > think.. Your bravado is telling. The issue you brought up, which YOU deleted out in replying to my response is best answered by reminding of the distributive property. So I answered you and you replied back leaving in one little piece taking away context. That's not only a sign of *your* intellectual weakness, it's a sign of immaturity. You can't handle the truth. > No I read the other comments and even *I* understand how you have to express > the a_i for all x and not calculate them for x=0 and then determine, > because they have constant factors there, that they are independent for all > x.. You constructed it in a manner that causes, confusion, but nevertheless > not enough to fool people here (and to be honest, they all got it the > minute they read your post) Well if you're telling the truth then you can find a mistake or a break in the logical chain. It's MATH, not fashion, so opinions don't count. If I'm wrong, find the mistake. Now here is the very basic argument again. Remember the distributive property, and read carefully this time. On of my important analysis tools is a simple technique to factor polynomials into non-polynomial factors. For instance, with the polynomial P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 that technique gives you P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 so I can factor to get P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). where the a's are the roots of the cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). Now despite the complexity, you can rely on *simple* ideas still, by noticing that setting x=0, pulls out constant terms, as P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's with x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3. You may not realize it, but what you just saw is revolutionary, both in the special techniques, and most importantly with the consequences that quickly follow. That's because P(x) has another special feature as P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) where that 49 is just begging to be divided off, which gives, of course, P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. But remember, my three factors with the a's from before had *constant* terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, 1, and 22, which is the result that is so earth shattering. Here the principle is like if you have S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) in that setting x=0 gives you *constant* terms within the expression, which you can conveniently, also look at to see how it works. S(0) = (7(0) + 7)(0 + 1) = 7(1). The point is that the 7 is constant, so x's value means nothing to it. So from before with P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) I know that dividing through by 49, it must go like P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) and as the constant terms are *independent* of the value of x, it MUST be that in general P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). James Harris === Subject: Re: Explaining math definition problem Adjunct Assistant Professor at the University of Montana. [.snip.] >If I'm wrong, find the mistake. Been there, done that. >Now here is the very basic argument again. Remember the distributive >property, and read carefully this time. The distributive property is a red herring. >On of my important analysis tools is a simple technique to factor >polynomials into non-polynomial factors. >For instance, with the polynomial >P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 >that technique gives you >P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 >so I can factor to get >P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). >where the a's are the roots of the cubic >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). >Now despite the complexity, you can rely on *simple* ideas still, by >noticing that setting x=0, pulls out constant terms, as >P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) >as the cubic defining the a's with x=0 is >a^3 - 3a^2, which has roots, 0, 0 and 3. >You may not realize it, but what you just saw is revolutionary, both >in the special techniques, and most importantly with the consequences >that quickly follow. >That's because P(x) has another special feature as >P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) >where that 49 is just begging to be divided off, which gives, of >course, >P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. >But remember, my three factors with the a's from before had *constant* >terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, >1, and 