Mathematics 210  Spring term 2003  Fourth assignment
This assignment requires you to submit several spreadsheets concerned
with (a) the machine representation of real numbers;
(b) probability distributions. It is due next Monday, February 10.
I repeat the instructions from the first lab:

Go to the MathSheet home page
and then to the applet page. Open a running
copy of the spreadsheet and return to this page.

Log in immediately: File/Log in. Your login id is your
Mathematics Department login name, and your password
is your student number. This allows you to save
and load spreadsheet files. You should save your work
frequently.
The graph signature
mechanism should now be working  please use this feature.

Question 1.
I have said in class that a positive real number y
in the computer is stored in the form y = 2^{e} x
where 1 <= x < 2 and takes up
53 bits in its base 2 expression. In the next few questions you
are going to use the spreadsheet to explore this.
The base 2 expression for 1 is 1,
that for 3 is 11 = 2 + 1, that for 7 is
111 = 4 + 2 + 1.
Work out by hand or using the spreadsheet for
scratch work the base 2 expressions for
15, 31, 63, 26, 121, 1023.
Put these expressions as strings like "11 in cells b0
down the second column
with the corresponding number itself in the first column.
Keep in mind that it should be easier to check your answer
than to find it.
If y is an arbitrary positive number, here is how to find its base 2
expression: (1) find e such that 2^{e} <= y < 2^{e+1}. (2) If x = y/2^{e}, then
1 <= x < 2. You now want to express x
as a possibly infinite sequence of bits b_{i} equal to
0 or 1:
x = b_{0}.b_{1}b_{2} ...
If x has an expression like this with b_{0} equal to
0 or 1, then either 0 <= x < 1 or 1 <= x < 2.
In the first case b_{0} = 0
and in the second b_{0} = 1.
In either case,
the first bit b_{0} is the integer just below x,
which is calculated as the floor of x in the spreadsheet.
Thus floor(1 + 1/4 + 1/8) = 1 and the bit expression
for it is 1.011. Then
x  b_{0} = .b_{1}b_{2} ...
2(x  b_{0}) = b_{1}.b_{2} ...
and we can now calculate b_{1} as the floor of this new number.
Etc. Calculate e and the first 10 bits of the base 2 expression for
the fraction 1/3 (by hand, or using the spreadsheet for scratch work).
Put the string for this in
c0 in the form "1/3 = ... .
Then do the same for 1/5, in cell c1,
and 5/7, in cell c2.
Save this sheet as m210.4.1.ms.

Question 2.
Clear your spreadsheet.
In this question you are to build a spreadsheet
with the following property:
when you enter a number y in cell b0,
it displays in cell b1 the exponent e,
in cell b2 the number x,
and in cells c3 down the next 60 bits in the
base 2 expression for x. The cells b3
down should be used to derive these,
and in the cells a3 down are the integers 0, 1
etc. indexing the bits in column c.
Format columns a and c for 0 digits
to make them easier to read. Make a bar graph that shoows the bits in
column c.
Check your spreadsheet by looking at 1/3 and
the largest integer the machine can store exactly,
which is 2^^{53}1. When the grader looks at this,
he will try other numbers at random.
Save this sheet as m210.4.2.ms.

Question 3.
Clear.
Build a spreadsheet with this property:
In cells a10 down through a60
are the numbers 0 through
50.
Suppose that in cells b10 through b20 are arbitrary
probabilities p_{i} of getting
i on some experiment. Their sum should always be 1.
Then in cells c10 down through c30 will
be the probabilities of getting i
if we choose pairs randomly from this set of values
and add them together.
In cells d10 down through d40 will
be the probabilities of getting a sum of i
if we choose triples randomly from this set of values,
and in columns e and f the same for quadruples and quintuples.
Graph these by bar graphs, starting with one item,
then two on top of that, and so on. For dice
the graphs look like something like this (only
the quintuples have been left out; the colours are your choice):
For submission, set p_{i} = 1/11.
Scale the graph so it looks good.
Save this sheet as m210.4.3.ms.
That's it!
If you find these questions confusing, please
write me.
