A type of algebraic expression that befuddles legions of students is the following:
6x – (2x + 3)
Read more »
Tampilkan postingan dengan label algebra. Tampilkan semua postingan
Tampilkan postingan dengan label algebra. Tampilkan semua postingan
Selasa, 10 Juli 2012
Selasa, 29 Mei 2012
Wholesale whole-number murder and redemption
An extended mathematics metaphor:
Flowing under the pre-K through high school curriculum, like the ever-widening Mississippi, is a steady expansion of the number system and its corresponding basic operations.
Important milepost concepts and mathematical problems which are posed, deconstructed and solved throughout those years are like the flatboats and steamboats of Mark Twain's era which floated upon the Mississippi's waters.
One doesn't seek to control the river, because the number system and its operations exist in nature, but we can select what floats on it and where to travel. Choosing when and how to introduce concepts, what problems to pose, and where they fit is the foremost responsibility of a standards and curriculum developer.
Read more »
Flowing under the pre-K through high school curriculum, like the ever-widening Mississippi, is a steady expansion of the number system and its corresponding basic operations.
Important milepost concepts and mathematical problems which are posed, deconstructed and solved throughout those years are like the flatboats and steamboats of Mark Twain's era which floated upon the Mississippi's waters.
One doesn't seek to control the river, because the number system and its operations exist in nature, but we can select what floats on it and where to travel. Choosing when and how to introduce concepts, what problems to pose, and where they fit is the foremost responsibility of a standards and curriculum developer.
Read more »
Kamis, 10 Mei 2012
Guess My Rule
Some activities seem ageless. They work in elementary, middle and beyond. Guess my rule I have played with very young students, and this week I got to try it with my summer intermediate algebra course.
Today I started with the number times two minus three. (Highlight to see it, or guess from the table.) Inputs came from all over. 12, 5, 10, 1... some shock at a negative answer. I encourage the students to record the results, hoping that it will lead to some ideas about what inputs to suggest. Organizing data is not a natural tendency. At one point they started asking 30, 40, 50...
Oooh! That's a pattern. It goes up 20 when the input goes up 10. So they could predict the answer for multiples of ten. Thinking about that, they suggested a rule that worked. We looked at the rule to see from where the up by 20 pattern came.
The solver was not comfortable coming up with his own rule, so I chose another: eight minus the number.
\(7 \to 1 \) then \(1 \to 7\) drew an audible gasp/hmm. There was one student who loved asking for 12. The third multiple of 10 got the rule again. Subtract eight and then take the opposite of your answer. One of the reasons I love this game is that it always brings up equivalent expressions. I shared my original rule and we talked out the equivalence.
Now they were ready to suggest a few rules. A couple like my rules, then input times 5 divided by 2, which got a nice check of "oh, that is the same as the number plus half of it," and "I had the number times one and a half." Then one student came up with a stumper. When guesses were not focusing, I started taking notes on the board.
I also asked for what patterns they noticed. Some good stuff. We got to the idea of asking inputs in a pattern, and I asked them to predict the output from 2 before Edras gave us the actual. (There's some really nice research on the power of prediction in math, some by my colleague Lisa Kasmer.) We spent some time on different expressions of the rule, like ___ x ___ + \(\frac{1}{2}\) ___ x ___, (___)^2 + ( ___)^2/2 and \( 1.5 x^2\).
For a last question, I asked them to find what input would give 100, and got unexpected riches. Students really worked hard, consulting group members. Several tried to solve symbolically, but didn't know what to do with the \( x^2 \). One student had an excellent guess and check. We spent some time discussing that, including what an excellent problem solving method it is, and how when I use it, using numbers gives me a much better feel of what's going on. Maybe they've been discouraged to use it by previous teachers, but it's a great strategy. We did discuss how to make our guess more accurate and efficient. Like was 8 or 9 closer to 100 in output?
Another then shared a traditional symbolic solution. One student asked, "why did you divide by 1.5 before you took the square root?"
"Because you have to."
"It has something to do with the order of operations..."
"Doesn't PEMDAS tell you what to do?"
"But you have to reverse it maybe..."