22, which is the result that is so earth shattering. Hyperbole. >Here the principle is like if you have >S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) This example is worthless; it is a red herring; your polynomial has constant coefficients and has been factored over Q into polynomials with integer coefficients. You've been told time and again, and it has been explained to you, that the key feature that affects the divisibility is the IRREDUCIBILITY of the polynomial over Q. Your example is reducible at all values, hence useless. >in that setting x=0 gives you *constant* terms within the expression, >which you can conveniently, also look at to see how it works. >S(0) = (7(0) + 7)(0 + 1) = 7(1). >The point is that the 7 is constant, so x's value means nothing to it. >So from before with >P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) >I know that dividing through by 49, it must go like >P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22) >and as the constant terms are *independent* of the value of x, it MUST >be that in general >P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7). That's the mistake. Here it is properly calculated. Since you are cutting and pasting, I will do the same. Try not to stop it midflow and say here's a good place to stop this time. -- BEGIN INSERTED TEXT -- Let g_1(m) = 5a_1(m) + 7 g_2(m) = 5a_2(m) + 7 g_3(m) = 5a_3(m) + 7. We have P(m) = g_1(m)*g_2(m)*g_3(m). So g_1(m) = h_1(m) + 7 g_2(m) = h_2(m) + 7 g_3(m) = h_3(m) + 22 where h_1(m) = g_1(m)-g_1(0) = 5a_1(m) h_2(m) = g_2(m)-g_2(0) = 5a_2(m) h_3(m) = g_3(m)-g_3(0) = 5a_3(m)-15 = 5(a_3(m)-3). Note what h_3(m) is: it does not LOOK like something independent of m at all: it is h_3(m) = 5a_3(m) - 15; we have that -15! But surely the part that is not independent of m cannot have a part which IS independent of m! What that means, of course, is that just LOOKING at the expression does not tell you what the independent term is. To figure out the indepdendent term, we MUST evaluate at 0. That's why h_3(m) has no independent term even though there is that -15 there: because h_3(0) = 5(a_3(0)-3) = 5(3-3) = 0. So remember: we don't just LOOK at the expression to find the independent term; we must EVALUATE AT m=0 to get it. That is: IT IS NOT TRUE THAT THE CONSTANT TERM NECESSARILY CONSISTS OF EVERY SUMMAND WHICH DOES NOT CHANGE AS m CHANGES. IT IS NOT TRUE THAT IN THE NON-CONSTANT TERM EVERY SUMMAND MUST BE MULTIPLIED BY SOMETHING THAT CHANGES AS m CHANGES. That is important. Remember it. Now, for each value of m, we know we can factor out factors of 7 out of each of g_1(m), g_2(m), g_3(m). So that P(m)/49 = (1/49) g_1(m)*g_2(m)*g_3(m) Let w_1(m) be the factor we take out of g_1(m); let w_2(m) be the factor we take out of g_2(m); and let $w_3(m) be the factor we take out of g_3(m). You claim that w_1(m)=7 for all m; w_2(m)=7 for all m; and w_3(m)=1 for all m. You claim this follows from the constant terms. We have the following: g_1(m)/w_1(m) = (h_1(m) + 7)/w_1(m) = (5a_1(m)/w_1(m)) + (7/w_1(m)) g_2(m)/w_2(m) = (h_2(m) + 7)/w_2(m) = (5a_2(m)/w_2(m)) + (7/w_2(m)) g_3(m)/w_3(m) = (h_3(m) + 22)/w_3(m) We can rewrite the last one if you want, as g_3(m)/w_3(m) = (5a_3(m)/w_3(m)) + (7/w_3(m)) so it looks like the others, but this is important to bear in mind: we will have not written it the sam as the others! g_1 and g_2 are written as the stuff that depends on m plus the stuff that is independent of m, but with g_3 we will not have done the same thing. So it would be misleading to write it that way; and we should certainly not write g_3(m)/w_3(m) = (h_3(m)/w_3(m)) + (22/w_3(m)) So we have a choice: we either write it as a single fraction, g_3(m)/w_3(m) = (h_3(m)+22)/w_3(m), or else we write it as g_3(m)/w_3(m) = (5a_3(m)/w_3(m)) + (7/w_3(m)) and remember that 7/w_3(m) is NOT the constant term. Let's given them new names to avoid typing extra: H_1(m) = g_1(m)/w_1(m) = (5a_1(m)/w_1(m)) + (7/w_1(m)) H_2(m) = g_2(m)/w_2(m) = (5a_2(m)/w_2(m)) + (7/w_2(m)) H_3(m) = g_3(m)/w_3(m) = (h_3(m)+22)/w_3(m) = (5a_3(m)/w_3(m)) + (7/w_3(m)). So P(m)/49 = H_1(m)*H_2(m)*H_3(m). The constant terms of these are: H_1(0) = (5a_1(0)/w_1(0)) + (7/w_1(0)) = 0/7 + 7/7 = 1. H_2(0) = (5a_2(0)/w_2(0)) + (7/w_2(0)) = 0/7 + 7/7 = 1. H_3(0) = (5a_3(0)/w_3(0)) + (7/w_3(0)) = 15/1 + 7/1=22. All this is correct. But now comes your mistake. Your error is NOT that H_1(0)=1; your error is that you think that 7/w_1(m) is the constant term. It is NOT. H_1(m) - H_1(0) = (5a_1(m)/w_1(m))+(7/w_1(m)) - 1 H_2(m) - H_2(0) = (5a_2(m)/w_2(m))+(7/w_2(m)) - 1 H_3(m) - H_3(0) = (h_3(m)+22)/w_3(m)) - 