- Welcome!
- I'm not using Canvas, so you'll have to make sure that you note (and bookmark) the course webpage:
http://www.nku.edu/~longa/classes/mat115r
- Some thoughts for the day, and for the course:
- For you to be successful, your attitude is as important as your ability.
- What you do shows what you are.
- Wherever you are, be all there. Jim Elliot
- Most days there will be a "Question of the day": today's is
What do you get when you put math and art together? - 3x5 cards: Please fill out the card, indicating
- Name
- Hometown
- What is your calling? ("a calling has to do with one's larger purpose, personhood, deepest values, and the gift one wishes to give the world.... A calling is about the use one makes of a career." David Orr, Earth in Mind)
- What is your dream job?
- What is something special about you?
- I am hearing-impaired (wear hearing aids): teaching has become
even trickier, given the masking. But if I don't understand
your question, please be patient with me; or, if I
misunderstand what you've said, just let me know that I'm off
track!
You are encouraged to help me out if I misunderstand what your classmates say.
- The syllabus.
- Class will be organized around discussions of material,
and activities. I hope that you will contribute! Chime in!
- I find that students are best at explaining ideas to
other students. So what I shoot for is some students
to get the idea in class, and then for them to
"infect" the rest of the students.
No Covid, please, but infections of ideas are encouraged!
- This to not going to be a "practical math" course: you
will not learn how to balance a checkbook.
This is a course incorporating material for a broad range of liberal arts disciplines. Some of them will be interesting to you, some of them may not. But this is not another algebra course. You may never have seen any mathematics quite like the things that we're going to study in this course!
- Class will be organized around discussions of material,
and activities. I hope that you will contribute! Chime in!
- There is no formal text; but you will be expected to read a
variety of on-line materials.
- The schedule and assignments.
- Because of scenes like this:
I ask that you keep the phones out of sight. If, for some reason, you need to use your phone, please leave the classroom.
It's really distracting for me to see someone playing on their phone during class. And if I get too distracted, I absent-mindedly begin making your quizzes and tests a lot harder....:)
- What is this "Logo"? Here are a couple that former students have done:
- Our goal is more than just mathematics -- there are "Lessons for life"
that we should focus on:
- Keep an open mind.
- Just do it. Jump in. Make mistakes and fail, but never give up.
- Often when we've done it once, we do it again. Follow up one good deed with another!
- Understand simple things deeply.
- Break a difficult problem into easier ones.
- Look for patterns and similarities.
- Explore the consequences of new ideas. Generalize.
- Examine issues from several points of view.
Don't be a turkey -- be a dog! (this one may require a little explanation....)
- To answer that, let's begin with a little field trip...
- Upon our return we'll take up an activity related to one aspect of
the "textbook": prime numbers. In particular, we'll learn about
the first (of three) important number decomposition, and I will
introduce you to mathematical "trees".
- First off, let's examine a Map of Mathemalchemy; it's the sort of thing that you can find at the Mathemalchemy website.
- Those chipmonks, what were they up to?
We're going to start today harkening back to the primes, which you no doubt encountered somewhere along your mathematical journey through school.
Definitions:
- The counting numbers are the positive integers: 1, 2, 3, .... These are also known as the natural numbers.
- A counting number is prime if and only if it has two distinct divisors among the counting numbers: numbers 1 and itself.
- How are primes identified?
Historically, the "Sieve of Eratosthenes" is the tool that was used (and that we'll use today).
You might guess that Eratosthenes is a Greek mathematician, and you'd be right (actually born in Libya): but he was quite the scientist, too, and gave one of the first careful measurements of the Earth's diameter (even back around 200 BCE folks knew that the Earth was a ball...).
- Now what are those primes good for? We break numbers down using
primes as the building blocks, using trees to "expose" the
unique "prime factorization" of composite numbers.
Theorem (prime decomposition): every natural number (other than 1) is either prime, or can be written as a product of primes in a unique way, from smallest to largest.
Let's build some trees for a few numbers, and talk about how we can associate a unique tree with a unique counting number (this is the key idea between one-to-oneness -- a perfect dance, with everyone matched up with just one partner).So 6 has a unique factorization (2*3 -- ordered from smallest to largest), and a unique binary tree (created by factoring the number -- in this case, 6 -- by primes, from smallest to largest). The root of the tree is the number itself, and the leaves of the tree dangle at the bottom of the prime factorization:
In the end, there's this notion: there is a one-to-one correspondence between counting numbers and their prime factorizations (with primes as their own partners), and their labelled trees (and the tree seems to summarize -- or contain -- the other two!):
6 $\iff$ 2*3 $\iff$ But the tree also contains an algorithm for finding the prime factorization, based on checking to see if smaller primes are factors of the given natural number. In a moment, we'll try another one; but first let's see what this sieve of Eratosthenes is all about....
- Let's use the Sieve of
Eratosthenes to generate all primes less than 100. While you do it,
look for
- The only two adjacent primes,
- The triplet primes,
- The twin primes,
- "Spirals", and
- The unprimiest looking primes!:)
