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 one-third, 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
- Implement a working version of the algorithm from Exercise 9.
- Identify a fraction a/b such that
(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
252. (Most computers like powers of 2.)
Wednesday, 31 January 2001 [Samuel A. Rebelsky]
Monday, 9 September 2002 [Samuel A. Rebelsky]
Thursday, 30 January 2003 [Samuel A. Rebelsky]
;
Check with Bobby