Day 17, MAT115

• Your multiplication homework is returned.
• Your division homework is returned (Kaitlyn).
• I hope that you had a good and safe spring break!
• We do have a test Thursday of next week....

• The prime numbers are an infinite subset of the natural numbers (What are natural numbers?).

• 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 -- order of the product aside).
  • Examples:
    • 42=2*21=2*3*7
    • 8=2*2*2 (prime factors can repeat)
    • 1729 = 7*247 = 7*13*19
    • If you can't factor by any prime less than or equal to the number's square root, then the number is prime.
    • 127: its square root is approximately 11.27. We don't need to go past 11 to find its factors.
  • Upshot: We can "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.

• 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? We can prove this theorem!)
  • 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 that, to show that there are infinitely many primes (by contradiction).

• 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!;)

Hermann Weyl (German Mathematician; 1885 - 1955)

• Where have we used symmetry already?
  • Yanghui's triangle had an error in it....
  • There's a certain symmetry between Egyptian multiplication and division.
  • There's a type of symmetry in the golden rectangle: within a golden rectangle is found a perfect scale copy of itself.

  Where else does symmetry appear, and how do we use it?

  Are humans symmetric? Turns out that we're purported to have a "good side" and an "evil side"!

• Symmetry can be used in two different ways:
  • to simplify, and
  • to complexify!
  We can use symmetry to solve some problems, and we can use symmetry to make things more interesting. Let's start with the latter:
  • Contradancing
  • Haeckel and the radiolaria: On the Discovery Docket: Proteus

• Now to help us break down some of the most important elements of these topics, I want to distribute this worthy handout, or four pages of it at any rate.

  It covers two kinds of symmetry that are very important: rotational and reflective.

  So let's take a look at the definitions.

  Your homework: to do the problems on the first three pages of this symmetry handout. This is due on Tuesday, of next week.

• In the plane of 2-Dimensions,
  • What is the most symmetric rectangle?
  • What are regular polygons?
  • How many regular polygons are there?
  • What kinds of symmetry do they possess?
  • Which regular polygons can be used exclusively to "tile" the plane, like a bathroom floor?