22. So we have: nonconstant term constant term P(m)/49 = [ (5a_1(m)/w_1(m) + (7/w_1(m)) -1) + 1 ] * [ (5a_2(m)/w_2(m) + (7/w_2(m)) -1) + 1 ] * [ ( ( (h_3(m)+22)/w_3(m) ) -22 ) + 22 ] As you can see, the constant terms are 1, 1, and 22. But the nonconstant terms are NOT what you claim they are. Note that even though there are those -1 and -22 in the nonconstant term, that's not a problem. Just as we had above, when we noticed that the nonconstant term of g_3(m) had a -15 in it; not every summand has to be mulitplied by something that changes with m for something to be the nonconstant term. Remember: IT IS NOT TRUE THAT THE CONSTANT TERM NECESSARILY CONSISTS OF EVERY SUMMAND WHICH DOES NOT CHANGE AS m CHANGES. IT IS NOT TRUE THAT IN THE NON-CONSTANT TERM EVERY SUMMAND MUST BE MULTIPLIED BY SOMETHING THAT CHANGES AS m CHANGES. You error is in thinking that 5a_1(m)/w_1(m) + 7/w_1(m) - 1 is equal to 5a_1(m)/w_1(m); but you can ONLY CONCLUDE THAT IF YOU ASSUME THAT w_1(m)=7 for all m; and that is what you want to ->conclude<-. That is, you are engaging in a circular argument. As you can see, even if w_1(m) is not equal to 7, the constant term does not change. It is still 1; the constant term is NOT (7/w_1(m)), and it is not [(7/w_1(m)) -1] + 1. It is 1. Say w_1(5) = sqrt(7) [it's not, but let's assume it is for argument's sake]; you claimed in the Hong Kong forum that this would imply that the constant term changes: (message 9 in http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782 ) Well, let's see: nonconstant term constant term P(5)/49 = [ (5a_1(5)/w_1(5) + (7/w_1(5)) -1) + 1 ] * [ (5a_2(5)/w_2(5) + (7/w_2(5)) -1) + 1 ] * [ ( ( (h_3(5)+22)/w_3(5) ) -22 ) + 22 ] = [ (5a_1(5)/(sqrt(7)) + (sqrt(7) -1) + 1 ] * [ (5a_2(5)/w_2(5) + (7/w_2(5)) -1) + 1 ] * [ ( ( (h_3(5)+22)/w_3(5) ) -22 ) + 22 ] Nope. The constant term of the H_1(m) is STILL 1; the constant term did not change at all. --- End Inserted Text --- Your error is that you think that the constant term is BOTH 1 and 7/w_1(m). It is not. The constant term is 1. The 7/w_1(m) is part of the nonconstant term, which is equal to (5a_1(m)/w_1(m)) + (7/w_1(m)) - 1. When you accuse me and others of saying that the constant term changes, this is based on your MISTAKE of thinking that the constant term is always equal to 7/w_1(m). It is not. Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to this extent, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Explaining math definition problem > With the polynomial P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078 that technique gives you P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 so I can factor to get P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7). where the a's are the roots of the cubic a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). Now despite the complexity, you can rely on *simple* ideas still, by > noticing that setting x=0, pulls out constant terms, as P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22) as the cubic defining the a's with x=0 is a^3 - 3a^2, which has roots, 0, 0 and 3. You may not realize it, but what you just saw is revolutionary, both > in the special techniques, and most importantly with the consequences > that quickly follow. That's because P(x) has another special feature as P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) where that 49 is just begging to be divided off, which gives, of > course, P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22. But remember, my three factors with the a's from before had *constant* > terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, > 1, and 22, which is the result that is so earth shattering. Here the principle is like if you have S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1) in that setting x=0 gives you *constant* terms within the expression, > which you can conveniently, also look at to see how it works. S(0) = (7(0) + 7)(0 + 1) = 7(1). The point is that the 7 is constant, so x's value means nothing to it. > I don't understand what you mean by this. Your usages of 7 is confusing > for me. Could you please restate this with the example > S(x) = 7x^2 + 19x + 10 = (7x+5)(x+2) > S(0) = (7(0)+5)(0+2) = 5(2) > And tell me what is constant and thus independent of what? I just wish to > understand what all this fuss is about. > Well you pointed them out, as 5 and 2 are constant in the factors of > your S(x), while its constant term, of course, is 10. > My examples have the further distinction that the polynomial has 7 as > a factor in general, as, for instance, I have > P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22) > which you'll notice has 49 as a factor. > The math I can use is VERY simple, thankfully, as mathematicians are > fighting this result, so I use very basic math to give them less room > to obfuscate. > Here what I'm using is the distributive property. Remember that one? > Distributive Property: > a(b+c) = ab + ac Completely untrue. I disproved the distributive property 7 years ago, which is also the 4th prime, and therefore what you are saying cannot possibly be false. === Subject: Z-scores If snow falls on a normally distributed curve and a hypothetical city averages 100 cm of snow over the last 40 yr period how many standard deviations is 50 cm from the mean? TIA Dan === Subject: Re: Z-scores >If snow falls on a normally distributed curve and a hypothetical city >averages 100 cm of snow over the last 40 yr period how many standard >deviations is 50 cm from the mean? Answer: There is not enough information given to determine this. === Subject: !!#- Need help with one problem please! I'm trying to figure out how to find the dimensions that minimizes surface area of a cylinder that has a volume of 398mL. How do u do this kind of question? Please help. === Subject: Re: !!#- Need help with one problem please! > I'm trying to figure out how to find the dimensions that minimizes surface > area of a cylinder that has a volume of 398mL. > How do u do this kind of question? > Please help. Assume it's a right circular cylinder, with height h and radius r. I think it's reasonably clear that a right circular cylinder will minimize surface area for a given volume. So what is the surface area of the cylinder? Ans: the area of the bases (two of them -- bottom and top) plus the area of the side A = 2*pi*r^2 + 2*pi*r*h And what is the volume? Ans: V = pi*r^2*h The formula for V allows you to solve for h in terms of r. Replace that value of h in the surface area formula and find the point where dA/dr = 0. Remember, V is a constant in this problem. That should be more than enough help. Good luck. .... Bob === Subject: Re: !!#- Need help with one problem please! > I'm trying to figure out how to find the dimensions that minimizes surface > area of a cylinder that has a volume of 398mL. > How do u do this kind of question? > Please help. > Assume it's a right circular cylinder, with height h and radius r. I think > it's reasonably clear that a right circular cylinder will minimize surface > area for a given volume. And the proof is much too difficult for an intro calc class. Jon Miller === Subject: Re: !!#- Need help with one problem please! U must minimise the function f(R,h) = 2 pi R^2 + 2 pi R h With fi(R,h) = pi R^2 h - 398 = 0 as a condition Use Lagrange multiplier lambda and the lagrangean function L( R, h, lambda) = f(R,h) + lambda * fi ( R,h) the partial derivatives of L in order at R, h and Lambda must vanish DR (L) = 0 Dh(L) = 0 Dlambda(L) = fi = 0 solve the sistem and U have points of possible maximum, minimum or sela. U must then decide where is the minimum from the Hessian matrix. ----sorry about my english > I'm trying to figure out how to find the dimensions that minimizes surface area of a cylinder that > has a volume of 398mL. > How do u do this kind of question? > Please help. === Subject: Re: !!#- Need help with one problem please! > U must minimise the function > f(R,h) = 2 pi R^2 + 2 pi R h > With fi(R,h) = pi R^2 h - 398 = 0 as a condition > Use Lagrange multiplier lambda and the lagrangean function > L( R, h, lambda) = f(R,h) + lambda * fi ( R,h) > the partial derivatives of L in order at R, h and Lambda must vanish > DR (L) = 0 > Dh(L) = 0 > Dlambda(L) = fi = 0 > solve the sistem and U have points of possible maximum, minimum or sela. saddle point. > U must then decide where is the minimum from the Hessian matrix. No problem. But I think in this case, it's easier to just set h=398/(pi*R^2), and substitute in the equation for surface area, so it's just a 1-variable calculus problem. Jon Miller === Subject: Pretty Calculus I Problem I'm trying to create a function in this form. f(x) = ax^3 + bx^2 + cx + d These are the prerequisites: a, b, c, d are all integers. If ax^3 + bx^2 + cx + d = 0, then the factors are also all integers. In other words, each of the x-intercepts of the functions consists out of integers. When the derivative 3ax^2 + 2bx + c = 0, the solution to x is also an integer. When I set the second derivative 6ax + 2b to zero, the solution of x is also an integer. In other words, this is a pretty Calculus 1 problem. Is there a way outside of the trial and error method that I