Ok, I've been in a lot of pain over the last few days and have been up to literally no good. I'm normally hazy and recently it's worse with undue pain. So I got to thinking because I have nothing to do aside from laying around nursing a fvcked up leg. I got my sis to give my her laptop for about a few hours so I could check the e-mails, and maybe cruise for '90-year-old hot mamas and you know the other fetish I'm not telling you about.
But as I lay there thinking early in the day, I had some of my daughter's toys to play with. Observing the curves and lines and angles of the different toys, how are the different properties of math the same for X whether X is 9 or 99.
Let's take the commutative property - 5 + 6 = 11 or 6 + 5 = 11 or 5-(-6)=11 and so on.
But how this work when X+Y = Z when the numbers are so huge, irrational, Geometrical (I guess this would be in a sense irrational), or something else...) . You obviously can't work them out on hand, calculator, mind or page so simply. Maybe others can, but I can't.
It is understood that what works for 2 works for 2.453678523 times 10^e or the algebraic representation of a solar system. What I want to know is how all these theorems, properties, or what not are co-relational.
Take any property. A Simple one if need be. And teach me something. Please?
Math Theory Abstraction
Moderator: Moderators
Math Theory Abstraction
Ancient History wrote:We were working on Street Magic, and Frank asked me if a houngan had run over my dog.
You're basically looking for real analysis and/or abstract algebra. I can mail you, physically, my old textbook for real analysis if you want (being a draft copy, it's not like I can sell it; unfortunately, I sold my abstract algebra textbook a long time ago). It's pretty hardcore. If you don't want that, I can try to explain something.
The commutative property of addition is not always true. The real numbers under addition form something known as an Abelian group. Abelian groups are basically a set and an operator (we'll call it +), and have a few special properties:
* Closure: For any element a and b in the set, the element a+b is also in the set.
* Associativity: For elements a, b, and c in the set, (a+b)+c = a+(b+c)
* Identity: There is some element i in the set such that a + i = a, for any a in the set.
* Inverse: For any element a in the set, there is an element b such that a + b = i.
* Commutivity: We already know what that is.
It's the commutivity that makes it an "Abelian" group, the rest just make it a group.
We already know this is true for the real numbers (why that is so goes into real analysis, not abstract algebra) -- but what is it not true for? Groups can represent more than just what we commonly think of as sets of numbers -- they can also represent things such as permutations of objects (such as the rotation, translation, and mirror of a square) -- these groups are often not abelian. An example of such a group is the Dihedral group of order 6. Wikipedia has a good page on it.
I'm not a very good math teacher, so I think I'll stop here. But I'll send you my old Real Analysis textbook if you want -- which talks about constructing the real numbers in the first chapter.
The commutative property of addition is not always true. The real numbers under addition form something known as an Abelian group. Abelian groups are basically a set and an operator (we'll call it +), and have a few special properties:
* Closure: For any element a and b in the set, the element a+b is also in the set.
* Associativity: For elements a, b, and c in the set, (a+b)+c = a+(b+c)
* Identity: There is some element i in the set such that a + i = a, for any a in the set.
* Inverse: For any element a in the set, there is an element b such that a + b = i.
* Commutivity: We already know what that is.
It's the commutivity that makes it an "Abelian" group, the rest just make it a group.
We already know this is true for the real numbers (why that is so goes into real analysis, not abstract algebra) -- but what is it not true for? Groups can represent more than just what we commonly think of as sets of numbers -- they can also represent things such as permutations of objects (such as the rotation, translation, and mirror of a square) -- these groups are often not abelian. An example of such a group is the Dihedral group of order 6. Wikipedia has a good page on it.
I'm not a very good math teacher, so I think I'll stop here. But I'll send you my old Real Analysis textbook if you want -- which talks about constructing the real numbers in the first chapter.