"Let's try it." (OK that was me.) They told me the steps and I wrote it down. Amazed to find the same answer. A good moment to point out to them that are almost always multiple methods in math. Teachers may have told them that there is only one way to do things in the past, but that is bullsh*t.
I should not swear in class. (In response to my usual 'jot down one thing you want to remember' there were many repetitions of that statement. In my defense, it is a college class. And the comedic potential was ripe.)
The next activity is adapted from Pam Wells' adaptation of an activity from the excellent Mathscape curriculum.
(Here's the Word file if you want to edit - it wasn't appearing correctly in the embed.)
It dovetailed very well into the Guess My Rule, allowing us time and grist to find many connections amongst tabular, symbolic and visual representations, reiterating the equivalent expressions and multiple solutions themes, and giving us a launching point for a definition. Using the Y pattern rule, I made a table for S=1, 4, 7 and 11. The changes in output were 9, 9 and 12... what went wrong? They found that the difference in the inputs was not constant, and when we changed 10 to 11, the output pattern worked. To me, that's the moment for the generalization.
We went on to launch Linear War, which gave me a chance to share why we use the term linear and some of what this stuff has to do with lines. Next class, we play!
All in all, it was a fun class. I was impressed with their willingness to try non-traditional problems, and they're gaining rapidly in ability to work cooperatively and make conjectures. The meta-messages about mathematics seem to be gaining some traction, too.
Guess My RuleI don't know who originated this game; it feels ancient and right, so kudos to whomever developed it. I let rulekeepers use a calculator, and I usually start out with a few rules until the guesser feels comfortable making up a rule.
Any number of players
A rulekeeper makes up a rule that gives a number output for a number input. (It should be a function.) Players take turns giving an input, and the rulekeeper tells them the output. If a player wants to guess the rule, they tell an input and what the output would be. If they're right, they guess the rule. Then that player makes the next rule.
Today I started with the number times two minus three. (Highlight to see it, or guess from the table.) Inputs came from all over. 12, 5, 10, 1... some shock at a negative answer. I encourage the students to record the results, hoping that it will lead to some ideas about what inputs to suggest. Organizing data is not a natural tendency. At one point they started asking 30, 40, 50...
| inputs | 12 | 5 | 10 | 1 | 30 | 40 | 50 |
| outputs | 21 | 7 | 17 | -1 | 57 | 77 | 97 |
Oooh! That's a pattern. It goes up 20 when the input goes up 10. So they could predict the answer for multiples of ten. Thinking about that, they suggested a rule that worked. We looked at the rule to see from where the up by 20 pattern came.
The solver was not comfortable coming up with his own rule, so I chose another: eight minus the number.
| inputs | 7 | 1 | 10 | 12 | 20 | 30 | 40 |
| outputs | 1 | 7 | -2 | -4 | -12 | -22 | -32 |
I also asked for what patterns they noticed. Some good stuff. We got to the idea of asking inputs in a pattern, and I asked them to predict the output from 2 before Edras gave us the actual. (There's some really nice research on the power of prediction in math, some by my colleague Lisa Kasmer.) We spent some time on different expressions of the rule, like ___ x ___ + \(\frac{1}{2}\) ___ x ___, (___)^2 + ( ___)^2/2 and \( 1.5 x^2\).
For a last question, I asked them to find what input would give 100, and got unexpected riches. Students really worked hard, consulting group members. Several tried to solve symbolically, but didn't know what to do with the \( x^2 \). One student had an excellent guess and check. We spent some time discussing that, including what an excellent problem solving method it is, and how when I use it, using numbers gives me a much better feel of what's going on. Maybe they've been discouraged to use it by previous teachers, but it's a great strategy. We did discuss how to make our guess more accurate and efficient. Like was 8 or 9 closer to 100 in output?Another then shared a traditional symbolic solution. One student asked, "why did you divide by 1.5 before you took the square root?"
"Because you have to."
"It has something to do with the order of operations..."
"Doesn't PEMDAS tell you what to do?"
"But you have to reverse it maybe..."
"Let's try it." (OK that was me.) They told me the steps and I wrote it down. Amazed to find the same answer. A good moment to point out to them that are almost always multiple methods in math. Teachers may have told them that there is only one way to do things in the past, but that is bullsh*t.
