Wednesday, September 3, 2008
Summary of recent homework
Hi Emile!
I sum up here your recent work at home, and post next assignments.
In Algebra, it took you only a few minutes to do section 11,
and we've finally completed Problem 31 from section 10 about use of parentheses. I'll write down the answer so that we don't forget.
The problem is about the number of ways to put parentheses into a product of several factors. Your answer was
for # of factors 1, 2, 3, 4, 5, 6, 7, ...
# of the ways is 1, 1, 2, 5,14,42,???, ...
(these are called Catalan numbers )
For instance, abc can be computed as either (ab)c or a(bc),
so for 3 factors the number of ways is 2. Likewise, the product abcd of 4 factors can be computed in 5 different ways. Namely, designating the last multiplication to be: a x bcd or ab x cd or abc x d , we find 5=1x2+1x1+2x1. Here 1, 1 and 2 are the numbers of ways to compute the product of 1, 2, and 3 factors. Similarly, for 5 factors abcde, the number of ways is: 1x5+1x2+2x1+5x1=14, and so on.
Can you compute (correctly!) the 7th Catalan number?
Your next assignment in Algebra is to work on section 12 "Letters in Algebra." Plan to do this in class on Fri, Sept. 5.
In Geometry, you've finished the previous assignment, but there is a "research" question left: Out of 3 altitudes of a triangle, how many can lie outside the triangle, and for which triangles this can happen?
The next assignment, to work in class on Th, Sept. 4, will be Exercises 60, 56, 55, 54 (in this order).
Looking forward to your solutions!
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4 comments:
Yes because the plane is never ending.
Hi Emi,
I have no idea what question you are answering. Could you please at least write the number of the exercise?
Thanks,
OK, you are talking about 54, i.e. you claim that the plane can be tiled by polygons all of whose angles have 140 degrees each. Then, in order to justify
your answer you need to exhibit a picture explaining how exactly you propose to tile
the plane with such polygons.
Also, I suggested to do the problems in a certain order, and this one was the last one, because I was not even sure you would understand what it asks. Do you?
Definition of altitude in a triangle:
An altitude is the perpendicular dropped from a vertex to the opposite side or its continuation.
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