(exclusive) to (inclusive).
log 0 is undefined.
With log defined this way, the values of sin-1 z, cos-1 z,
and tan-1 z are according to the following formulæ:
This chapter describes Scheme's built-in procedures. The initial (or ``top level'') Scheme environment starts out with a number of variables bound to locations containing useful values, most of which are primitive procedures that manipulate data. For example, the variable abs is bound to (a location initially containing) a procedure of one argument that computes the absolute value of a number, and the variable + is bound to a procedure that computes sums. Built-in procedures that can easily be written in terms of other built-in procedures are identified as ``library procedures''.
A program may use a top-level definition to bind any variable. It may subsequently alter any such binding by an assignment (see 4.1.6). These operations do not modify the behavior of Scheme's built-in procedures. Altering any top-level binding that has not been introduced by a definition has an unspecified effect on the behavior of the built-in procedures.
A predicate is a procedure that always returns a boolean value (#t or #f). An equivalence predicate is the computational analogue of a mathematical equivalence relation (it is symmetric, reflexive, and transitive). Of the equivalence predicates described in this section, eq? is the finest or most discriminating, and equal? is the coarsest. Eqv? is slightly less discriminating than eq?.
The eqv? procedure defines a useful equivalence relation on objects. Briefly, it returns #t if obj1 and obj2 should normally be regarded as the same object. This relation is left slightly open to interpretation, but the following partial specification of eqv? holds for all implementations of Scheme.
The eqv? procedure returns #t if:
obj1 and obj2 are both #t or both #f.
obj1 and obj2 are both symbols and
(string=? (symbol->string obj1)
(symbol->string obj2))
===> #t
Note: This assumes that neither obj1 nor obj2 is an ``uninterned symbol'' as alluded to in section 6.3.3. This report does not presume to specify the behavior of eqv? on implementation-dependent extensions.
obj1 and obj2 are both numbers, are numerically equal (see =, section 6.2), and are either both exact or both inexact.
obj1 and obj2 are both characters and are the same character according to the char=? procedure (section 6.3.4).
both obj1 and obj2 are the empty list.
obj1 and obj2 are pairs, vectors, or strings that denote the same locations in the store (section 3.4).
obj1 and obj2 are procedures whose location tags are equal (section 4.1.4).
The eqv? procedure returns #f if:
obj1 and obj2 are of different types (section 3.2).
one of obj1 and obj2 is #t but the other is #f.
obj1 and obj2 are symbols but
(string=? (symbol->string obj1)
(symbol->string obj2))
===> #f
one of obj1 and obj2 is an exact number but the other is an inexact number.
obj1 and obj2 are numbers for which the = procedure returns #f.
obj1 and obj2 are characters for which the char=? procedure returns #f.
one of obj1 and obj2 is the empty list but the other is not.
obj1 and obj2 are pairs, vectors, or strings that denote distinct locations.
obj1 and obj2 are procedures that would behave differently (return different value(s) or have different side effects) for some arguments.
(eqv? 'a 'a) ===> #t
(eqv? 'a 'b) ===> #f
(eqv? 2 2) ===> #t
(eqv? '() '()) ===> #t
(eqv? 100000000 100000000) ===> #t
(eqv? (cons 1 2) (cons 1 2)) ===> #f
(eqv? (lambda () 1)
(lambda () 2)) ===> #f
(eqv? #f 'nil) ===> #f
(let ((p (lambda (x) x)))
(eqv? p p)) ===> #t
The following examples illustrate cases in which the above rules do not fully specify the behavior of eqv?. All that can be said about such cases is that the value returned by eqv? must be a boolean.
(eqv? "" "") ===> unspecified
(eqv? '#() '#()) ===> unspecified
(eqv? (lambda (x) x)
(lambda (x) x)) ===> unspecified
(eqv? (lambda (x) x)
(lambda (y) y)) ===> unspecified
The next set of examples shows the use of eqv? with procedures that have local state. Gen-counter must return a distinct procedure every time, since each procedure has its own internal counter. Gen-loser, however, returns equivalent procedures each time, since the local state does not affect the value or side effects of the procedures.
(define gen-counter
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) n))))
(let ((g (gen-counter)))
(eqv? g g)) ===> #t
(eqv? (gen-counter) (gen-counter))
===> #f
(define gen-loser
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) 27))))
(let ((g (gen-loser)))
(eqv? g g)) ===> #t
(eqv? (gen-loser) (gen-loser))
===> unspecified
(letrec ((f (lambda () (if (eqv? f g) 'both 'f)))
(g (lambda () (if (eqv? f g) 'both 'g))))
(eqv? f g))
===> unspecified
(letrec ((f (lambda () (if (eqv? f g) 'f 'both)))
(g (lambda () (if (eqv? f g) 'g 'both))))
(eqv? f g))
===> #f
Since it is an error to modify constant objects (those returned by literal expressions), implementations are permitted, though not required, to share structure between constants where appropriate. Thus the value of eqv? on constants is sometimes implementation-dependent.
(eqv? '(a) '(a)) ===> unspecified
(eqv? "a" "a") ===> unspecified
(eqv? '(b) (cdr '(a b))) ===> unspecified
(let ((x '(a)))
(eqv? x x)) ===> #t
Rationale: The above definition of eqv? allows implementations latitude in their treatment of procedures and literals: implementations are free either to detect or to fail to detect that two procedures or two literals are equivalent to each other, and can decide whether or not to merge representations of equivalent objects by using the same pointer or bit pattern to represent both.
