Numeric Values

Computer scientists write algorithms for a variety of problems. Some types of computation, such as representation of knowledge, use symbols and lists. Others, such as the construction of Web pages, may involve the manipulation of strings (sequences of alphabetic characters). However, as you've seen with some of your initial experiments with images, a significant amount of computation involves numbers.

One advantage of doing numeric computation with a programming language, like Scheme, is that you can write your own algorithms to make the computer automate repetitive tasks. As you do numeric computation in any language, you must first discover what types of numbers the language supports (some languages support only integers, some only real numbers, some combinations) and what numeric operations the language supports. Fortunately, Scheme supports many types of numbers (as you may have discovered in the first few labs) and a wide variety of operations on those numbers.

As you probably learned in secondary school, there are a variety of kinds of numbers. The most common types are the integers (numbers with no fractional component), rational numbers (numbers that can be expressed as the ratio of two integers), and real numbers (numbers that can be plotted on a number line). Script-Fu, the Scheme in GIMP, supports only integers and real numbers, so DrFu supports only integers and real numbers.

In some Scheme implementations, other numeric types are available, such as the complex numbers (numbers with a possible imaginary component). While you will not use those other types in this course, we alert you to their availability so that you can think of applications outside of the primary focus of this course. Why does Script-Fu leave out some kinds of numbers? Because the implementers did not see a need for them in the kinds of applications Script-Fu was intended for. The standard language definition for Scheme says that an implementation of the language does not have to support all categories of numbers.

Scheme provides two predicates that let us check whether or not a value has a particular type: integer? and real?.

> (integer? 2)
#t
> (real? 2)
#t
> (integer? 2.5)
#f
> (real? 2)
#t
> (integer? "two")
#f

DrFu uses integers to represent other kinds of values, such as images and RGB colors, so integer? will return true for them, too. (Don't worry that you haven't seen RGB colors yet; we'll get to them in a few days.)

> (integer? (rgb-new 0 0 0))
#t
> (integer? (image-load "/home/rebelsky/glimmer/samples/rebelsky-stalkernet.jpg"))
#t
> (integer? "/home/rebelsky/glimmer/samples/rebelsky-stalkernet.jpg")
#f

We will return to predicates when we consider conditionals.

Within each category 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?. Real numbers are never represented exactly, and integers can be represented exactly or inexactly. You can convert between the two representations with exact->inexact and inexact->exact.

> (exact? 2)
#t 
> (exact? 2.0) 
#f
> (inexact->exact 2.0)
2

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.

Section 6.2.5 of the “Revised⁵ report on the algorithmic language Scheme” lists Scheme's primitive procedures for numbers. Read through the list at this point to get a feel for what Scheme supports. The following notes explain some of the subtler features of commonly used numerical procedures. As you read about procedures, think about how you might use them in writing color filters or in other graphical algorithms.

Warning! The output from DrFu is not always consistent with our expectations. DrFu is also evolving. In a few cases, you may see slightly different responses than appear in this reading.

As you've already seen, 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 (its negative), the result of subtracting it from 0.

The max procedure returns the largest of its parameters and the min procedure returns the smallest of its parameters. As we've already seen, max can be useful when you want to ensure that a computation returns a value no smaller than a certain value and min can be useful when you want to ensure that a computation returns a value no larger than a desired maximum value.

There are four procedures that relate to division (/, quotient, remainder, and modulo). The / procedure returns the multiplicative inverse of a single argument (its reciprocal), the result of dividing 1 by it. The quotient and remainder procedures apply only to integers and perform the kind of division you learned in elementary school, in which the quotient and the remainder are separated: “Four goes into thirteen three times with a remainder of one”:

> (quotient 13 4)
3
> (remainder 13 4)
1
> (quotient 1 2.5)
Error: quotient: argument 2 must be: integer

As the final example suggests, quotient can only be applied to integers. The / procedure, on the other hand, can be applied to numbers of any kind (except that you can't use zero as a divisor) and yields a single result.

The modulo procedure is like remainder, except that it always yields a result that has the same sign as the divisor. In particular, this means that when the divisor is positive and the dividend is negative, modulo yields a positive (or zero) result. (When can a remainder be negative? Consider -7 divided by 3. Do we think of -7 as -2*3-1 or -3*3+2? Scheme makes the former decision for remainder and the latter decision for modulo.)

> (remainder -13 4)
-1
> (modulo -13 4)
3

The modulo procedure can be particularly useful when you want to ensure that a value falls in a certain range, and you don't just want higher values to map to the highest value in the range. For example, you'll find many times this semester that you want to compute a number between 0 and 255, but end up computing something out of that range. we can ensure that they fall within the appropriate range with max and min. We can get somewhat different effects by using (modulo computed-value 256). This expression ensures that the value is between 0 and 255, but causes larger numbers to wrap-around to become smaller numbers.

> (define blue-component 250)
> (min 255 (+ 32 blue-component))
255
> (modulo (+ 32 blue-component) 256)
26

At times, we will have a real number and will want to convert it to a nearby integer. For example, if you are working with images, the components of an RGB should be integers; weird things can happen if you try to use real numbers (not always, but sometimes). Scheme provides four basic procedures for this conversion: round, truncate, floor, and ceiling. You will explore the differences between these procedures in the corresponding lab.

Warning! At times, DrFu will complain that it is expecting an integer but sees a real value, even when you think you have a real value. The problem is not with you, but with the error messages. Most of the time that DrFu says that it wants an integer, it really wants an exact integer, so use inexact->exact to get the number in the correct form.

> (quotient 3 (round 2.5))
Error: quotient: argument 2 must be: integer 
> (quotient 3 (inexact->exact (round 2.5)))
1

Scheme provides five basic predicates for comparing numeric values, < (less than), <= (less than or equal to), = (equal to), >= (greater than or equal to), and > (greater than). When given two arguments, they return #t if the indicated relation holds between the two arguments.

> (< 5 10)
#t
> (> 5 10)
#f

These predicates and also take more than two arguments. The predicate returns #t only if the relation holds between each pair of adjacent arguments.

> (< 2 3 4)
#t
> (< 2 3 1)
#f

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. In case you've forgotten, the common logarithm of n is “the power to which you raise 10 in order to get n”.

> (log 100)
4.605170185988092
> (/ (log 100) (log 10))
2.0

Scheme provides the standard host of trigonometric functions, which include sin, cosine, and tan. When using these functions, remember that all angles are measured in radians, not degrees.

 
> (sin 90)
0.8939966636005579
> (cos 90)
-0.4480736161291701 
> pi 
3.141592654
> (exact? pi)
#f 
> (sin (/ pi 2)) 
1.0
> (cos (/ pi 2))
6.123031769e-17 

You may wonder why the cosine of pi-over-2 (a right angle) is not 0. It's because pi is not exactly the value of pi. However, as scientific notation indicates, the value is pretty close to 0. (There are sixteen leading 0's.)

We can use the trigonometric functions when we start doing more involved drawings (e.g., they can help us draw polygons). The trigonometric functions also provide the opportunity to do some interesting color transformations.

Many students are puzzled by both the modulo and remainder procedures. For remainder, you really should think back to middle-school math: the remainder is what's left after whole-number division. Since modulo is the same as remainder for positive numbers, you can think of it that way.

More importantly, modulo provides an interesting way of counting. Most of the time you add 1, you follow standard protocols (1 plus 1 is 2, 2 plus 1 is 3, ...). However, when you reach the modulus value, you go back to zero.

The following table shows the value of remainder and modulo for a variety of values.