As I understand it, the commutative property for numbers is part of the definition of addition (e: it's not). Addition is defined so that A + B = B + A = C, and for all A and B there exists a C to fulfill it. On the whole numbers it's defined as repetitions of the "increment" function, with similar definitions expanded for rationals and reals. On the complex numbers its two orthogonal real additions ( [A + Bi] + [C + Di] = [A + C] + [B + D]i )
Actually, a lot of these basic properties come down to definitions, and they don't have to be defined that way. I think commutativity is part of the general definition of addition on all sets (that is, if an operation does not have a commutative property, then it's not addition, no matter what you're doing it to), but it's not for, for instance, multiplication.
Let's take Vectors. A vector is essentially a point in space, or a quantity and a direction. You can think of it as an arrow that you're allowed to move anywhere, but not rotate; it points in a specific direction and has a specified length. If you have a defined origin (zero point), then any point in space can be converted into a vector by just drawing that arrow from the origin to the point.
To add vectors, there are a few mathematically equivalent ways. Geometrically, you just put the tail of one arrow at the origin and the tail of the other on the head of the first, and draw a new vector from the origin to the head of the second. You can draw these on a piece of paper, and drawing addition, commuting it, and drawing that gives you a parallelogram with one corner at the origin and the new vector being the diagonal from the origin. Arithmetically, you represent the vector as a point in terms of x, y, and maybe z, and just add corresponding components (X + x, Y + y, Z + z) and leave them in the same spots.
But when you get to multiplication, vectors get weirder. First off, you have to decide exactly what it means to multiply two vectors (multiplying a vector by a scalar is easy; just multiply the length). Mathematicians defined two major ways to do this, which give different results; which one you use depends on what you're trying to do. The dot product (named after its notation), also called the inner product, is a single number biggest for two vectors pointed in the same direction, zero if the two vectors are perpendicular, and the largest negative when the two vectors are pointed exactly opposite. It commutes. Geometrically, you take the lengths of the two vectors, multiply them together, and then multiply by the cosine of the angle between them (1 when they point the same way, 0 perpendicular, -1 when they point opposite ways). Algebraically, you multiply and add xX + yY + zZ.
There's another kind, though, the cross product (not called an outer product). This produces another vector perpendicular to both previous vectors (in a 3-dimensional space); it's largest for two perpendicular vectors, and zero if they're parallel or antiparallel. Geometrically, its length is equal to the lengths of the two vectors, times the absolute value of the sine of the angle between them. Algebraically, (x, y, z) * (X, Y, Z) = (yZ - Yz, Xz - xZ, xY - Xy). An interesting thing happens when you commute the two vectors here: you end up with a vector of the same length, pointing in the opposite direction.
Matrices are even weirder with their multiplication. A matrix is a two-dimensional list of numbers of a set number of rows and columns. For instance:
a b c
d e f
is a 2x3 matrix.
To multiply two matrices, the number of columns in the first matrix has to be equal to the number of rows in the second matrix. Matrix multiplication is not only noncommutative, but if you take a defined multiplication and try to commute it you might actually get an undefined result, and, unless your matrices are all square, you'll get a matrix of different dimensions. To multiply an m x n matrix by an n x o matrix, create a new m x o matrix. Pick an entry in it. That entry is the dot product of its corresponding row in the first matrix with its corresponding column in the second, as though they were vectors.
As an aside, if you were wondering what a outer product is, an outer product of two vectors (x, y, z) and (X, Y, Z) is matrix that looks like:
xX xY xZ
yX yY yZ
zX zY zZ
So, yeah, the commutative property is defined to hold for all addition, but not for multiplication.
If I remember right, multiplication is always associative (it is for all the above), though (x*y)*z = x*(y*z) for all x, y, and z, even if their vectors, or matrices, or functions, provided it's defined every step of the way (for vectors, using * for dot, x for cross, and nothing for scalar multiplication, for instance, (a*b)c is defined; a*bc is not; (axb)*c is defined; ax(b*c) is not. But if you're not switching kinds of multiplication, here, using only cross products, you'll be fine to associate them), and is always distributive over addition, by definition: a*(b+c) = a*b + a*c, no matter what you're working in.
e: Okay, for nonabelian groups, addition isn't always commutative either. I don't have as much math as some other people here; I'm just going off the Linear Algebra class I took last spring.