Eq? is similar to eqv? except that in some cases it is capable of discerning distinctions finer than those detectable by eqv?.
Eq? and eqv? are guaranteed to have the same behavior on symbols, booleans, the empty list, pairs, procedures, and non-empty strings and vectors. Eq?'s behavior on numbers and characters is implementation-dependent, but it will always return either true or false, and will return true only when eqv? would also return true. Eq? may also behave differently from eqv? on empty vectors and empty strings.
(eq? 'a 'a) ===> #t
(eq? '(a) '(a)) ===> unspecified
(eq? (list 'a) (list 'a)) ===> #f
(eq? "a" "a") ===> unspecified
(eq? "" "") ===> unspecified
(eq? '() '()) ===> #t
(eq? 2 2) ===> unspecified
(eq? #\A #\A) ===> unspecified
(eq? car car) ===> #t
(let ((n (+ 2 3)))
(eq? n n)) ===> unspecified
(let ((x '(a)))
(eq? x x)) ===> #t
(let ((x '#()))
(eq? x x)) ===> #t
(let ((p (lambda (x) x)))
(eq? p p)) ===> #t
Rationale: It will usually be possible to implement eq? much more efficiently than eqv?, for example, as a simple pointer comparison instead of as some more complicated operation. One reason is that it may not be possible to compute eqv? of two numbers in constant time, whereas eq? implemented as pointer comparison will always finish in constant time. Eq? may be used like eqv? in applications using procedures to implement objects with state since it obeys the same constraints as eqv?.
Equal? recursively compares the contents of pairs, vectors, and strings, applying eqv? on other objects such as numbers and symbols. A rule of thumb is that objects are generally equal? if they print the same. Equal? may fail to terminate if its arguments are circular data structures.
(equal? 'a 'a) ===> #t
(equal? '(a) '(a)) ===> #t
(equal? '(a (b) c)
'(a (b) c)) ===> #t
(equal? "abc" "abc") ===> #t
(equal? 2 2) ===> #t
(equal? (make-vector 5 'a)
(make-vector 5 'a)) ===> #t
(equal? (lambda (x) x)
(lambda (y) y)) ===> unspecified
Numerical computation has traditionally been neglected by the Lisp community. Until Common Lisp there was no carefully thought out strategy for organizing numerical computation, and with the exception of the MacLisp system [20] little effort was made to execute numerical code efficiently. This report recognizes the excellent work of the Common Lisp committee and accepts many of their recommendations. In some ways this report simplifies and generalizes their proposals in a manner consistent with the purposes of Scheme.
It is important to distinguish between the mathematical numbers, the Scheme numbers that attempt to model them, the machine representations used to implement the Scheme numbers, and notations used to write numbers. This report uses the types number, complex, real, rational, and integer to refer to both mathematical numbers and Scheme numbers. Machine representations such as fixed point and floating point are referred to by names such as fixnum and flonum.
Mathematically, numbers may be arranged into a tower of subtypes in which each level is a subset of the level above it:
number
complex
real
rational
integer
For example, 3 is an integer. Therefore 3 is also a rational, a real, and a complex. The same is true of the Scheme numbers that model 3. For Scheme numbers, these types are defined by the predicates number?, complex?, real?, rational?, and integer?.
There is no simple relationship between a number's type and its representation inside a computer. Although most implementations of Scheme will offer at least two different representations of 3, these different representations denote the same integer.
Scheme's numerical operations treat numbers as abstract data, as independent of their representation as possible. Although an implementation of Scheme may use fixnum, flonum, and perhaps other representations for numbers, this should not be apparent to a casual programmer writing simple programs.
It is necessary, however, to distinguish between numbers that are represented exactly and those that may not be. For example, indexes into data structures must be known exactly, as must some polynomial coefficients in a symbolic algebra system. On the other hand, the results of measurements are inherently inexact, and irrational numbers may be approximated by rational and therefore inexact approximations. In order to catch uses of inexact numbers where exact numbers are required, Scheme explicitly distinguishes exact from inexact numbers. This distinction is orthogonal to the dimension of type.
Scheme numbers are either exact or inexact. A number is exact if it was written as an exact constant or was derived from exact numbers using only exact operations. A number is inexact if it was written as an inexact constant, if it was derived using inexact ingredients, or if it was derived using inexact operations. Thus inexactness is a contagious property of a number. If two implementations produce exact results for a computation that did not involve inexact intermediate results, the two ultimate results will be mathematically equivalent. This is generally not true of computations involving inexact numbers since approximate methods such as floating point arithmetic may be used, but it is the duty of each implementation to make the result as close as practical to the mathematically ideal result.
Rational operations such as + should always produce exact results when given exact arguments. If the operation is unable to produce an exact result, then it may either report the violation of an implementation restriction or it may silently coerce its result to an inexact value. See section 6.2.3.
With the exception of inexact->exact, the operations described in this section must generally return inexact results when given any inexact arguments. An operation may, however, return an exact result if it can prove that the value of the result is unaffected by the inexactness of its arguments. For example, multiplication of any number by an exact zero may produce an exact zero result, even if the other argument is inexact.
