Fundamentals of Computer Science 1 (CS151 2003S)
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Number predicates used in this lab:
complex?
,
exact?
,
inexact?
,
number?
,
rational?
, and
real?
,
Other number procedures used in this lab:
denominator
,
numerator
, and
sqrt
.
You may want to refer to the reading on numbers before or while you work on this lab.
Start DrScheme.
Have DrScheme confirm that 3/4 is a rational number but not an integer and that the square root of 1 is a complex number but not a real number.
Confirm that the value DrScheme computes for (sqrt 2)
is an
inexact real that is also rational.
As you've just seen, some kinds of numbers are subsets of other kinds of numbers. Determine the relationships between the various kinds of numbers.
Write a Scheme numeral for 1.507 times ten to the fifteenth power, as an exact number. Have Scheme evaluate the numeral.
a. Have DrScheme find the square of the square root of 2 and subtract 2 from the result.
b. Ideally, the difference should be 0; why isn't it?
c. How big is the difference?
d. Will you have the same problem if you start with 4? Why or why not?
Write a Scheme numeral for onethird, as an inexact number. Have Scheme evaluate the numeral.
Scheme provides a number of numerical procedues that can
produce integer results.
We've already explored
expt
,
abs
,
+
,

, and
*
.
Here are some others. For each, try to figure (by experimentation, by discussing results with other students, and, eventually, by reading documentation) out how many parameters each procedure can take and what the procedure does. Make sure to try a variety of values for each procedure, including positive and negative, integer and real.
Warning: You may not be able to figure all of them out.
a. quotient
b. remainder
c. modulo
d. max
e. min
f. numerator
g. denominator
h. gcd
i. lcm
j. floor
k. ceiling
l. truncate
m. round
Since you've found that DrScheme seems to represent every real number as a rational, it might be worth finding a way to see what that rational number is.
a. Determine the numerator and denominator of the rational representation of the square root of 2.
b. Determine the numerator and denominator of the rational representation of 1.5.
c. Determine the numerator and denominator of the rational representation of 1.2.
If you're puzzled by the last answer, you may want to read the notes on this problem.
For small numbers, the exact>inexact
procedure produces lots and lots of digits after the decimal point.
Figure out how to get just two digits after the decimal point. You may
need to use multiplication, division, and some of the last procedures
from the previous exercise.
You need not implement your algorithm; simply come up with one you think will work.
We've already seen a variety of predicates (procedures that return
true or false) that can be applied to numbers. These predicates
include
exact?
,
integer?
, and
real?
.
By reading the Scheme documentation, identify other predicates that can be applied to numbers.
If you finish early, you might
(numerator a/b)
is not the same as (numerator #ia/b)
.
When you get stuck on this problem, it's probably worth skimming through DrScheme's Help Desk. The numeric operations are documented in section 6.2.5 of the Revised(5) Report on the Algorithmic Language Scheme. (I have not yet put all of these procedures in the Glimmer Scheme Reference.
DrScheme seems to represent the fractional part of many numbers as the ratio of some number and 4503599627370496, which happens to be 2^{52}. (Most computers like powers of 2.)
Wednesday, 31 January 2001 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2001S/Labs/numbers.html
.
Monday, 9 September 2002 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2002F/Labs/numbers.html
.
Thursday, 30 January 2003 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2003S/Labs/numbers.html
.
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Misc:
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Disclaimer:
I usually create these pages on the fly
, which means that I rarely
proofread them and they may contain bad grammar and incorrect details.
It also means that I tend to update them regularly (see the history for
more details). Feel free to contact me with any suggestions for changes.
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.
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