- You have an assignment due Thursday -- see the assignment page, and make sure to keep
up!
- Notice that this assignment is partly about writing -- about expressing yourself. One of our goals in this course is to encourage you to become better writers, ironically enough -- so we're going to ask you to write about mathematics. Pay attention to spelling, grammar, etc. -- and write your work up cleanly. Typed is best. I don't want to struggle to decipher your handwriting....
- This problem moves us away from probability, and into the realm of
number representation, which is our next topic.
You rub an old oil lamp and a genie appears. You ask for a wish - since you let him out of the lamp, after all. Reluctantly the genie agrees. You tell him you want the Royal Black Diamond - a famous gem worth millions of dollars. At your feet appear 27 identical - looking gems. You ask Mr Genie what is going on and he replies that you may take any one of the stones with you. They are identical in appearance, but the genie explains that the Royal Black is slightly heavier than the 26 fakes. At this you say, " Great I wish I had a way to weigh them!" The genie grants your wish and a magic balance scale appears at your feet. The genie explains that upon being used three times the scale will disappear. Can you find a way to choose the right gem from the 27?
- We could do this problem if...if there were only three stones! How?
- For which numbers could we solve the problem, and how can we use
the answers to these simpler problems to approach the case of 27?
- How does this solution illustrate some of our strategies, including
- Look at things from different perspectives.
- Break difficult problems into simpler ones.
- Look for patterns and similarities.
- Explore the consequences of new ideas. Generalize.
- Do it again...
- This solution also illustrates the value in breaking down numbers in different ways. In this case, it is sums (which represent weighings) that are particularly useful.
- Of course we'll start with the simplest numbers, which are the natural numbers:
- We can think about defining the natural numbers this way:
- 1 is a natural number.
- For each natural number n there is another natural number n+1.
- The natural numbers are thus the counting numbers: 1, 2, 3, ....
- All natural numbers are interesting, and we can prove it!
- Is 1 interesting?
- Is 2 interesting?
- Suppose there were a first uninteresting natural number....
- The number 1729 is especially interesting: the story of
Ramanujan and Hardy:
This story illustrates that we can break down numbers in interesting ways.
- We can think about defining the natural numbers this way:
- Now, zero didn't make it right into the mix: it just confused the
heck out of everyone. Next came
- "rational numbers" -- ratios of natural numbers
- "irrational numbers" -- e.g. the side length of a square with sides of length 1
- negative numbers
- imaginary numbers (square roots of negative numbers)
- But finally zero had its day -- what's its history?
- Used by Mayans and in India in the 7th C.
- First used in the west as a place holder -- punctuation -- in examples like 100 and 504 (the zeros are just filling spaces.)
- Zero -- as a number -- came late to the West - Fibonacci's Book of Counting ("Liber Abaci") in 1202 brought Arabic-Indian numerals to Europe. This includes zero. (Previously math was done using Roman numerals and Abacus-like tools.)
- How to divide by zero? You can't! Can you explain why?
- Now Prime Numbers are very old: they're useful for breaking
numbers down by multiplication.
- 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.
- The general procedure (and one that we'll see
repeated time and again in mathematics):
do it once, then do it again, and again, and again.... until done (or tired). I'll call this out often, saying simply "do it again...." I mean that you can apply exactly the same procedure just used, often on a simpler problem.
- Upshot: We can "decompose" (or factor) natural numbers using prime numbers and products.
- Euclid - he of geometry fame -- concluded that there are infinitely many prime numbers.
- Twin primes conjecture -- To this day no one knows if there are an infinite number of primes p and p+2. For example 17 and 19, or 29 and 31, are "twin primes"
- Goldbach conjecture - Every even number (greater than two) is the sum of exactly two prime numbers. For example 18=11+7
- Primes of the form 4k+1 can be written a the sum of two squares in exactly
one way: so for k=3,
- Primes of the form 4k-1 can never be written as the sum of two squares. So for k=3, 11=what?? Try the few possibilities.
- 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).
- What is a prime number?