Implementations of Scheme are not required to implement the whole tower of subtypes given in section 6.2.1, but they must implement a coherent subset consistent with both the purposes of the implementation and the spirit of the Scheme language. For example, an implementation in which all numbers are real may still be quite useful.
Implementations may also support only a limited range of numbers of any type, subject to the requirements of this section. The supported range for exact numbers of any type may be different from the supported range for inexact numbers of that type. For example, an implementation that uses flonums to represent all its inexact real numbers may support a practically unbounded range of exact integers and rationals while limiting the range of inexact reals (and therefore the range of inexact integers and rationals) to the dynamic range of the flonum format. Furthermore the gaps between the representable inexact integers and rationals are likely to be very large in such an implementation as the limits of this range are approached.
An implementation of Scheme must support exact integers throughout the range of numbers that may be used for indexes of lists, vectors, and strings or that may result from computing the length of a list, vector, or string. The length, vector-length, and string-length procedures must return an exact integer, and it is an error to use anything but an exact integer as an index. Furthermore any integer constant within the index range, if expressed by an exact integer syntax, will indeed be read as an exact integer, regardless of any implementation restrictions that may apply outside this range. Finally, the procedures listed below will always return an exact integer result provided all their arguments are exact integers and the mathematically expected result is representable as an exact integer within the implementation:
+ - *
quotient remainder modulo
max min abs
numerator denominator gcd
lcm floor ceiling
truncate round rationalize
expt
Implementations are encouraged, but not required, to support exact integers and exact rationals of practically unlimited size and precision, and to implement the above procedures and the / procedure in such a way that they always return exact results when given exact arguments. If one of these procedures is unable to deliver an exact result when given exact arguments, then it may either report a violation of an implementation restriction or it may silently coerce its result to an inexact number. Such a coercion may cause an error later.
An implementation may use floating point and other approximate representation strategies for inexact numbers. This report recommends, but does not require, that the IEEE 32-bit and 64-bit floating point standards be followed by implementations that use flonum representations, and that implementations using other representations should match or exceed the precision achievable using these floating point standards [12].
In particular, implementations that use flonum representations must follow these rules: A flonum result must be represented with at least as much precision as is used to express any of the inexact arguments to that operation. It is desirable (but not required) for potentially inexact operations such as sqrt, when applied to exact arguments, to produce exact answers whenever possible (for example the square root of an exact 4 ought to be an exact 2). If, however, an exact number is operated upon so as to produce an inexact result (as by sqrt), and if the result is represented as a flonum, then the most precise flonum format available must be used; but if the result is represented in some other way then the representation must have at least as much precision as the most precise flonum format available.
Although Scheme allows a variety of written notations for numbers, any particular implementation may support only some of them. For example, an implementation in which all numbers are real need not support the rectangular and polar notations for complex numbers. If an implementation encounters an exact numerical constant that it cannot represent as an exact number, then it may either report a violation of an implementation restriction or it may silently represent the constant by an inexact number.
The syntax of the written representations for numbers is described formally in section 7.1.1. Note that case is not significant in numerical constants.
A number may be written in binary, octal, decimal, or hexadecimal by the use of a radix prefix. The radix prefixes are #b (binary), #o (octal), #d (decimal), and #x (hexadecimal). With no radix prefix, a number is assumed to be expressed in decimal.
A numerical constant may be specified to be either exact or inexact by a prefix. The prefixes are #e for exact, and #i for inexact. An exactness prefix may appear before or after any radix prefix that is used. If the written representation of a number has no exactness prefix, the constant may be either inexact or exact. It is inexact if it contains a decimal point, an exponent, or a ``#'' character in the place of a digit, otherwise it is exact. In systems with inexact numbers of varying precisions it may be useful to specify the precision of a constant. For this purpose, numerical constants may be written with an exponent marker that indicates the desired precision of the inexact representation. The letters s, f, d, and l specify the use of short, single, double, and long precision, respectively. (When fewer than four internal inexact representations exist, the four size specifications are mapped onto those available. For example, an implementation with two internal representations may map short and single together and long and double together.) In addition, the exponent marker e specifies the default precision for the implementation. The default precision has at least as much precision as double, but implementations may wish to allow this default to be set by the user.
3.14159265358979F0
Round to single --- 3.141593
0.6L0
Extend to long --- .600000000000000
The reader is referred to section 1.3.3 for a summary of the naming conventions used to specify restrictions on the types of arguments to numerical routines. The examples used in this section assume that any numerical constant written using an exact notation is indeed represented as an exact number. Some examples also assume that certain numerical constants written using an inexact notation can be represented without loss of accuracy; the inexact constants were chosen so that this is likely to be true in implementations that use flonums to represent inexact numbers.
These numerical type predicates can be applied to any kind of argument, including non-numbers. They return #t if the object is of the named type, and otherwise they return #f. In general, if a type predicate is true of a number then all higher type predicates are also true of that number. Consequently, if a type predicate is false of a number, then all lower type predicates are also false of that number. If z is an inexact complex number, then (real? z) is true if and only if (zero? (imag-part z)) is true. If x is an inexact real number, then (integer? x) is true if and only if (= x (round x)).
