- Announcements:
- Would you be so kind.... as to permit me to move our test of
Friday, 2/15, to Monday 2/18? I'm scheduled to be out of town on
Monday, and it's a lot easier to get proctors than it is to get people
to actually lead a class....
- Your quizzes are returned. The quizzes (and homeworks) give you
some idea of what the test questions will be like.
Here's a "key".
- Hopefully you've read chapter 6 for today.
- Here's our "Question of the day":
How did Babylonians represent numbers?
- We'll begin by wrapping up our introduction to prime numbers. Then
we'll move into chapter six.
- Primes (and composites): A summary
- Upshot: We can "decompose" (or factor) natural numbers
greater than one using prime numbers and
products, and we can do it in a unique way (so
says the prime factorization theorem).
- 1 is neither
prime or composite (which is why it's so lonely).
- What is a prime number?
- A counting number (natural number, 1, 2, 3, 4, ....) that
can only be expressed as a rectangle in one way: as a single
row or column.
- A natural number that can be divided
evenly (that is, without remainder) by
only two distinct natural numbers: 1 and
itself.
- We found primes up to 100 using the
Sieve of Eratosthenes
- Composite numbers can be written as rectangles with more
than one row or column. They are "broken down" starting from
small primes and moving up.
Examples:
- 42=2*21=2*3*7
- 8=2*2*2 (prime factors can repeat)
- 84=22*3*7 (we usually write the factors
in order, from smallest to largest)
- 1729 = 7*247 = 7*13*19 (Now you can see that it's probably time to
get out that calculator!)
- Here's a rule you can count on (which we didn't
talk about last time): "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. So we
don't need to check primes past 11 to find its
factors. Are there any?
- Is 931 prime?
- 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).
- Twin primes: these are special pairs of primes, separated by 2:
- 3 and 5
- 5 and 7
- 11 and 13
- 17 and 19
- 41 and 43
- and on and on, forever and ever, amen? We don't know....
Our author personifies these as special beautiful pairs, fading
off into the distance, primes farther and farther apart, yet
occasionally a beautiful pair falls side by side.
- Preamble to our next topic:
- Primitive folks might have been able to write their numbers, as
knotted cords, representing each number uniquely and writing
numbers as high as they want using only two different types of
knots.
- We've learned that we can count forever using our counting
numbers, and we have a way of writing our numbers uniquely: we
can write any number using the digits 0 through 9, and we can
write any number uniquely as a product of primes.
- We're now going to take a look at two other cultures, and see how
they wrote their numbers: the Babylonians and the Mayans.
- First up: the Babylonians.... I want to introduce you to the
Babylonians via a "detective story": I hope that you all like
detective stories.... The story is called Detective
in the times of Babylon. Let's see if we can solve this mystery
before the end of class:
What is that table, and how does their number
system work?
Website maintained by Andy Long.
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