Scheme treats numbers slightly differently depending on whether they are integers (whole numbers), rational numbers (expressible as a ratio of integers), real numbers (corresponding to points on a number line), or complex numbers (corresponding to points on the plane determined by a real-number line and a perpendicular line for ``imaginary numbers'' -- the square roots of negative numbers). From the Scheme programmer's point of view, these categories of numbers are nested: all integers also qualify as rational numbers (5 is the same thing as 5/1); all rationals count as real numbers, and all real numbers as complex numbers. But, mathematically speaking, the converse inclusions do not generally hold. (3/4 is rational but not an integer, the square root of 2 is real but not rational, and the square root of -1 is complex but not real.)
Scheme supplies a predicate for each of these categories of numbers:
integer?, rational?, real?, and
complex?.
Start DrScheme and have it 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.
Within each of these categories of numbers, Scheme distinguishes between exact numbers, which are guaranteed to be calculated and stored internally with complete accuracy (no rounding off), and approximations, also called inexact numbers, which are stored internally in a form that conserves the computer's memory and permits faster computations, but allows small inaccuracies (and occasionally ones that are not so small) to creep in. Since there's no great advantage in obtaining an answer quickly if it may be incorrect, we shall avoid using approximations in this course, except when the data for our problems are themselves obtained by inexact processes of measurement.
To determine whether Scheme is representing a particular number exactly or
inexactly, use one of the predicates exact? and
inexact?.
> (exact? 5/9) #t > (exact? 13.2) #f > (inexact? 13.2) #t
DrScheme happens to store real numbers in such a way that any real number
that can be named or computed also counts as rational. For instance, when
DrScheme computes (sqrt 2), the value it returns is an inexact
approximation to the correct value, and it turns out that DrScheme uses
only rational numbers, even when trying to approximate irrational ones.
Confirm that the value DrScheme computes for (sqrt 2) is an
inexact real that is also rational.
The standard language definition for Scheme says that an implementation of the language does not have to support all these categories of numbers; it would be legal, for instance, to leave out complex numbers or to treat all numeric values as inexact. However, most implementations (including DrScheme) support all the kinds of numbers described here.
The built-in Scheme procedure exact->inexact takes an exact
number as its argument and returns an inexact approximation to it:
> (exact->inexact 12/7) 1.7142857142857142
Since DrScheme uses fractional notation to print out exact numbers, but renders approximations as decimals, invoking this procedure is a simple way to determine the general format in which results are printed. As we'll see later in the semester, however, there are better ways that give the programmer finer control over the format.
The Scheme standard does not directly support the familiar category of natural numbers, but we can think of them as being just the same things as Scheme's exact non-negative integers.
Define and test a predicate natural-number? that takes one
argument, which might be anything, and determines whether that argument is
an exact non-negative integer.
When you write a numeral into a Scheme program or type one in as part of a definition or command to the interactive interface, the structure of the numeral you type determines the category of the number represented.
One basic rule is that a numeral that contains a decimal point normally
stands for an approximation rather than an exact number. Scheme assumes
that you may have rounded off the last decimal place and takes this as
implicit permission to use a rounded-off representation. If you want
Scheme to interpret the numeral as an exact number, you can either convert
it to a fraction -- for instance, changing 1.732 to
1732/1000 -- or attach the exactness prefix
#e at the beginning of the numeral, making it
#e1.732.
Conversely, a number written such as a sequence of digits (possibly with a
sign at the beginning) or as a fraction normally stands for an exact
number. If you want an approximation instead, use an equivalent numeral
with a decimal point or attach the inexactness prefix
#i. (So 23/70 is an exact number, but
#i23/70 is an approximation.)
Scheme permits the use of a version of scientific notation, in
which a real number is expressed as the product of some coefficient and
some integer power of 10. For instance, the numeral 3.17e8
denotes the real number three hundred and seventeen million -- that is,
3.17 times ten to the eighth power. The part of the numeral that precedes
the e is the coefficient; the part that follows indicates the
power of ten by which the coefficient should be multiplied. A number
expressed in scientific notation is also inexact unless preceded by the
exactness prefix. Chez Scheme uses scientific notation when printing out
an inexact number if its absolute value is either very large or very small.
Write a Scheme numeral for 1.507 times ten to the fifteenth power, as an exact number. Have Scheme evaluate the numeral.
Have DrScheme find the square of the square root of 2 and subtract 2 from the result. Ideally, the difference should be 0; why isn't it? How big is the difference?
Write a Scheme numeral for one-third, as an inexact number. Have Scheme evaluate the numeral.
The Revised5 report on the algorithmic language Scheme contains descriptions of more than fifty procedures for operating on numbers. (Figure 3.3 on page 76 of the textbook provides a more manageable list of the ones that are most commonly used.) I'll assume that you're more or less familiar with the list and just comment on a few features of them that you may want to exploit:
The addition and multiplication procedures, + and
*, accept any number of arguments. You can, for instance,
ask Scheme to imitate a cash register with a command like this one:
> (+ 1.19
.43
.43
2.59
.89
1.39
5.19
.34
)
12.45
You can call the - procedure or the / procedure
to operate on a single argument. The - procedure returns the
additive inverse of a single argument (that is, its negative, the
result of subtracting it from 0), and the / procedure returns
the multiplicative inverse of a single argument (its reciprocal,
the result of dividing 1 by it).
Use Scheme to find the reciprocal of 3/5.
The log procedure, despite its name, computes natural (base
e) logarithms rather than common (base ten) logarithms. You can
convert a natural logarithm into a common logarithm by dividing it by the
natural logarithm of 10.
Define a Scheme procedure log10 that takes any positive real
number as argument and returns its common logarithm, computing it by the
strategy just suggested. Use your procedure to confirm that the common
logarithm of one million is 6.
My daughter's arithmetic textbooks are full of tedious-looking exercises
like ``Find the smallest number that is an exact multiple of 1732, 680, and
2520.'' Scheme has built-in gcd (``greatest common divisor''
and lcm (``least common multiple'') procedures for such
computations.
Have Scheme find the smallest number that is an exact multiple of 1732, 680, and 2520.
If you're planning to use any of the trigonometric functions, bear in mind that Scheme measures all angles in radians, not degrees.
Once around the circle is an angle of 360 degrees or, equivalently,
2pi radians. Define a Scheme procedure
degrees->radians that takes the measure of an angle in
degrees and converts it to radians (by multiplying or dividing by an
appropriate conversion factor).
Note: Since pi is irrational, the value of a call to this procedure
is almost always an approximation rather than an exact value. You'll
probably find, however, that (degrees->radians 0) is exactly
0.
Scheme doesn't provide a cotangent procedure cot. Define one,
using the fact that the cotangent of a number is the reciprocal of its
tangent.
An integer m evenly divides an integer n
if the remainder left over when n is divided by m
is zero. Define a Scheme predicate evenly-divides? that takes
two arguments, both assumed to be integers (and the second assumed to be
non-zero), and returns #t if the first evenly divides the
second and #f if it does not. (Hint: If you haven't yet
studied the list of numerical procedures on page 76 of the text, this would
be a good time to do so.)
This document is available on the World Wide Web as
http://www.cs.grinnell.edu/~stone/courses/scheme/numbers.xhtml
created September 4, 1997
last revised March 17, 2000