(complex? 3+4i) ===> #t
(complex? 3) ===> #t
(real? 3) ===> #t
(real? -2.5+0.0i) ===> #t
(real? #e1e10) ===> #t
(rational? 6/10) ===> #t
(rational? 6/3) ===> #t
(integer? 3+0i) ===> #t
(integer? 3.0) ===> #t
(integer? 8/4) ===> #t
Note: The behavior of these type predicates on inexact numbers is unreliable, since any inaccuracy may affect the result.
Note: In many implementations the rational? procedure will be the same as real?, and the complex? procedure will be the same as number?, but unusual implementations may be able to represent some irrational numbers exactly or may extend the number system to support some kind of non-complex numbers.
These numerical predicates provide tests for the exactness of a quantity. For any Scheme number, precisely one of these predicates is true.
These procedures return #t if their arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically nondecreasing, or monotonically nonincreasing.
These predicates are required to be transitive.
Note: The traditional implementations of these predicates in Lisp-like languages are not transitive.
Note: While it is not an error to compare inexact numbers using these predicates, the results may be unreliable because a small inaccuracy may affect the result; this is especially true of = and zero?. When in doubt, consult a numerical analyst.
These numerical predicates test a number for a particular property, returning #t or #f. See note above.
These procedures return the maximum or minimum of their arguments.
(max 3 4) ===> 4 ; exact
(max 3.9 4) ===> 4.0 ; inexact
Note: If any argument is inexact, then the result will also be inexact (unless the procedure can prove that the inaccuracy is not large enough to affect the result, which is possible only in unusual implementations). If min or max is used to compare numbers of mixed exactness, and the numerical value of the result cannot be represented as an inexact number without loss of accuracy, then the procedure may report a violation of an implementation restriction.
These procedures return the sum or product of their arguments.
(+ 3 4) ===> 7
(+ 3) ===> 3
(+) ===> 0
(* 4) ===> 4
(*) ===> 1
With two or more arguments, these procedures return the difference or
quotient of their arguments, associating to the left. With one argument,
however, they return the additive or multiplicative inverse of their argument.
(- 3 4) ===> -1
(- 3 4 5) ===> -6
(- 3) ===> -3
(/ 3 4 5) ===> 3/20
(/ 3) ===> 1/3
Abs returns the absolute value of its argument.
(abs -7) ===> 7
These procedures implement number-theoretic (integer)
division. n2 should be non-zero. All three procedures
return integers. If n1/n2 is an integer:
(quotient n1 n2) ===> n1/n2
(remainder n1 n2) ===> 0
(modulo n1 n2) ===> 0
From this we can conclude that for integers n1 and n2 with
n2 not equal to 0,
(= n1 (+ (* n2 (quotient n1 n2))
(remainder n1 n2)))
===> #t
(modulo 13 4) ===> 1
(remainder 13 4) ===> 1
(modulo -13 4) ===> 3
(remainder -13 4) ===> -1
(modulo 13 -4) ===> -3
(remainder 13 -4) ===> 1
(modulo -13 -4) ===> -1
(remainder -13 -4) ===> -1
(remainder -13 -4.0) ===> -1.0 ; inexact
These procedures return the greatest common divisor or least common
multiple of their arguments. The result is always non-negative.
(gcd 32 -36) ===> 4
(gcd) ===> 0
(lcm 32 -36) ===> 288
(lcm 32.0 -36) ===> 288.0 ; inexact
(lcm) ===> 1
These procedures return the numerator or denominator of their
argument; the result is computed as if the argument was represented as
a fraction in lowest terms. The denominator is always positive. The
denominator of 0 is defined to be 1.
(numerator (/ 6 4)) ===> 3
(denominator (/ 6 4)) ===> 2
(denominator
(exact->inexact (/ 6 4))) ===> 2.0
These procedures return integers. Floor returns the largest integer not larger than x. Ceiling returns the smallest integer not smaller than x. Truncate returns the integer closest to x whose absolute value is not larger than the absolute value of x. Round returns the closest integer to x, rounding to even when x is halfway between two integers.
Rationale: Round rounds to even for consistency with the default rounding mode specified by the IEEE floating point standard.
Note: If the argument to one of these procedures is inexact, then the result will also be inexact. If an exact value is needed, the result should be passed to the inexact->exact procedure.
(floor -4.3) ===> -5.0
(ceiling -4.3) ===> -4.0
(truncate -4.3) ===> -4.0
(round -4.3) ===> -4.0
(floor 3.5) ===> 3.0
(ceiling 3.5) ===> 4.0
(truncate 3.5) ===> 3.0
(round 3.5) ===> 4.0 ; inexact
(round 7/2) ===> 4 ; exact
(round 7) ===> 7
Rationalize returns the simplest rational number differing from x by no more than y. A rational number r1 is simpler than another rational number r2 if r1 = p1/q1 and r2 = p2/q2 (in lowest terms) and |p1| < |p2| and |q1| < |q2|. Thus 3/5 is simpler than 4/7. Although not all rationals are comparable in this ordering (consider 2/7 and 3/5) any interval contains a rational number that is simpler than every other rational number in that interval (the simpler 2/5 lies between 2/7 and 3/5). Note that 0 = 0/1 is the simplest rational of all.