can do this? Kees === Subject: Re: Pretty Calculus I Problem > I'm trying to create a function in this form. > f(x) = ax^3 + bx^2 + cx + d > These are the prerequisites: > a, b, c, d are all integers. > If ax^3 + bx^2 + cx + d = 0, then the factors are also all integers. In > other words, each of the x-intercepts of the functions consists out of > integers. > When the derivative 3ax^2 + 2bx + c = 0, the solution to x is also an > integer. > When I set the second derivative 6ax + 2b to zero, the solution of x is also > an integer. > In other words, this is a pretty Calculus 1 problem. > Is there a way outside of the trial and error method that I can do this? Without loss of generality, assume roots of f are -p, 0, and q. Then you get f(x) = x(x+p)(x-q) = x^3 + (p-q)x^2 -pqx Then the first derivative is f'(x) = 3x^2 + 2(p-q)x - pq whose roots are x = (q - p +- sqrt(p^2 + q^2 + pq)) / 3 Let's assume p is even: p = 2e x = (q - 2e +- sqrt ((q+e)^2 + 3e^2)) / 3 (q+e)^2 + 3e^2 has to be the square of an integer, so let's say it's (q+e+g)^2. Then x = (q - 2e +- (q + e + g)) / 3 x1 = (2q - e + g / 3) x2 = (-3e - g) / 3 = -e - g/3 Therefore, g is divisible by 3. Let g = 3h: x1 = (2q - e + 3h) / 3 x2 = (-3e - 3h) / 3 = -e - h Now, 3e^2 = g(2q + 2e + 3h), so let g = 3 (h = 1) and e^2 = 2q + 2e + 3h. Then q = (e^2 - 2e - 3h)/2 x1 = 1/3 * e^2 - e So now e has to be divisible by three: e = 3i q = (9i^2 - 6i - 3)/2 x1 = 3i^2 - 3i = 3i(i-1) x2 = -3i - 1 Going back to f: f(x) = 6x + 2(p-q) = 0 x3 = 2(q-p) / 6 = (q-p) / 3 = (3i^2 - 6i - 1) / 2 = q - (3i^2 - 1) Now you are in a better position for trial and error. For any values of i and h, the coefficients of f', the coefficients of f, the roots of f and the roots of f' are integer. If q is an integer, then the root of f is also an integer, so all you have to do is find values i for which q is integer. The only requirement there is that 9i^2 - 3 is even, or, in other words, that i is odd. For example: h = 1, i = 3, e = 9: p = 18 q = 30 f(x) = x^3 - 12x^2 - 540x, roots: -18, 0, 30 f'(x) = 3x^2 - 24x - 540, roots: 18, -10 f(x) = 6x - 24, roots: 4 h = 1, i = 5, e = 15 p = 30 q = 96 f(x) = x^3 - 66x^2 - 2880x, roots = -30, 0, 96 f'(x) = 3x^2 - 132x - 2880, roots = 60, -16 f(x) = 6x - 132, root = 22 Of course, you can always slide the graph by an integer constant left or .9784 === Subject: Re: Simple principle, core error proven >> good demonstration, Randy Poe. >> >> Algebraic integers are defined to be roots of monic polynomials with >> integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where >> monic refers to the leading coefficient. >> >> If we are interested in the set of numbers which are >> roots of monic polynomials with integer coefficients, >> what does it mean that the set should include numbers >> that are not such roots? >> >> You've also never answered this one: It is possible to >> form a product ab = c with a and c in the set Z but b >> not in the set Z. Does that mean the set Z is incomplete >> and there is an error in the definition? (Ex: a=3, c=5, >> b = 5/3). >What I've found is a problem with their set of algebraic integers, as >unfortunately, despite what many mathematicians think, it's too small. >That's it. The definition they use is too small to do what they think >it does, which is include all these interesting numbers with special >properties. The first problem with your assertion is that you have demonstrated that you have no idea what these interesting properties that number theorists are actually interested in ->are<-. The properties YOU think are needed (which are that they include constants to make your factorizations work the way you think they should work) are not what they were interested in. Since it was not what they were interested in, then how can it be twoo small? What they were interested in, what Dedekind was interested in, was being able to extend Kummer's Unique Factorization Theorem from the cyclotomic fields to arbitrary finite extensions of Q. That is the Unique Factorization into Prime Ideals, a major theorem of Dedekind. Are you claiming this theorem is false? That would be quite a laugh, since you have no idea what it says. So, no, you are wrong. The definition is not too small to do what they think it does. The definition is too small to do what YOU think you should be able to do, but that's a different story altogether. [.rest deleted.] Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can criticize. A great many people are staggered to this extent, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan Arturo Magidin magidin@math.berkeley.edu