I should not swear in class. (In response to my usual 'jot down one thing you want to remember' there were many repetitions of that statement. In my defense, it is a college class. And the comedic potential was ripe.)
The next activity is adapted from Pam Wells' adaptation of an activity from the excellent Mathscape curriculum.
(Here's the Word file if you want to edit - it wasn't appearing correctly in the embed.)
We went on to launch Linear War, which gave me a chance to share why we use the term linear and some of what this stuff has to do with lines. Next class, we play!
All in all, it was a fun class. I was impressed with their willingness to try non-traditional problems, and they're gaining rapidly in ability to work cooperatively and make conjectures. The meta-messages about mathematics seem to be gaining some traction, too.
Sabtu, 31 Maret 2012
What is a lattice? - Or lattices in M336
#openuniversity #m336
The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).
In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.
Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.
A catologue of lattices.
Junkyard article about lattices and geometry of numbers.
The mathematical universe is expanding with tremendous speed.
The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).
 |
| Click to enlarge |
In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.
Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.
A catologue of lattices.
Junkyard article about lattices and geometry of numbers.
The mathematical universe is expanding with tremendous speed.
Rabu, 21 Maret 2012
M336 - Group Theory - Fundamental Theorem of Abelian Groups
#openuniversity #m336 #video
One of the theorems that is discussed in the group theory track in the Open University Course 'M336 Groups and Geometry' is the Fundamental Theorem of Abelian Groups. Early on in Group Theory it becomes clear that there is a connection between group theory and number theory in Langrange's theorem and the Sylow Theorems ( also part of M336 ) but only after studying the Fundamental Theorem of Abelian Groups you'll get a notion of the depth of the connection between Group Theory and Number Theory.
MathDoctorBob ( his YouTube alias ) made a short video lecture on the topic. Precise as always.
One of the theorems that is discussed in the group theory track in the Open University Course 'M336 Groups and Geometry' is the Fundamental Theorem of Abelian Groups. Early on in Group Theory it becomes clear that there is a connection between group theory and number theory in Langrange's theorem and the Sylow Theorems ( also part of M336 ) but only after studying the Fundamental Theorem of Abelian Groups you'll get a notion of the depth of the connection between Group Theory and Number Theory.
MathDoctorBob ( his YouTube alias ) made a short video lecture on the topic. Precise as always.
Minggu, 11 Maret 2012
Sets and multisets
A set is a collection of well defined and distinct objects. I remember it as I have learned the Set interface in Java, a Set has no duplicates and is not sorted: 'it models the mathematical set abstraction'.
But what if we want to study collections of well defined but not necessarily distinct objects? The easy way out is to simply define another base abstraction. The beauty of mathematics is that you don't have to. The body of mathematical knowledge is built from a minimal number of base abstractions. Then how should we define a multi-set?
Definition: Let S be a nonempty set. A multi-set M with underlying set S is a set of ordered pairs: $$M=\left\{ (s_i,n_i) | s_i \in S, n_i \in \mathbb{Z}^+ \right\},$$ where $n_i$ is the multiplicity of the element $s_i$.
A multi-set defined as, or using, a set.
But what if we want to study collections of well defined but not necessarily distinct objects? The easy way out is to simply define another base abstraction. The beauty of mathematics is that you don't have to. The body of mathematical knowledge is built from a minimal number of base abstractions. Then how should we define a multi-set?
Definition: Let S be a nonempty set. A multi-set M with underlying set S is a set of ordered pairs: $$M=\left\{ (s_i,n_i) | s_i \in S, n_i \in \mathbb{Z}^+ \right\},$$ where $n_i$ is the multiplicity of the element $s_i$.
A multi-set defined as, or using, a set.
Sabtu, 03 Maret 2012
An algebraic proof of Fermat's Little Theorem
Let $G$ be an abelian group. Define a scalar multiplication over $\mathbb{Z}$ as follows: $$n \cdot g = \underbrace{g+g+\cdots+g}_{n \ \text{times}}.$$ Note that in this case $|G| \ g=0$. ( We turned $G$ into a $\mathbb{Z}$-module. )
For primes, the multiplicative group $\mathbb{Z}_p$ is abelian, $| \mathbb{Z}_p | = p-1$ and the identity element is $1$. Let $a \in \mathbb{Z}_p$ and the multiplicative notation of $|G| \ g=0$ becomes $a^{p-1} \equiv 1 \bmod{p}$. But this is just Fermat's Little Theorem!
