Wednesday, May 31, 2006
Monday, May 29, 2006
Sunday, May 28, 2006
Saturday, May 27, 2006
Friday, May 26, 2006
The job market around here is the pits unless I want to do what is euphemistically referred to as "light assembly." I have nothing against doing this, but my royally screwed up back does. I don't think that getting a job that would leave me pretty much disabled within 6 months is a good career choice. And I passed on a part-time job through the temp agency. It seemed wiser not to lock myself into something with no defined end date, but to wait for something better. I think I started second-guessing and regretting that as soon as I got off the phone. I did manage to get my resume in on a couple of jobs, but I'm not holding my breath. (Well, I am holding my breath. Just not in expectation of getting an interview for either job.)
I have been reading a lot this week. And I've set-up a separate blog for my honor's research stuff - not that there's anything to see there yet. Zephyr came over with The Other Sister Person yesterday. TOSP brought lunch. Zephyr cut my hair.
No links this week since the time I've spent on-line has been mostly job searches and psychology stuff.
That's all, folks.
Tuesday, May 23, 2006
I'm back to reading Analysis With an Introduction to Proof, 4th Ed. by Steven R. Lay. This is the book with the proof of the proof by the method of induction that I mentioned the other day. So I've made it through chapter 3, section 11: Ordered Fields relatively quickly largely because this is pretty much all review.
The interesting thing about ordered fields is that all of you are familiar from high school algebra with the things that determine whether something is a field and whether the field is ordered. It's just that nobody bothered to tell you this. An oversight I'm about to correct.
"What," you ask, "are these ordinary things that define fields?" Axioms. The very same axioms you learned in high school. And, yes, I'm going to tell you what they are, but first you need to learn a couple of mathy symbols. ℝ - The set of real numbers, and -the symbol meaning "is a member of" so x ℝ reads "x is a member of the set of real numbers. That's all you need to know.
On to the axioms.
There are 5 axioms for addition.
There are also five axioms for multiplication that will look pretty much like the addition axioms.
Now for the last 5 axioms. You should recognize the first of these since it is the distributive law. This is pretty cool because it shows how addition and multiplication are related to each other.
Then you have the 4 axioms that define order.
Actually, any set of numbers or mathematical system where the first 11 axioms are true is a field. If all 15 axioms are true, you have an ordered field. The set of rational numbers makes up an ordered field, but the set of integers does not (do you see why?).
Reference: Lay, S.R. (2005). Analysis With an Introduction to Proof, 4th Ed. Pearson-Prentice Hall: Upper Saddle River, NJ. pp 108 - 112.Math
Saturday, May 20, 2006
Technorati Tags : blogging stuff
Friday, May 19, 2006
I like to check stat counter to see where people have come from on their way to my blog. When I did that tonight, I found my way to an interesting blog - A Sweet, Familiar Dissonance. And I got as close to a rave review as I'm ever going to get. *grin*
Two links for tonight.
The first is about the "discovery" of a possibly habitable planetary system at MSNBC. It's really cool how planets are found - lots of math.
The second (which is really three) is at Science and Politics - Teaching the bare bones of Biology and Teaching Update. I wish I'd had this guy for a biology teacher. He also has a really interesting post about Lyme Disease here.
If any of you think that I've run out of things to say about order and mathematics, think again. There will be more order coming to you. *hehehe*
Wednesday, May 17, 2006
Tuesday, May 16, 2006
Sunday, May 14, 2006
Blog mom talks about the rain by her here and Yahoo has the AP story here. We're lucky this time around with only a flood watch, but no warning for the county I live in. The New Hampshire TV station spent an hour tonight on flood news. It was nice to hear towns with evacuations telling residents to bring their pets because arrangements had been made for them. Frustrating to hear that people were out "sight seeing" when the they're being told to stay home because of the danger with roads washed out.
I know it's selfish, but I'm glad that the probability of repeating last fall's washout is relatively slim.
Friday, May 12, 2006
That said, some linky stuff to annoy, amuse, or inform you.
From Mixing Memory: an interesting post on a study of about how Craving a Cigarette Warps Your Sense of Time. Having sat through all the speeches at a college graduation (which by themselves warp time), I can confidently say that craving a cigarette when you have to pee and can't do either really warps time. I swear it took at least 5 or 10 minutes for the speaker to utter a one syllable word. You can imagine how long the bigger words took. *grin*
From Science and Politics: a link to a letter to the editor about faith vs. ethics. Love this part of the letter - "The irrational mind knows no boundaries and has no tolerance for self-restraint. It is the rational mind that can set boundaries and control impulses."
And a new blog to check out - Heo Cwaeth. You'll have to read it to find out what the blog is all about. I found it via Mixing Memory.
I could rant for months on this story on Yahoo - No States Meet Teacher Quality Goal.
And check out blog daughter Zephyr's post about being a student at a beauty academy.
Misc and life woes
Thursday, May 11, 2006
The Well-Ordering Property: Every non-empty set of positive integers has a least element.
That's it. Notice that it says nothing about the numbers being in order (ie., 1, 2, 3, 4). Only that a set with at least one element (non-empty) will have a least or smallest element.