(rationalize
(inexact->exact .3) 1/10) ===> 1/3 ; exact
(rationalize .3 1/10) ===> #i1/3 ; inexact
These procedures are part of every implementation that supports general real numbers; they compute the usual transcendental functions. Log computes the natural logarithm of z (not the base ten logarithm). Asin, acos, and atan compute arcsine (sin-1), arccosine (cos-1), and arctangent (tan-1), respectively. The two-argument variant of atan computes (angle (make-rectangular x y)) (see below), even in implementations that don't support general complex numbers.
In general, the mathematical functions log, arcsine, arccosine, and
arctangent are multiply defined.
The value of log z is defined to be the one whose imaginary
part lies in the range from - (exclusive) to (inclusive).
log 0 is undefined.
With log defined this way, the values of sin-1 z, cos-1 z,
and tan-1 z are according to the following formulæ:
| sin-1 z = - i log (i z + (1 - z2)1/2) |
| cos-1 z = / 2 - sin-1 z |
| tan-1 z = (log (1 + i z) - log (1 - i z)) / (2 i) |
The above specification follows [27], which in turn cites [19]; refer to these sources for more detailed discussion of branch cuts, boundary conditions, and implementation of these functions. When it is possible these procedures produce a real result from a real argument.
Returns the principal square root of z. The result will have either positive real part, or zero real part and non-negative imaginary part.
Returns z1 raised to the power z2. For z1 
0
| z1z2 = ez2 log z1 |
0z is 1 if z = 0 and 0 otherwise.
These procedures are part of every implementation that supports general complex numbers. Suppose x1, x2, x3, and x4 are real numbers and z is a complex number such that
| z = x1 + x2i = x3 · ei x4 |
Then
(make-rectangular x1 x2) ===> z
(make-polar x3 x4) ===> z
(real-part z) ===> x1
(imag-part z) ===> x2
(magnitude z) ===> |x3|
(angle z) ===> xangle
< xangle < with xangle = x4 + 2 n
for some integer n.
Rationale: Magnitude is the same as abs for a real argument, but abs must be present in all implementations, whereas magnitude need only be present in implementations that support general complex numbers.
Exact->inexact returns an inexact representation of z. The value returned is the inexact number that is numerically closest to the argument. If an exact argument has no reasonably close inexact equivalent, then a violation of an implementation restriction may be reported.
Inexact->exact returns an exact representation of z. The value returned is the exact number that is numerically closest to the argument. If an inexact argument has no reasonably close exact equivalent, then a violation of an implementation restriction may be reported.
These procedures implement the natural one-to-one correspondence between exact and inexact integers throughout an implementation-dependent range. See section 6.2.3.
Radix must be an exact integer, either 2, 8, 10, or 16. If omitted,
radix defaults to 10.
The procedure number->string takes a
number and a radix and returns as a string an external representation of
the given number in the given radix such that
(let ((number number)
(radix radix))
(eqv? number
(string->number (number->string number
radix)
radix)))
If z is inexact, the radix is 10, and the above expression can be satisfied by a result that contains a decimal point, then the result contains a decimal point and is expressed using the minimum number of digits (exclusive of exponent and trailing zeroes) needed to make the above expression true [3, 5]; otherwise the format of the result is unspecified.
The result returned by number->string never contains an explicit radix prefix.
Note: The error case can occur only when z is not a complex number or is a complex number with a non-rational real or imaginary part.
Rationale: If z is an inexact number represented using flonums, and the radix is 10, then the above expression is normally satisfied by a result containing a decimal point. The unspecified case allows for infinities, NaNs, and non-flonum representations.
Returns a number of the maximally precise representation expressed by the given string. Radix must be an exact integer, either 2, 8, 10, or 16. If supplied, radix is a default radix that may be overridden by an explicit radix prefix in string (e.g. "#o177"). If radix is not supplied, then the default radix is 10. If string is not a syntactically valid notation for a number, then string->number returns #f.
(string->number "100") ===> 100
(string->number "100" 16) ===> 256
(string->number "1e2") ===> 100.0
(string->number "15##") ===> 1500.0
Note: The domain of string->number may be restricted by implementations in the following ways. String->number is permitted to return #f whenever string contains an explicit radix prefix. If all numbers supported by an implementation are real, then string->number is permitted to return #f whenever string uses the polar or rectangular notations for complex numbers. If all numbers are integers, then string->number may return #f whenever the fractional notation is used. If all numbers are exact, then string->number may return #f whenever an exponent marker or explicit exactness prefix is used, or if a # appears in place of a digit. If all inexact numbers are integers, then string->number may return #f whenever a decimal point is used.
This section describes operations on some of Scheme's non-numeric data types: booleans, pairs, lists, symbols, characters, strings and vectors.
The standard boolean objects for true and false are written as #t and #f. What really matters, though, are the objects that the Scheme conditional expressions (if, cond, and, or, do) treat as true or false. The phrase ``a true value'' (or sometimes just ``true'') means any object treated as true by the conditional expressions, and the phrase ``a false value'' (or ``false'') means any object treated as false by the conditional expressions.
Of all the standard Scheme values, only #f counts as false in conditional expressions. Except for #f, all standard Scheme values, including #t, pairs, the empty list, symbols, numbers, strings, vectors, and procedures, count as true.
