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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.
We can carry out the "Fraudini process" using a binary tree, like we're making change.
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.
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....
Today's Question of the Day:
(and what can we learn from it? what does it do for us?)
Let's try to answer some questions: Mathematicians look for patterns.
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.