For primes, the multiplicative group $\mathbb{Z}_p$ is abelian, $| \mathbb{Z}_p | = p-1$ and the identity element is $1$. Let $a \in \mathbb{Z}_p$ and the multiplicative notation of $|G| \ g=0$ becomes $a^{p-1} \equiv 1 \bmod{p}$. But this is just Fermat's Little Theorem!
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| Fermat |
Rabu, 22 Februari 2012
Abstract Algebra E-222 video 26 Rings 2
#maths
Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?
These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.
Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?
 |
| Prof. Gross ... "ideals in the Gaussian integers $\mathbf{Z}\left[i\right]$ of type $\mathbf{Z}/p\mathbf{Z}$". |
These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.
Senin, 20 Februari 2012
Abstract Algebra E-222 video 25 Rings 2
Every word counts in mathematics.
The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.
In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.
Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )
The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.
 |
| $x^2-1$ has $4$ roots... |
In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.
Sabtu, 18 Februari 2012
Abstract Algebra E-222 video 24 Rings 1
#M336
Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.
But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).
At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.
Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.
In the business of commercial education, in programming for example, teachers are often confronted with students ( sent by their employers ) who expect to leave as a qualified programmer just by hanging in their chairs during the course. Needless to say they leave as empty headed as they came in.
But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).
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| $(\mathbf{Z/nZ})^{\times}$ has $\phi(n)$ elements |
At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.
Rabu, 15 Februari 2012
Polynomials
I am studying some more about polynomials, the topic of symmetric polynomials for example, is an interesting one.
Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask...
To be continued.
Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask...
...which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?
To be continued.
Selasa, 10 Januari 2012
About the Open University
Last year's problems seem simple with what I am going through now. What is going on? I have to withdray money from my student account which I hold with the Open University to pay for this year's courses, the procedure is simple and fast, really. First you call the course registration line, then you simply ask the student advisor to arrange it for you. But...
... I can't get through.
Not that I haven't been trying. Last week I gave up and wrote an e-mail asking for advice on how to establish contact. They say that it can take 'up to three working days before you get a reply'. It's past three working days already. But...
... no reply.
What's going on? Does it have to do with budget cuts? Queries about the new fees? I never had a problem to get through. The Open University is not just any university. They are =huge=. On a piece on the OU site I read that the OU is the biggest university in the UK with
- 250,000 students
- 7000 tutors ( Associate Lecturers )
- 1200 academic staff, and
- 3500 support staff
It is also an -international- university, they have 3500 students in Ireland, 9000 in the EU and another 7500 outside the EU.
In the meantime I am still waiting. I'll have to figure out something because I can't wait to start studying again. When everything is OK I do the reflection post first.
I haven't dropped the Fearless Symmetry series, on the contrary: I am styding algebra again. I have posted a question about Galois Theory here. Basically a course in Galois Theory makes you understand -why- polynomials with rational coefficients and degree five or higher can't be solved by radicals. At least not in general, if the corresponding Galois group of the polynomial is soluble however then there is a solutiuon. - Which you won't find in the textbooks. And that's what I find disappointing, to say the least. Maybe this is why a study in mathematics never seems to stop. A course answers a few of your questions but you'll have more questions after the course than you had before.
If Galois Theory is a blank for you, then this article ( pdf) might fill it, a bit. ;-)
... I can't get through.
Not that I haven't been trying. Last week I gave up and wrote an e-mail asking for advice on how to establish contact. They say that it can take 'up to three working days before you get a reply'. It's past three working days already. But...
... no reply.
What's going on? Does it have to do with budget cuts? Queries about the new fees? I never had a problem to get through. The Open University is not just any university. They are =huge=. On a piece on the OU site I read that the OU is the biggest university in the UK with
- 250,000 students
- 7000 tutors ( Associate Lecturers )
- 1200 academic staff, and
- 3500 support staff
It is also an -international- university, they have 3500 students in Ireland, 9000 in the EU and another 7500 outside the EU.
