Day 16, MAT115

• Sorry, I didn't get your multiplication homework graded yet.
• You have a homework due today.
• You have NO homework due Tuesday after the break!
• Have a good and safe spring break!

Well, sort of....

• Let's start by recapping that tricky one that Buddy forced us to do. Remember that I had no part in that....:)

  $\frac{17}{25}$

  • = 1/3 + 1/5 + 1/15 + 1/25 + 1/50 + 1/75 + 1/150

    By the unit fraction table

  • = 1/2 + 1/8 + 1/25 + 1/100 + 1/200

    By the multiplication table

  • = 1/3 + 1/5 + 1/75 + 1/10 + 1/30

    By a method I've not shown you yet!

    And finally,

  • = 1/2 + 1/6 + 1/75

    By a method due to Fibonacci, in the Liber Abaci

• Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?

  (Think about what the answer means, in terms of bread.)

• A little trickier:
  1. How would you divide 5 by 7? (use the multiplication table backwards)
  2. How can we use the unit fraction table to get the same answer?

• How would you like to do story problems like this one?!

  Notice the numerals and fractional notation, as we see in our text on pages 13 and 29.

• Fibonacci appears again, being the first European mathematician to use the fraction bar as it is now used.
• Decimals came late to Europe (1585AD), even while they'd been in use in the Arabic world since about 1000AD.
• If you travel in Europe, you'll find countries (e.g. France) where they use a comma for decimals, rather than a period!

  So number notation is still evolving.

• Scientific notation (based on powers, like we're using in our bases), is the preferred way of writing big numbers. Have you heard of a Googol, and Googolplex?
  • Googol: $10^{100}$
  • Googolplex: $10^{Googol}$

• In the life of Pascal, you might have noticed that he invented an early computing machine. He really was an amazing scientist.

• The pages 48-59 speak of numbers, unusual and mysterious. Some of these are based on primes, which we'll study in a moment.
  • Polygonal numbers (p. 52)
  • "Perfect" numbers (p. 51)
  • "Amicable" numbers (p. 52)
  • Magic Squares (p. 53/57):
    

    (or diagonals!)

  • $\sqrt{2}$ -- the number that killed a man (called "irrational")
  • Battle between Arabic and Roman numbers
  • $\pi$: Archimedes got awfully close....
    

  • $e$

• 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 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.

  • We're going to show that there is a prime number bigger than any number you can give.

• 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!;)