Note: Programmers accustomed to other dialects of Lisp should be aware that Scheme distinguishes both #f and the empty list from the symbol nil.
Boolean constants evaluate to themselves, so they do not need to be quoted in programs.
#t ===> #t
#f ===> #f
'#f ===> #f
Not returns #t if obj is false, and returns #f otherwise.
(not #t) ===> #f
(not 3) ===> #f
(not (list 3)) ===> #f
(not #f) ===> #t
(not '()) ===> #f
(not (list)) ===> #f
(not 'nil) ===> #f
Boolean? returns #t if obj is either #t or #f and returns #f otherwise.
(boolean? #f) ===> #t
(boolean? 0) ===> #f
(boolean? '()) ===> #f
A pair (sometimes called a dotted pair) is a record structure with two fields called the car and cdr fields (for historical reasons). Pairs are created by the procedure cons. The car and cdr fields are accessed by the procedures car and cdr. The car and cdr fields are assigned by the procedures set-car! and set-cdr!.
Pairs are used primarily to represent lists. A list can be defined recursively as either the empty list or a pair whose cdr is a list. More precisely, the set of lists is defined as the smallest set X such that
The empty list is in X.
If list is in X, then any pair whose cdr field contains list is also in X.
The objects in the car fields of successive pairs of a list are the elements of the list. For example, a two-element list is a pair whose car is the first element and whose cdr is a pair whose car is the second element and whose cdr is the empty list. The length of a list is the number of elements, which is the same as the number of pairs.
The empty list is a special object of its own type (it is not a pair); it has no elements and its length is zero.
Note: The above definitions imply that all lists have finite length and are terminated by the empty list.
The most general notation (external representation) for Scheme pairs is the ``dotted'' notation (c1 . c2) where c1 is the value of the car field and c2 is the value of the cdr field. For example (4 . 5) is a pair whose car is 4 and whose cdr is 5. Note that (4 . 5) is the external representation of a pair, not an expression that evaluates to a pair.
A more streamlined notation can be used for lists: the elements of the list are simply enclosed in parentheses and separated by spaces. The empty list is written () . For example,
(a b c d e)
and
(a . (b . (c . (d . (e . ())))))
are equivalent notations for a list of symbols.
A chain of pairs not ending in the empty list is called an improper list. Note that an improper list is not a list. The list and dotted notations can be combined to represent improper lists:
(a b c . d)
is equivalent to
(a . (b . (c . d)))
Whether a given pair is a list depends upon what is stored in the cdr field. When the set-cdr! procedure is used, an object can be a list one moment and not the next:
(define x (list 'a 'b 'c))
(define y x)
y ===> (a b c)
(list? y) ===> #t
(set-cdr! x 4) ===> unspecified
x ===> (a . 4)
(eqv? x y) ===> #t
y ===> (a . 4)
(list? y) ===> #f
(set-cdr! x x) ===> unspecified
(list? x) ===> #f
Within literal expressions and representations of objects read by the read procedure, the forms '<datum>, `<datum>, ,<datum>, and ,@<datum> denote two-element lists whose first elements are the symbols quote, quasiquote, unquote, and unquote-splicing, respectively. The second element in each case is <datum>. This convention is supported so that arbitrary Scheme programs may be represented as lists. That is, according to Scheme's grammar, every <expression> is also a <datum> (see section 7.1.2). Among other things, this permits the use of the read procedure to parse Scheme programs. See section 3.3.
Pair? returns #t if obj is a pair, and otherwise returns #f.
(pair? '(a . b)) ===> #t
(pair? '(a b c)) ===> #t
(pair? '()) ===> #f
(pair? '#(a b)) ===> #f
Returns a newly allocated pair whose car is obj1 and whose cdr is obj2. The pair is guaranteed to be different (in the sense of eqv?) from every existing object.
(cons 'a '()) ===> (a)
(cons '(a) '(b c d)) ===> ((a) b c d)
(cons "a" '(b c)) ===> ("a" b c)
(cons 'a 3) ===> (a . 3)
(cons '(a b) 'c) ===> ((a b) . c)
Returns the contents of the car field of pair. Note that it is an error to take the car of the empty list.
(car '(a b c)) ===> a
(car '((a) b c d)) ===> (a)
(car '(1 . 2)) ===> 1
(car '()) ===> error
Returns the contents of the cdr field of pair. Note that it is an error to take the cdr of the empty list.
(cdr '((a) b c d)) ===> (b c d)
(cdr '(1 . 2)) ===> 2
(cdr '()) ===> error
Stores obj in the car field of pair.
The value returned by set-car! is unspecified.
(define (f) (list 'not-a-constant-list))
(define (g) '(constant-list))
(set-car! (f) 3) ===> unspecified
(set-car! (g) 3) ===> error
Stores obj in the cdr field of pair. The value returned by set-cdr! is unspecified.
: These procedures are compositions of car and cdr, where for example caddr could be defined by
(define caddr (lambda (x) (car (cdr (cdr x))))).
Arbitrary compositions, up to four deep, are provided. There are twenty-eight of these procedures in all.
Returns #t if obj is the empty list, otherwise returns #f.
Returns #t if obj is a list, otherwise returns #f. By definition, all lists have finite length and are terminated by the empty list.
