Your first homework assignment (fun and games) returned.
Your grade is out of 10.
Some of you did problems from the wrong section, or the
wrong problem numbers. Be careful! Read carefully!
Some of you didn't type your solutions. Follow instructions!
Watch your writing. Be careful about spelling, grammar, and punctuation.
Some nice solns for 2, 5, and
8. I didn't get any really clean solutions for 9 and
14.
Your homework assignment on Estimation is due, but let's put it
off until next time because of the missed class session.
For Thursday, read sections 2.3 (Prime Factorizations) of our text and 2.2 (Numerical Patterns in
Nature -- Fibonacci numbers).
Today:
Some last examples on Estimation:
How many ordinary dice would it take to fill this
classroom? (your estimates)
Do you have a temporal twin? That is, is there someone who
was born on the same date as you (i.e. within 24 hours
of you), and will die on the same date (within 24 hours
of you)?
Is there anyone who obviously does?
Is there anyone who obviously doesn't?
Number Systems:
Bases:
By the end of today, you should know a "magic trick" that you can use to
entertain friends young and old, using these
cards.
Let's start with the natural number 1729, which was the
subject of an interesting story in section 2.1.
What did Hardy think of 1729?
What did Ramanujan tell Hardy is so special about 1729?
Our authors gave a comical "proof" of something after
this story -- what was it?
What is the meaning of each digit (the numbers 0 through 9) of the
number 1729?
One of the things we want to do with natural numbers is break them
down into simpler components. We will do this in various ways, using
powers, Fibonacci numbers, and Prime numbers.
What are some special numbers in base 10?
How would we write 1729 in base 8?
How about 1729 in base 12?
And now: a Magic Trick, performed by the Great Fraudini
How does the trick work?
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.
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).
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):
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!;)
Links:
From the same author who brought you the bases paper above, we have a paper on the importance of units.
Website maintained by Andy Long.
Comments appreciated.
