Day 07: Heart of Mathematics

• I graded
  • How many times will you breathe today?
  • How tall is the Lucas Administration building?

  Here's a nice solution.

• When you estimate, you need to think about
  • Questions: what are the issues to consider?
  • Data -- where you got it, quality, etc.
  • Units
  • How you do your calculations
  • Make sure you use bounds -- that is, make a box! -- in addition to giving your best estimate. "Roughly" is not a box: it depends on each person's definition of "rough", doesn't it?
  • If you don't show your work -- just report an answer --
• Be neat, and use complete sentences to answer your questions.
• For the breathing example:
  • Question/issues: are all minutes the same?
  • Data: if you collected data on yourself, how reliable was it? What's the slowest you'd breathe in a day, and what's the fastest? How variable is your breathing? Did you consider any research? Maybe even a websearch.... One person confessed to "cheating" -- said s/he'd looked it up -- and that is, in a sense, a cheat: I want you to see where these estimates come from!
  • Units: Most of you did breaths/minute, and figured out how many minutes in a day.
  • How you do your calculations: Multiply those together, and you get units of breaths! If you give a breathing range (e.g. 12-20) use those numbers in your box!
  • If you use your extreme measurements, you'll get a nice box for your results.
• For Lucas:
  • Questions: how is height measured? From the foundation, or from the plaza level, or ??
  • Data: Some of you did height estimates of a particular floor. How good were your estimates? Some of you checked out the Lucas building on NKU's website -- good idea!
  • I contacted Facilities Management at NKU: access.nku.edu/FacilitiesManagement! They say it's 81' from the plaza level to the clock tower; the clock tower adds another 14'8". Now what's your estimate?

• We can "decompose" (or factor) natural numbers using prime numbers and products.

• A natural number that can be divided evenly (that is, without remainder) by only two distinct natural numbers: 1 and itself.
• Note: therefore 1 is not a prime number.
• We can find the primes using the Sieve of Eratosthenes (here's an animation)

• 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).
• Example: #2, p. 77

• 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?)
• Example: #35, p. 80
• Conclusion: gaps without primes of any size exist in the natural numbers!

• 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):

  The division algorithm says:

  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.

• 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!;)

How many bunny pairs at the end of a year?

• Oddly enough, Fibonacci numbers appear frequently in nature (e.g. in pine cones).

• dandelions (Here's a scanned dandelion)
• Spirals in daisies:
  • 21 spirals
  • 34 spirals
  • together

• Rules:
  • Two players, n sticks/coins/bottlecaps/etc.
  • First player must take anywhere from 1 to (n-1) sticks
  • From then on, the next player may take from 1 up to twice what the previous player took.
  • Winner is the player who picks up the last stick
• Is there a winning strategy?
• Is there a perfect defensive strategy?

every natural number is either

1. Fibonacci, or
2. can be written as a sum of non-consecutive Fibonacci's in a unique way.

Examples?

Can we justify this?

• If a number is Fibonacci, then we're done.
• Assume a number is not Fibonacci:
  • then there is a largest Fibonacci that "fits inside it" -- e.g. 24 = 21 + 3.
  • If the remainder (3, in the example above), after subtraction, is Fibonacci, how do we know that it is not consecutive (that is, that it is not 13)? That is, how do we know that the two Fibonacci numbers are not successive Fibonacci numbers?
  • If it is not Fibonacci, can we not simply repeat the current process of looking for a sum for a non-Fibonacci number, but using the remainder instead of the original number? (i.e. can we not simply "recurse" -- that is, do it again, and so construct a chain of numbers leading down towards 1.

    Example: 33=21+12=21+8+4=21+8+3+1