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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Discussion 1 -- Mathematical Discovery}
\centerline{September 17, 2010}
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\noindent
{\it Background}
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There is no doubt that mathematics requires careful, 
precise, logical thinking (at least if you want to 
produce reliable, correct results!)  But is that (i.e. logical
thinking) the {\it whole story} as Kline sometimes seems to 
imply?  Can there also be {\it creativity} or the 
{\it excitement of finding out something new} (or at least new to you) in doing 
mathematics?  We'll try to see today by looking at 
a couple of questions -- one from geometry, and one 
from arithmetic.  Don't be afraid to get interested/excited 
if you feel like it!   Note:  Each group will be assigned {\it one}
of the following questions and work on it for the full period.
We will discuss the results on Monday.  
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\noindent
{\it Questions}
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\noindent
I.  Geometric figures with four straight line segment edges 
(``quadrilaterals'') in the plane can have pretty ``random'' shapes.  
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\item{1)}  On the graph paper supplied, draw the quadrilaterals
with corners (``vertices'') at the points given.
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\itemitem{(a)} $(0,0), (1,0), (2,3), (1,5)$ (connect the points
with straight line segments in the order given and then connect $(1,5)$
back to $(0,0)$ to close up the figure).  
\itemitem{(b)} Do the same for $(0,0), (1,0), (5,1), (3,6)$.
\itemitem{(c)} Do the same for $(0,0), (2,-1), (3,0), (0,5)$.
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\item{2)}  For each of the quadrilaterals from question 1,
figure out where the {\it midpoints} of the 4 edges are, and
plot them in your figures.  Make a new quadrilateral in
each case by connecting the midpoints in order, going around
the figure in the same counter-clockwise way as in the original quadrilateral.  
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\item{3)}  Do you notice something (surprising) about
the quadrilaterals with corners at the midpoints?  Are their
shapes as ``random-looking'' as those of the original quadrilaterals?
Are these all some particular type of quadrilateral?  How can you 
tell?  
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\item{4)}  (Harder)  Can you prove a {\it general statement} about what
happens with the process described in part 2)?   Suggestion:  Try using
coordinates with the first two vertices at $(0,0),(1,0)$ (In case you 
are worried, there is no loss of generality
in doing this since we can shift and rotate any quadrilateral to put
one side along the positive $x$-axis starting at $(0,0)$, and then
we can choose our unit of length to put the other endpoint of that
first edge at $(1,0)$).   Then call the third
and fourth vertices $(a,b), (c,d)$ in general.  Where are the midpoints?
What are the edges of the midpoint quadrilateral?
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\item{5)}  All of the figures in part 1) are {\it convex} quadrilaterals
(this means that none of the edges cross, and none of the vertices
is inside the triangle formed by the other three vertices).  But you 
can certainly get nonconvex quadrilaterals with the coordinates from part 4).
For instance, take $(a,b) = (0,3)$, $(c,d) = (1,1)$.  Is the general statement
still true in this case?  What about $(a,b) = (0,3)$, $(c,d) = (1,3)$ where
edges of the original quadrilateral will cross when you connect the 
vertices?  
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\noindent
II.  In class earlier this week we talked about base 2 (binary) and 
base 8 (octal) as ways to represent whole numbers (and also fractions, etc.)
These seem quite different -- are they related somehow?
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\item{1)}  Find the base 2 and base 8 forms for the numbers $89$, $497$, and
$1045$.  Any obvious pattern?  (Your answer should be no!)
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\noindent
If you take the base 2 form of a number, you can split it
up into 3-digit blocks starting from the 1's digit.  For instance 
$1101101010 \leftrightarrow 001\ 101\ 101\ 010$ -- note that I inserted
two zeros at the start of the leftmost block so that there would always
be exactly three binary digits in each block.  Now, a three-digit
binary number is the same as a base 10 number 0, 1, 2, 3, 4, 5, 6, or 7.
These are exactly the same as the base 8 digits!  So we can ``read a 
base 2 number as a base 8 number'' if we convert each 3-binary digit
block to the corresponding base 8 digit.  For instance:
$$001\ 101\ 101\ 010 \mapsto 1\ 5\ 5\ 2$$
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\item{2)}  What base 10 number
is $(11011010101)_2$.  What base 8 number do we get from the 3-digit
blocks above -- $(1552)_8$?  Are these two numbers the same?  
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\item{3)}  What happens if you compare the base 2 and base 8 forms
of the three numbers from part 1)?  Is it the same pattern?
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\item{4)}  (Harder)  Can you prove a {\it general statement} about
the relation between the base 2 and base 8 forms of a whole number
along these lines?   Start by trying to show exactly what happens
if there are two blocks of three binary digits(!)
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\item{5)}  Another common number system in the computer world
is base 16 (``hexadecimal'').  The ``digits'' in base 16 are written as 
$$0,1,2, \ldots, 9, 10 = A, 11 = B, 12 = C, 13 = D, 14 = E, 15 = F.$$
If you have a number written in base 2, how can you get its
base 16 form (without too much calculation!)?
\bye
