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Wednesday, April 26, 2006

Scribe ?

Hello. That's enough introduction.
For today's class, Mr.K put three problems on the board that we were to solve.

Expand and Simplify

(a+b)^o= 1

(a+b)^1= a+b

(a+b)^2= a^2+2ab+b^2

(a+b)^3= a^3+3a^2b+3ab^2+b^3



Fill in the next rows
(Look for patterns)

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1



Evaluate each Expression

oC0
1Co 1C1
2Co 2C1 2C2
3Co 3C1 3C2 3C3
4Co 4C1 4C2 4C3 4C4

*There is a pattern. It has a connection with the first two problems.
oCo is the same thing as 1. 1Co and 1C1 is the same thing as 1. 2Co is the same thing as 1. 2C1 is the same thing as 2. 2C2 is the same thing as 1. See the pattern?

In the first problem, (a+b)^0 is the same thing as 1. (a+b)=a+b. Their coefficients are 1.

In the second problem, the first row is 1, the second row has two 1's. The third row has 1,2, and 1, and so on.

Point is from all that rambling, is that there is a pattern.


He also said something about a guy named Pascal and how he used the triangle but there was actually a chinese dude who discovered it first. His name was Chi Che? Later on, another French guy used the triangle and made triangles inside the pascal triangle. And if you shade all the odd numbers in the pascal triangle, you'll see what that french guy was talking about.


Now, Mr. K gave us a hand out about Pascal's Triangle. Sorry guys. I don't got a printer right now so you had to be in class to see this. Anyways, there were many patterns and encodings we found out from this triangle.

One significant one was Fibonacci's numbers. This is when you add the sum of the two previous numbers in a number sequence.

Ex.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55....etc..

Bees populate this way. Female bees have two parents and male bees have one. If you create a family tree, you'll see that it follows fibonacci's numbers. Same with how rabbits populate, the petals on a flower, how trees grow, our body structure, etc.

I think Mr.K said that dividing these numbers get you the golden ratio. 1.6-ish. And he went on about how people like 6 by 4 pictures because of the golden ratio.

And that's about all I remember from today's class. It was fun listening.


Oh yeah. The next scribe will be Michael.



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