(list? '(a b c)) ===> #t
(list? '()) ===> #t
(list? '(a . b)) ===> #f
(let ((x (list 'a)))
(set-cdr! x x)
(list? x)) ===> #f
Returns a newly allocated list of its arguments.
(list 'a (+ 3 4) 'c) ===> (a 7 c)
(list) ===> ()
Returns the length of list.
(length '(a b c)) ===> 3
(length '(a (b) (c d e))) ===> 3
(length '()) ===> 0
Returns a list consisting of the elements of the first list followed by the elements of the other lists.
(append '(x) '(y)) ===> (x y)
(append '(a) '(b c d)) ===> (a b c d)
(append '(a (b)) '((c))) ===> (a (b) (c))
The resulting list is always newly allocated, except that it shares structure with the last list argument. The last argument may actually be any object; an improper list results if the last argument is not a proper list.
(append '(a b) '(c . d)) ===> (a b c . d)
(append '() 'a) ===> a
Returns a newly allocated list consisting of the elements of list in reverse order.
(reverse '(a b c)) ===> (c b a)
(reverse '(a (b c) d (e (f))))
===> ((e (f)) d (b c) a)
Returns the sublist of list obtained by omitting the first k elements. It is an error if list has fewer than k elements. List-tail could be defined by
(define list-tail
(lambda (x k)
(if (zero? k)
x
(list-tail (cdr x) (- k 1)))))
Returns the kth element of list. (This is the same as the car of (list-tail list k).) It is an error if list has fewer than k elements.
(list-ref '(a b c d) 2) ===> c
(list-ref '(a b c d)
(inexact->exact (round 1.8)))
===> c
These procedures return the first sublist of list whose car is obj, where the sublists of list are the non-empty lists returned by (list-tail list k) for k less than the length of list. If obj does not occur in list, then #f (not the empty list) is returned. Memq uses eq? to compare obj with the elements of list, while memv uses eqv? and member uses equal?.
(memq 'a '(a b c)) ===> (a b c)
(memq 'b '(a b c)) ===> (b c)
(memq 'a '(b c d)) ===> #f
(memq (list 'a) '(b (a) c)) ===> #f
(member (list 'a)
'(b (a) c)) ===> ((a) c)
(memq 101 '(100 101 102)) ===> unspecified
(memv 101 '(100 101 102)) ===> (101 102)
Alist (for ``association list'') must be a list of pairs. These procedures find the first pair in alist whose car field is obj, and returns that pair. If no pair in alist has obj as its car, then #f (not the empty list) is returned. Assq uses eq? to compare obj with the car fields of the pairs in alist, while assv uses eqv? and assoc uses equal?.
(define e '((a 1) (b 2) (c 3)))
(assq 'a e) ===> (a 1)
(assq 'b e) ===> (b 2)
(assq 'd e) ===> #f
(assq (list 'a) '(((a)) ((b)) ((c))))
===> #f
(assoc (list 'a) '(((a)) ((b)) ((c))))
===> ((a))
(assq 5 '((2 3) (5 7) (11 13)))
===> unspecified
(assv 5 '((2 3) (5 7) (11 13)))
===> (5 7)
Rationale: Although they are ordinarily used as predicates, memq, memv, member, assq, assv, and assoc do not have question marks in their names because they return useful values rather than just #t or #f.
Symbols are objects whose usefulness rests on the fact that two symbols are identical (in the sense of eqv?) if and only if their names are spelled the same way. This is exactly the property needed to represent identifiers in programs, and so most implementations of Scheme use them internally for that purpose. Symbols are useful for many other applications; for instance, they may be used the way enumerated values are used in Pascal.
The rules for writing a symbol are exactly the same as the rules for writing an identifier; see sections 2.1 and 7.1.1.
It is guaranteed that any symbol that has been returned as part of a literal expression, or read using the read procedure, and subsequently written out using the write procedure, will read back in as the identical symbol (in the sense of eqv?). The string->symbol procedure, however, can create symbols for which this write/read invariance may not hold because their names contain special characters or letters in the non-standard case.
Note: Some implementations of Scheme have a feature known as ``slashification'' in order to guarantee write/read invariance for all symbols, but historically the most important use of this feature has been to compensate for the lack of a string data type.Some implementations also have ``uninterned symbols'', which defeat write/read invariance even in implementations with slashification, and also generate exceptions to the rule that two symbols are the same if and only if their names are spelled the same.
Returns #t if obj is a symbol, otherwise returns #f.
(symbol? 'foo) ===> #t
(symbol? (car '(a b))) ===> #t
(symbol? "bar") ===> #f
(symbol? 'nil) ===> #t
(symbol? '()) ===> #f
(symbol? #f) ===> #f
Returns the name of symbol as a string. If the symbol was part of an object returned as the value of a literal expression (section 4.1.2) or by a call to the read procedure, and its name contains alphabetic characters, then the string returned will contain characters in the implementation's preferred standard case -- some implementations will prefer upper case, others lower case. If the symbol was returned by string->symbol, the case of characters in the string returned will be the same as the case in the string that was passed to string->symbol. It is an error to apply mutation procedures like string-set! to strings returned by this procedure.