In the meantime I am still waiting. I'll have to figure out something because I can't wait to start studying again. When everything is OK I do the reflection post first.
I haven't dropped the Fearless Symmetry series, on the contrary: I am styding algebra again. I have posted a question about Galois Theory here. Basically a course in Galois Theory makes you understand -why- polynomials with rational coefficients and degree five or higher can't be solved by radicals. At least not in general, if the corresponding Galois group of the polynomial is soluble however then there is a solutiuon. - Which you won't find in the textbooks. And that's what I find disappointing, to say the least. Maybe this is why a study in mathematics never seems to stop. A course answers a few of your questions but you'll have more questions after the course than you had before.
If Galois Theory is a blank for you, then this article ( pdf) might fill it, a bit. ;-)
Selasa, 03 Januari 2012
Example of a Galois Group of order 8 ( Introducing Math Doctor Bob )
Regular readers must have noticed my interest in Abstract Algebra, of which I am currently studying, in different ways, the topic of Galois Theory. If you have chosen a different route in mathematics ( computation, statistics, and so forth ) or if you are at the early undergraduate level you may have difficulty picturing what Galois Theory is -all about-. I am trying to communicate that idea by summarizing the popular introduction to the field 'Fearless Symmetry' which basically introduces Galois Theory to the general ( but educated ) public. ( Currently working on part 7 out of 23). But as they say, one picture says more than a thousand words. For those that want to get an idea, fast and easy, and *now*, I recommend the following video ( mini lecture ). Don't expect you can master the subject by watching a ten minute video but the ten minutes are well worth it. The video lecturer is 'Math Doctor Bob', who uploaded about 600 mini lectures on various mathematical topics.
See also:
- Fearless Symmetry
See also:
- Fearless Symmetry
Senin, 26 Desember 2011
Sabtu, 24 Desember 2011
Fearless Symmetry 6/23: Equations and varieties
I read the sixth chapter of Fearless Symmetry.
To be continued with 7. Quadratic reciprocity
Part 1: Algebraic Preliminaries
Chapter 6: Equations and varietiesLogic of Equality
An equation is a statement, or assertion, that one thing is identical to another. In mathematics we replace is by = and use symbols that stand for the terms.History of equations
Long before algebra as we know it, ancient peoples were working with equations.
A triangle whose three sides have lengths 3, 4, and 5 is a right triangle which is an example of a Diophantic equation because the unknowns are restricted to integers. Around the late 1500s Descartes added the connection between algebra and geometry now known as analytic geometry. Descartes, as a philosopher believed that the physical universe was governed entirely by the laws of geometry. Newton ( and Leibniz ) discovered that this wasn't true, they had to invent calculus to solve their scientific problems mathematically.Z-Equations
A rational number is any number that can be expressed as the ratio of two integers. Real numbers are rational iff it is a terminating decimal or a repeating decimal. The set of all rational numbers is usually denoted as Q. We will deal mostly with equations where all the constants are integers. Or equations "defined over the integers". A Z-Equation is an equality of polynomials with integer coefficients. One of the main problems in number theory is finding and understanding all solutions of Z-equations.Varieties
Fix the attention on a particular Z-equation. Write S(Z) for the set of all integral solutions of that equation, S(Q) for the set of all rational solutions of it, and so on. We call S an "algebraic variety". The variety S defined by a Z-equation ( or a system of Z-equations ) is the function that assigns to any number system the set of solutions S(A) of the equation or system of the equations.
For example define the Variety S as x^2 + Y^2 = 1
Then
S(Z) = {{1,0),(0,1),(-1,0),(0,-1)}.
S(Q) = {t in Q | 1-t^2 / 1+t^2, 2t/1+t^2}.
We can reformulate Fermat's Last Theorem using varieties as follows.
For any positive integer n, let V_n be the variety defined by x^n + y_n = z^n. Then if n > 2, V_n(Z) contains only solutions where one or more of the variables is 0.Systems of equations
The system
x^2+y^2=1
x>0
is valid and has solutions, but it does NOT define an algebraic variety because inequalities are not defined in C nor in any of the finite fields.
