- You have a quiz Thursday over the Fraudini trick, binary factorization, primitive counting, and the reading. Also everything that we do in class last week! It might pay to review your agendas prior to a quiz, too.
- For the quiz, you may have only hand-written notes (e.g. example problems, summaries of readings, etc.). If you have a calculator, great; if you must use your phone, then you may only use it in calculator mode, and it will be on the table in front of you.
- I've assigned you some homework problems, just to check if you are getting it.
- Key: A sample of nice answers....
- Biggest problems:
- Did primitive counting, rather than prime factorization -- whoops!
- Didn't know how to test if a number is prime or not. Starts with a square root: you only need to check primes up to the square root (and I gave you all the primes you needed at the top of the page!).
- Pretty good work generally, with our median (middle score) of 8.
- We wrapped up the binary factorization:
Theorem: Every counting number is either a power of 2, or can be written as a sum of distinct powers of 2 in a unique way.
- Practiced our Fraudini and binary factorization skills: you should
know how to
- do the Fraudini trick, using the six cards which construct
any number from 1 through 63.
We can carry out the "Fraudini process" using a binary tree, like we're making change.
- write the number (in our base, base 10) as a binary number (using only the powers of 2).
- do primitive counting, by partitioning the
given set of objects into "halves", keeping track of
whether there is an object left over, or not.
If written down as we've done in class, this results in the writing of the binary representation of a number.
We can use ternary trees to carry out this process.
- recover the "base 10" number, if given its expression in binary.
Vi Hart has the most amazing ways of showing us interesting mathematics. She shows us that we can count to 31 on one hand. And, if you'll use the 10 fingers of two hands, you can get all the way up to $2^{10}-1=1023$...
Some people can only count to 10 with their fingers....
- do the Fraudini trick, using the six cards which construct
any number from 1 through 63.
Today's Question of the Day:
(and what can we learn from it? what does it do for us?)
- Today's new thing, the best new thing in the world, is an old
thing. It's called "Yanghui's triangle", and you should study it
carefully:
Let's try to answer some questions: Mathematicians look for patterns.
- What do you notice? What catches your attention?
- Can you understand the writing system for numbers?
- Can you find any number patterns?
- What is this thing? What is its purpose?
- There's an error in the table....
- When we're done with those questions, we're going to create a more modern version of
this triangle on the back of your sheet.
There's a rule that we want to decipher for this triangle; and we'll relate it to counting.
For example, Pascal used it to answer this question:
What are the chances that a family of X children will have Y girls?
Before we can do that, however, we need to recognize some features of the table.
- The resulting triangle contains some interesting number patterns:
- The counting numbers (natural numbers)
- Triangular numbers
- Tetrahedral numbers (next time)
- The powers of two....
- (And more! Your reading comments on all of these... and more!)