Before you ask, this property addresses only positive integers because the set of negative integers has no least element. Rememeber that the negative integers go -1, -2, -3, and that -1 is greater than -2 which is greater than -3 and so forth. So while the set of positive integers is well-ordered, the set of negative integers is not.
Tuesday, May 09, 2006
Psych of Learning - A
History & Systems - A
Honors Research - pass (pass/fail only)
Honors Research Seminar - pass (pass/fail only)
GPA is now at
(I was trying so hard to be "smarter" than I am, but I have no idea where the 5.67 came from. This time I copied/pasted so it's the correct number. *grin*)
I am just so good. ROFLMAO
1. Why does adding two even numbers give an even number?
2. Why does adding two odd numbers give an even number?
3. Why does adding an odd and an even number give an odd number?
4. Why aren’t there more even than odd numbers?
I’ll answer each one for you although the last is going to take the most explaining. *grin* So, here goes.
To start with, the answers to these questions are referring only to integers (…, -3, -2, -1, 0, 1, 2, 3, …).
There are several things to keep in mind.
When you add two integers, the answer is always another integer (this is a property of the set of integers called closure).
The associative property – for any integers m, n, p; m + n + p = (m + n) + p = m + (n + p).
The distributive property – for any integers a, b, c; ab + ac = a(b + c).
Now onto the first three questions.
1. Why does adding two even numbers give an even number?
Let’s start with a definition. We are defining an even number as a number that when divided by 2 has no remainder. We can write any even number as 2k where k is any integer. It doesn’t matter if it’s even or odd since multiplying by 2 immediately makes the result (2k) even.
Now add two even numbers. By the definition above, we can choose any two even numbers. Let’s call them 2k and 2m where k and m are integers.
Do the arithmetic: 2k + 2m = 2(k +m) by the distributive property. (k + m) is an integer because adding two integers always results in another integer. Since 2(k + m) is divisible by 2 with no remainder (see our original definition), adding 2k + 2m results in an even number.
2. Why does adding two odd numbers give an even number?
First we’ll define an odd number as a number that when divided by 2 results in a remainder of 1. We know that for any n = 2k, n is even from above. What we want is an n that is odd. By our definition of an odd number, n = 2k + 1 is odd because when we divide n by 2 we end up with a remainder of 1.
Now add two odd numbers. As above, we can choose any two odd numbers. Let’s choose n = 2k + 1 and m = 2r + 1 where k and r are integers.
Next the arithmetic: n + m = (2k + 1) + (2r + 1) = 2k + 2r + 2 (combining like terms).
2k + 2r + 2 = 2(k + r + 1) by the distributive property. Since k, r, and 1 are integers, (k + r + 1) is also an integer. So we now have an even number by our previous definition of an even number.
3. Why does adding an odd and an even number give an odd number?
Choose any odd number m = 2k + 1 (by our previous definition of an odd number) and any even number n = 2p (by our previous definition of an even number) where m, k, n, and p are integers. Now do the addition.
m + n = (2k + 1) + (2p) = 2k + 2p + 1 = (2k + 2p) + 1 (by the associative property) and
(2k + 2p) + 1 = 2(k + p) + 1 (by the distributive property). Since k and p are integers, k + p is an integer, and so we have by definition an odd number.
Now for the last question – why aren’t there more even than odd numbers?
I need to start by talking about infinite sets and how we tell their “size” or cardinality in math speak. The set of natural numbers (counting numbers – 1, 2, 3, …) is an infinite set. It’s a countably infinite set because (at least in theory) you could count all the members of the set. Of course, since there is no “largest” natural number, you’d never actually count to the end. Georg Cantor, when he described infinite sets, said that the set of natural numbers has a cardinality (size) of aleph-0. All countably infinite sets have this cardinality of aleph-0. So all countably infinite sets are the same size. Got that?
In order to show that a set is countably infinite, I have to find a way to match each member of the set to each member of the set of natural numbers. This is a one-to-one correspondence between the two sets. The good thing is that I can rearrange the numbers in the set I’m trying to put into one-to-one correspondence with the set of natural numbers – order, in the every day sense, doesn’t matter in this case.
Let’s start with the even numbers. The set of even numbers (I’m using integers) is (…, -6, -4, -2, 0, 2, 4, 6, …). This is a problem because the even numbers go on forever in the negative and in the positive direction. But I can reorder the set and end up with (0, -2, 2, -4, 4, -6, 6,...).
With this reordering, I can make the match to the set of natural numbers like this (my one-to-one correspondence) by lining up the natural numbers under the even numbers.
|0||-2||2||-4||4||-6||6||and so forth (the even numbers)|
|1||2||3||4||5||6||7||and so forth (the natural numbers)|
Since there is a one-to-one correspondence between the set of even numbers and the set of natural numbers, the set of even numbers is countably infinite and has the same cardinality (size) as the natural numbers – aleph-0.
The odd numbers work the same way.
Reorder the set of odd numbers (…, -5, -3, -1, 0, 1, 3, 5, …) to (0, -1, 1, -3, 3, -5, 5, …).