Actually, a lot of these basic properties come down to definitions, and they don't have to be defined that way. I think commutativity is part of the general definition of addition on all sets (that is, if an operation does not have a commutative property, then it's not addition, no matter what you're doing it to), but it's not for, for instance, multiplication.
Let's take Vectors. A vector is essentially a point in space, or a quantity and a direction. You can think of it as an arrow that you're allowed to move anywhere, but not rotate; it points in a specific direction and has a specified length. If you have a defined origin (zero point), then any point in space can be converted into a vector by just drawing that arrow from the origin to the point.
To add vectors, there are a few mathematically equivalent ways. Geometrically, you just put the tail of one arrow at the origin and the tail of the other on the head of the first, and draw a new vector from the origin to the head of the second. You can draw these on a piece of paper, and drawing addition, commuting it, and drawing that gives you a parallelogram with one corner at the origin and the new vector being the diagonal from the origin. Arithmetically, you represent the vector as a point in terms of x, y, and maybe z, and just add corresponding components (X + x, Y + y, Z + z) and leave them in the same spots.
But when you get to multiplication, vectors get weirder. First off, you have to decide exactly what it means to multiply two vectors (multiplying a vector by a scalar is easy; just multiply the length). Mathematicians defined two major ways to do this, which give different results; which one you use depends on what you're trying to do. The dot product (named after its notation), also called the inner product, is a single number biggest for two vectors pointed in the same direction, zero if the two vectors are perpendicular, and the largest negative when the two vectors are pointed exactly opposite. It commutes. Geometrically, you take the lengths of the two vectors, multiply them together, and then multiply by the cosine of the angle between them (1 when they point the same way, 0 perpendicular, -1 when they point opposite ways). Algebraically, you multiply and add xX + yY + zZ.
There's another kind, though, the cross product (not called an outer product). This produces another vector perpendicular to both previous vectors (in a 3-dimensional space); it's largest for two perpendicular vectors, and zero if they're parallel or antiparallel. Geometrically, its length is equal to the lengths of the two vectors, times the absolute value of the sine of the angle between them. Algebraically, (x, y, z) * (X, Y, Z) = (yZ - Yz, Xz - xZ, xY - Xy). An interesting thing happens when you commute the two vectors here: you end up with a vector of the same length, pointing in the opposite direction.
Matrices are even weirder with their multiplication. A matrix is a two-dimensional list of numbers of a set number of rows and columns. For instance:
a b c
d e f
is a 2x3 matrix.
To multiply two matrices, the number of columns in the first matrix has to be equal to the number of rows in the second matrix. Matrix multiplication is not only noncommutative, but if you take a defined multiplication and try to commute it you might actually get an undefined result, and, unless your matrices are all square, you'll get a matrix of different dimensions. To multiply an m x n matrix by an n x o matrix, create a new m x o matrix. Pick an entry in it. That entry is the dot product of its corresponding row in the first matrix with its corresponding column in the second, as though they were vectors.
As an aside, if you were wondering what a outer product is, an outer product of two vectors (x, y, z) and (X, Y, Z) is matrix that looks like:
xX xY xZ
yX yY yZ
zX zY zZ
So, yeah, the commutative property is defined to hold for all addition, but not for multiplication.
If I remember right, multiplication is always associative (it is for all the above), though (x*y)*z = x*(y*z) for all x, y, and z, even if their vectors, or matrices, or functions, provided it's defined every step of the way (for vectors, using * for dot, x for cross, and nothing for scalar multiplication, for instance, (a*b)c is defined; a*bc is not; (axb)*c is defined; ax(b*c) is not. But if you're not switching kinds of multiplication, here, using only cross products, you'll be fine to associate them), and is always distributive over addition, by definition: a*(b+c) = a*b + a*c, no matter what you're working in.
e: Okay, for nonabelian groups, addition isn't always commutative either. I don't have as much math as some other people here; I'm just going off the Linear Algebra class I took last spring.
Last edited by IGTN on Tue Jul 07, 2009 4:53 am, edited 1 time in total.
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