- Your homework assignment on Estimation is due (last chance!)
- For Today you should have read sections 2.3 (Prime
Factorizations) of our text and 2.2 (Numerical
Patterns in Nature -- Fibonacci numbers).
- I think that it goes without saying that when I hand out something
in class -- i.e., the paper
on bases -- you understand that it's material you're expected to
read. If not, I'll say it!
- Assignment 3 (due Monday, 2/16)
- Bases:
- Numbers are usually written with digits 0-9, where the place value represents the appropriate power of 10.
- So the fascinating natural number 1729, which was the subject of an interesting story in section 2.1, is actually
- We've broken down the number 1729 into powers of 10s. Ramanujan
broke it down into
- We could just as well have broken them down using powers of 8s, or 2s (which was the secret of The Great Fraudini's magic trick).
Let's see how that trick works, by writing 25 in binary (the way that a computer would think of it), and thinking about our cards.
- 1729 in base 8: (3 3 0 1)8
- And now a base 8 treat! New Math (Tom Lehrer) -- lyrics
- Some exercises:
- Write 172910 in binary.
- How about 172910 in base 16 (hexidecimal)?
(What's the special problem with this situation?)
- How do you go backwards? For example, what's (2 7 5 3)8 in base 6?
- Write 172910 in binary.
- Numbers are usually written with digits 0-9, where the place value represents the appropriate power of 10.
- Section 2.3: Prime Cuts of Numbers
- Decomposing numbers, part II:
- We can "decompose" (or factor) natural numbers using prime numbers and products.
- 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: therefore 1 is not a prime number.
- 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).
- Example: #2, p. 77
- 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?)
- Example: #35, p. 80
- Conclusion: gaps without primes of any size exist in the natural numbers!
- 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):
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.
- 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):
- 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!;)
- Goldbach Question: Can every even natural number
greater than 2 be written as the sum of two primes?
- Decomposing numbers, part II:
- From the same author who brought you the bases paper above, we have a paper on the importance of units.
- Another good video of a Tom Lehrer song: the Elements (of the periodic table).