Now you can make a one-to-one correspondence to the natural numbers in the same way as for the even numbers. With this one-to-one correspondence, the set of odd numbers is countably infinite and so has a cardinality (size) of aleph-0.
Since the set of odd numbers and the set of even numbers both have a cardinality of aleph-0, they are the same size.
BTW – the set of rational numbers (your fractions – ½, ¼) is also countably infinite which makes it the same size as the set of integers, the set of even numbers, and the set of odd numbers. The set of real numbers (includes the integers, the rationals, and the irrational numbers) though, is uncountably infinite and has a different cardinality. Proofs are here. I think the proof for the rational numbers is pretty slick.
Sunday, May 07, 2006
When I returned to college to complete my math degree, I had tons of math, science, and programming courses, but virtually none of the dreaded general education courses. As a result I took way more gen ed courses than math courses and met a lot of freshmen students who weren't math majors. Most of them were and are totally not memorable, but two became something very like little sisters to me.
My first semester I met Chick1. We were assigned to the same team for a major presentation. Chick1 was terrified of public speaking to the point that she'd pass out. Not good when a large chunk of your grade is dependent on one presentation. So the erstwhile housemate and I practiced with Chick1 until she knew her part of the presentation so well that nothing could shake her. And I kept on giving pep-talks, lots and lots of pep-talks. We aced the presentation and somehow Chick1 became the person I most wanted to see succeed in college. She's funny, smart, energetic, caring, and so much more. She was also having a tough time with her school work. The story of how she dealt with that is hers, but by the end of her second year she was on the dean's list every semester and graduated with membership in two honors societies. At today's graduation, Chick1 told me that she'd gotten accepted to grad school for her masters. I am so proud of her.
I met Chick2 my second semester. We might never have gone past the surface conversations between two people who are in the same class except that I needed to interview someone whose work was unusual in some way for a cultural anthropology class. I have to side track a bit here and tell you that I run screaming from anyone who tries to preach to me or shove their religion down my throat. As a result, I tend to avoid people who appear to be openly religious. (And I am NOT discussing this with anyone here.) In one of our earlier conversations, Chick2 had told me that she wanted to be a missionary. There were a few more conversations about this where I learned that she spent her school breaks doing missionary work. Okay, this pretty well covers unusual work in my book. Chick2 was nice enough to agree to be the subject of the paper. That interview lead to a friendship which I value deeply. Chick2 is one of the most giving people I know. She is also one of the few people I know who lives her religion rather than preaches it. The world needs more good people. I think Chick2 is one of these good people (even if she doesn't entirely approve of my often raunchy humor).
So the chicks left the nest today. Next year as I finish up the last couple of classes for my psych degree will feel very strange without these two young women on campus.
student and life woes
Saturday, May 06, 2006
A matrix, as opposed to the matrix of movie fame, is an array of numbers. You will often see a matrix referred to as an (m x n) matrix where m and n denote the number of rows and columns, respectively. Matrix multiplication is not commutative.
Here's your math lesson for the day.
In order to multiply 2 matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. If you have an (m x n) matrix multiplying an (r x s) matrix, you must have n = r. The funny thing is that if you switch the order so that you're multiplying the same (r x s) matrix by the same (m x n) matrix, then you need s = m.
Here's an example: if matrix A is a (3 x 3) matrix and matrix B is a (3 x 1) matrix, you can multiply A by B – a (3 x 3) multiplying a (3 x 1), where n and r both equal 3. But you can’t commute the two matrices. Multiplying B by A is undefined since you have a (3 x 1) multiplying a (3 x 3) and n = 1 while r = 3.
But what if you have two matrices with the same number of rows and columns? Shouldn't they be commutative? First, you need to know how matrix multiplication is performed. The simple (text book) definition looks like this:
For a matrix A = (aij) and a matrix B = (bij), the product AB is defined as
In math speak that makes very little sense. It looks something like this.
Nasty! Let’s try an example with numbers. I’m going to use (2 x 2) matrices A and B where
When you multiply AB, you get
And when you multiply BA, you get
So, for this example, you can see that AB is not equal to BA. There are occasional instances where AB will equal BA, but this is not generally true.
Just to briefly return to the Fundamental Theorem of Arithmetic where this started. The theorem orders the prime factors. This is important within the proof - order matters - but I think we’ll get to that tomorrow.
Reference: Johnson, L.W., Riess, R.D. & Arnold, J.T. (2002). Introduction to Linear Algebra (5th ed.). Boston, MA: Addison-Wesley.
Thursday, May 04, 2006
I'll know next week what my grades are. The honors project is pass/fail and that has already been posted - pass. 4 credits. Nothing that counts towards my GPA. What's really sad is that I'll be working on the blasted thing this summer without earning credit, but what the heck I don't have to pay for credits either.
Tomorrow I'll get back to all the fun stuff (like math) that got dropped while in the end of semester crunch. BTW - that fit learning theory into 4 page paper was an A. hehehe.
Monday, May 01, 2006
Like the terminator, I will be back. LOL