Take the system
x^2+y^2+z^2=w
w^4=1
x+y=z
then S(R) is the ellipse x^2+y^2+x y = 1/2.Finding roots of polynomials
The easiest general class of varieties to look at would be those defined by a single Z-equation in a single variable, for instance, x^3 + x - 2 = 0. The study of this type of variety is dominated by the concept of the Galois group. ( More in 8 and 13 ). If f(x) is a polynomial the roots of f(x) are the numbers c such that f(c)=0.Are There General Methods for Finding Solutions to Systems of Polynomial Equations?
On a purely number-theoretical level, leaving philosophy and logic behind, we also have the famous theorem of Abel and Ruffini: Unlike quadratic polynomials, for which we can use the quadratic formula, for polynomials f (x) of degree 5 or greater, there is no formula involving just addition, subtraction, multiplication, division, and nth roots (n = 2, 3, 4, . . .) that can solve f (x) = 0 in general.Deeper understanding is desirable
The amazing discovery of Galois is that there is more structure to S(A). As we shall see, S(A) is not just a set; it is the basis for defining a representation of a certain group, called the Galois group. We will look at another series of very interesting and very important, though not so very simple, Z-varieties: elliptic curves. These two kinds of varieties will give us some of our main examples to help us understand Galois groups and their representations.
To be continued with 7. Quadratic reciprocity
Minggu, 18 Desember 2011
Fearless Symmetry 5/23: Complex Numbers
I read the fifth chapter of Fearless Symmetry.
To be continued with 6. Equations and varieties
Part 1: Algebraic Preliminaries
Chapter 5: Complex Numbers
Well, I think it is safe to assume that readers of this blog know the complex numbers. This chapter is in fact about a subset of the Complex Numbers called the Algebraic Numbers. In FS they use $\mathbf{Q}^{Alg}$ as notation, whereas I have seen mostly the notation $A$ for the Algebraic Numbers.
Every algebraic number can be expressed as the root of of a polynomial equation with integer coefficients. So $\pi$ is not a member of $\mathbf{Q}^{Alg}$, but $\sqrt{2}$ is because $\sqrt{2}$ is a solution of $x^2 - 2 = 0$.
 |
| Visualisation of the (countable) field of algebraic numbers in the complex plane. ( From Wikipedia ) |
Jumat, 16 Desember 2011
Fearless Symmetry 4/24: Modular Arithmetic
I read the fourth chapter of Fearless Symmetry.
To be continued with 5. Complex Numbers
Part 1: Algebraic Preliminaries
Chapter 4: Modular Arithmetic
Chapter 4 is all about modular arithmetic.
Considering the goal of the book somewhere fields have to be introduced and in this chapter we find the first definition of a field.
Definition: A field is a number system where we can divide by anything nonzero.
Anything more precise would scare off the laymen casual reader for who the book is intended. I kind of like the definition myself. '... where you can divide anything by nonzero'.
Modular arithmetic is introduced as clock arithmetic of course with examples like: "Today is Tue. What day is it in 25 days?" or "The analog clock shows 8. What time will it show in 33 hours?"
Also, the extremely important concept of an equivalence relation is defined. There is much more about modular arithmetic in the book, of course.
To be continued with 5. Complex Numbers
Selasa, 13 Desember 2011
Fearless Symmetry 3/23: Permutations
I read the third chapter of Fearless Symmetry.
To be continued with 4. Modular Arithmetic
Part 1: Algebraic Preliminaries
Chapter 3: Permutations
In chapter 3 the concept of a permutation is explained and how they form groups.
Definition: A permutation is a one-to-one map from a set to itself.
Example 1:
Given the set {1,2} the possible one-to-one maps ( permutations ) are:
1->1, 2->2 and
1->2, 2->1.
Example 2:
For sets of three elements there are 6 = 3! possible permutations.
If we put all the permutations of a set in a set by itself and add the composition of permutations as the operation then this set becomes a group. Permutation groups are among the most important objects in group theory because every finite group is a subgroup of some permutation group.
