Next Time |
http://www.nku.edu/~longa/classes/mat115
You are encouraged to help me out if I misunderstand what your classmates say.
No Covid, please, but infections of ideas are encouraged!
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!
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....:)
We're going to start today harkening back to the primes, which you no doubt encountered somewhere along your mathematical journey through school.
Definitions:
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...).
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....