Day 5, MAT115

Last Time

Next Time

Announcements:
- Here's a video covering the material today.
- Your First homework was turned in on Monday (or supposed to be):
  - Homework Key
  - First Homework Video Key (I talk you through as I do them)
  Sorry I haven't gotten it graded yet.
  But inspection of some of your homeworks has shown me that a bunch of you don't seem to be following along with our website. Every Monday and Thursday I post new material, at
  
  http://www.nku.edu/~longa/classes/mat115/
  which is our class home page.
  I know some of you must have missed this, but Canvas is just a tool for me.
  - Number 1, I don't like to let someone else manage (maybe even own) my web stuff. And
  - Number 2, my materials are free to the whole world. Anyone can come here and see what we're doing. Canvas excludes people. I don't want to exclude anyone who wants to come and learn a little mathematics.
- Last time we learned about the Babylonians (or at least how they counted).
  We played Sherlock Holmes in order to solve the mystery of the clay tablet, and we were able to deduce that
  - the Babylonians used place value, like we do but with a "base" of 60 instead of 10 (maybe Babylonians had 60 fingers!). That means they use powers of 60 to represent numbers, rather than powers of 10 like we do.
  - The tablet was a times table for multiplying by 9.
  - the Babylonians were missing the number 0, which opened their number representation to possible confusion.
  - The Babylonians apparently read things the way we do, from top to bottom and left to right.
  - 60 is with us today in our clocks (60 minutes), and 360 degrees: we're all still a little Babylonian at heart.
  - And we saw how the Babylonians would write these numbers:
    1. $47 = (47)*60^0$
    2. $573 = (9)*60^1+(33)*60^0$
    3. $3600=(1)*60^2$
    4. $14001=(3)*60^2+(53)*60^1+(21)*60^0$
    5. $0=?$ (they can't do it!)
Today's new topic: well, there are two, actually.
1. Your reading: Early Concepts of Number and Counting
2. Next we focus on the old world, with the sieve of Eratosthenes.
Okay, let's have at it!
1. I hope that you enjoyed that reading as much as I do. I love history, and the history of how we counted is pretty fascinating. How about that body counting of Papua New Guinea? They counted to 41 with all parts of their body. (Who knew that a nose would be 11, and your left hip 26?)
  Or those South Africans using people as place values, or "registers" -- with two people you could get up to 99, and then, if you'll throw in a third person, you can cross over to 100, and then get all the way to 999! Keep going... add more people and you can get to 9999, then 99999, etc.
  But the key in this reading was the idea of that one-to-one correspondence: "More simply, each element in the first set corresponds to exactly one element in the second set and vice versa."
  There's an idea in there that's really important: "exactly one". This is called uniqueness -- you can't get confused. If I'm holding up five fingers, you don't get confused and say "4".
  One idea in this article which will come back to haunt us is knots. The author talks about societies where knotted strings are used to represent numbers (p. 6). We're going to be talking about knots a bunch more later on in the course. (Who knew that knots were mathematics?)
2. In "Rock groups", our author says that "some numbers, like 2, 3, 5, and 7, truly are hopeless. None of them can form any sort of rectangle at all, other than a simple line of rocks (a single row). These strangely inflexible numbers are the famous prime numbers."
  Prime numbers are not divisible by any other natural numbers other than 1 and themselves. So they can only form single file rock groups.
  - The prime numbers are an infinite subset of the natural numbers (What are natural numbers again? The counting numbers: 1, 2, 3, ....).
  - What is a prime number?
    - A natural number that can be divided evenly (that is, without remainder) by only two distinct natural numbers: 1 and itself.
    - NOTE: when mathematicians say "distinct" natural numbers, they mean that they need to be different; and so the number 1 is not prime.
    - We can find the primes using the Sieve of Eratosthenes (here's an animation)
  - Prime Factorization Theorem:
    - Every natural number greater than 1 is either prime, or it can be expressed as a product of prime numbers (in one and only one way -- a unique way -- order of the product aside).
    - Examples (and we can use trees to do these, too!):
      - 42=2*21=2*3*7
      - 8=2*2*2 (prime factors can repeat)
      - 1729 = 7*247 = 7*13*19
    - Here's a rule you can use, to try to decide quickly if a number is prime: If you can't factor by any prime less than or equal to the number's square root, then the number is prime.
      Example: 127.
      Its square root is approximately 11.27. We don't need to go past 11 to find its factors.
      - 2 doesn't divide 127
      - 3 doesn't divide 127
      - 5 doesn't divide 127
      - 7 doesn't divide 127
      - 11 doesn't divide 127
      - 127 must be prime!
    - Upshot: We can break down, "decompose", or "factor" natural numbers using prime numbers and products.
  - Of what value are prime numbers? Protecting your credit card!
    It turns out that it's very hard to factor huge numbers. These are the keys that credit card companies use to exchange information securely.
    If I multiply two huge prime numbers $p$ and $q$ together, to get \[ n=p*q \]
    Well I know the factorization -- but the evil-doers don't, and it would take them and their computers a very long time to "crack" $n$. It's that simple!
  - The distribution of primes
    - The first 100 Primes (primes come around a lot less often, the higher up you go in the natural numbers -- here are the the primes under 10000 -- is 1729 prime? Check out 9767 and its neighbor....)
  - How many primes are there? There are infinitely many -- they just don't stop! (But how do we know? Euclid -- remember him, from geometry? -- he proved that it is so.)
  - A natural number n may or may not divide natural number m evenly, but there's always a unique way of writing the attempt (that is, exactly one way -- no more):
    
    m=qn+r
    where 0 ≤ r ≤ n-1.
    Obviously, if r=0 then n divides m; otherwise n does not divide m.
  - It's not hard, given this fact, to show that there are infinitely many primes (which Euclid did by contradiction).
3. Unanswered questions about prime numbers:
  - Goldbach Question: Can every even natural number greater than 2 be written as the sum of two primes?
  - Twin Prime Question: are there infinitely many pairs of prime numbers that differ from one another by two? (3,5; 5,7; 11,13; etc. -- can you find another pair in the sieve?)
  Mathematicians don't know the answers yet!;) Another thing (besides patterns) that mathematicians love: mysteries....

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