There are two possible ways of notation when it comes to permutations. Let's consider the set {1,2,3,4} which has 24 possible permutations.
The permutation 1->1, 2->2, 3->4 and 4->3 can be written as [1 2 4 3] and also as (1)(2)(3 4) or short (3 4) this is the so called cycle notation. Thus (1 2 3) and [2 3 1] represent the same permutation. Clearly the cycle notation is more efficient, especially when considering permutations of large sets.
The composition of permutations means permuting one after the other. Unfortunately in some books it is done from left to right, in others from right to left.
Exercise 1:
Show that (ab)(cde)*(ae)(bc)(d)=(ac)(bde).
( I would solve it as follows: )Right hand side:
A B C D E
_ D _ E B Apply (bde)
C D A E B Apply (ac)
Left hand side:
A B C D E
_ _ _ D _ Apply (d)
_ C B D _ Apply (bc)
E C B D A Apply (ae)
E C B D A
C D _ E _ Apply (cde)
C D A E B Apply (ab)
And it shows that LHS = RHS
Sets of permution form a group under composition because:
- composition leads to a new permutation ( closure )
- the neutral element is the do-nothing permutation, i.e. (1)(2)(3).
- every permutation has an inverse because it can be permuted back to the original positions.
- composition of permutations is associative.
Note that the composition of permutations is NOT ( always ) commutative.
To be continued with 4. Modular Arithmetic
Minggu, 11 Desember 2011
Fearless Symmetry 2/23: Groups
I read the second chapter of Fearless Symmetry.
For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.
To be continued with 3. Permutations
Part 1: Algebraic Preliminaries
Chapter 2: Groups
Definition: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
1. For any three elements x,y,z in G: x*(y*z) = x*(y*z)
2. G contains an element e such that for all x in G: x*e = e*x = x.
3. For any element x in G, there is an element y in G such that x*y=e.
Example 1:
The group of rotations of the sphere in R3: SO(3) or the Special Orthogonal Group in 3 dimensions. -The set G is the collection of all rotational symmetries of the sphere, i.e. if we rotate the sphere by any angle, the sphere doesn't noticeably change. The group property basically means that if we rotate the sphere over any angle A, after this over an angle B, it is the same if we would have rotated it in one go, but over some different angle. Also any rotation has an inverse: rotating it over the opposite angle. This makes the rotations a group. SO(3) is in fact a Lie group because these rotations can be done arbitrary small which is not the case when considering the symmetry group of for example a cube. Lie groups capture the concept of "continuous symmetries".
For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.
To be continued with 3. Permutations
Sabtu, 10 Desember 2011
Fearless Symmetry 1/23: Representations
I read the first chapter of Fearless Symmetry. See: Reading ( conflicts time management ) for the history on this topic.
Part 1: Algebraic PreliminariesTo be continued with 2. Groups.
Chapter 1: Representations
The goal of the book is 'Mod p linear representations of Galois groups' and how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefficients.
It is very important to know that a mathematical definition can redefine a commonly word used elsewhere. ( A simple group is not 'simple' but complex. A tree is a graph, a 'tree' in the forest is not. ) Sometimes an object is defined by listing its properties and following that a proof is given of the existence of such an object.
Definition: A set is a collection of things which are the elements of the set.
Definition: A function f: A-> B from a set A to a set B is a rule that assigns to each element in A an element of B.
Definition: A morphism is a function from A to B that "captures at least part of the essential nature" of the set A in its image in B. ( Clearly "captures at least part of the essential nature" needs to be refined later. )
Definition: A representation is a morphism from a source object to a standard target object.
Example 1: Take A,B and the fact that B represents A. A may be a citizen, B her state rep and X the legal fact that B represents A by voting in the legislature on her behalf. A may be a(n abstract) group, B a group of matrices, and X a morphism from A to B. The ultimate in abstraction is representing A,B as dots and X as an arrow from A to B.
Example 2: In the context of counting, given any two finite sets A and B, a morphism is a one-to-one correspondence from A to B. A representation in this case is a morphism from a given finite set to one of the sets {1}, {1,2}, {1,2,3} and so on. So a flock of three sheep has the set {1,2,3} as its target.
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