The following examples assume that the implementation's standard case is lower case:
(symbol->string 'flying-fish)
===> "flying-fish"
(symbol->string 'Martin) ===> "martin"
(symbol->string
(string->symbol "Malvina"))
===> "Malvina"
Returns the symbol whose name is string. This procedure can create symbols with names containing special characters or letters in the non-standard case, but it is usually a bad idea to create such symbols because in some implementations of Scheme they cannot be read as themselves. See symbol->string.
The following examples assume that the implementation's standard case is lower case:
(eq? 'mISSISSIppi 'mississippi)
===> #t
(string->symbol "mISSISSIppi")
===> the symbol with name "mISSISSIppi"
(eq? 'bitBlt (string->symbol "bitBlt"))
===> #f
(eq? 'JollyWog
(string->symbol
(symbol->string 'JollyWog)))
===> #t
(string=? "K. Harper, M.D."
(symbol->string
(string->symbol "K. Harper, M.D.")))
===> #t
Characters are objects that represent printed characters such as letters and digits. Characters are written using the notation #\<character> or #\<character name>. For example:
|
Case is significant in #\<character>, but not in #\<character name>. If <character> in #\<character> is alphabetic, then the character following <character> must be a delimiter character such as a space or parenthesis. This rule resolves the ambiguous case where, for example, the sequence of characters ``#\space'' could be taken to be either a representation of the space character or a representation of the character ``#\s'' followed by a representation of the symbol ``pace.''
Characters written in the #\ notation are self-evaluating. That is, they do not have to be quoted in programs. Some of the procedures that operate on characters ignore the difference between upper case and lower case. The procedures that ignore case have ``-ci'' (for ``case insensitive'') embedded in their names.
Returns #t if obj is a character, otherwise returns #f.
These procedures impose a total ordering on the set of characters. It is guaranteed that under this ordering:
The upper case characters are in order. For example, (char<? #\A #\B) returns #t.
The lower case characters are in order. For example, (char<? #\a #\b) returns #t.
The digits are in order. For example, (char<? #\0 #\9) returns #t.
Either all the digits precede all the upper case letters, or vice versa.
Either all the digits precede all the lower case letters, or vice versa.
Some implementations may generalize these procedures to take more than two arguments, as with the corresponding numerical predicates.
These procedures are similar to char=? et cetera, but they treat upper case and lower case letters as the same. For example, (char-ci=? #\A #\a) returns #t. Some implementations may generalize these procedures to take more than two arguments, as with the corresponding numerical predicates.
These procedures return #t if their arguments are alphabetic, numeric, whitespace, upper case, or lower case characters, respectively, otherwise they return #f. The following remarks, which are specific to the ASCII character set, are intended only as a guide: The alphabetic characters are the 52 upper and lower case letters. The numeric characters are the ten decimal digits. The whitespace characters are space, tab, line feed, form feed, and carriage return.
Given a character, char->integer returns an exact integer representation of the character. Given an exact integer that is the image of a character under char->integer, integer->char returns that character. These procedures implement order-preserving isomorphisms between the set of characters under the char<=? ordering and some subset of the integers under the <= ordering. That is, if
(char<=? a b) ===> #t and (<= x y) ===> #t
and x and y are in the domain of integer->char, then
(<= (char->integer a)
(char->integer b)) ===> #t
(char<=? (integer->char x)
(integer->char y)) ===> #t
These procedures return a character char2 such that (char-ci=? char char2). In addition, if char is alphabetic, then the result of char-upcase is upper case and the result of char-downcase is lower case.
Strings are sequences of characters. Strings are written as sequences of characters enclosed within doublequotes ("). A doublequote can be written inside a string only by escaping it with a backslash (\), as in
"The word \"recursion\" has many meanings."
A backslash can be written inside a string only by escaping it with another backslash. Scheme does not specify the effect of a backslash within a string that is not followed by a doublequote or backslash.
A string constant may continue from one line to the next, but the exact contents of such a string are unspecified. The length of a string is the number of characters that it contains. This number is an exact, non-negative integer that is fixed when the string is created. The valid indexes of a string are the exact non-negative integers less than the length of the string. The first character of a string has index 0, the second has index 1, and so on.
In phrases such as ``the characters of string beginning with index start and ending with index end,'' it is understood that the index start is inclusive and the index end is exclusive. Thus if start and end are the same index, a null substring is referred to, and if start is zero and end is the length of string, then the entire string is referred to.
Some of the procedures that operate on strings ignore the difference between upper and lower case. The versions that ignore case have ``-ci'' (for ``case insensitive'') embedded in their names.
Returns #t if obj is a string, otherwise returns #f.
Make-string returns a newly allocated string of length k. If char is given, then all elements of the string are initialized to char, otherwise the contents of the string are unspecified.
Returns a newly allocated string composed of the arguments.
Returns the number of characters in the given string.
k must be a valid index of string. String-ref returns character k of string using zero-origin indexing.
k must be a valid index of string.
String-set! stores char in element k of string
and returns an unspecified value.
(define (f) (make-string 3 #\*))
(define (g) "***")
(string-set! (f) 0 #\?) ===> unspecified
(string-set! (g) 0 #\?) ===> error
(string-set! (symbol->string 'immutable)
0
#\?) ===> error
Returns #t if the two strings are the same length and contain the same characters in the same positions, otherwise returns #f. String-ci=? treats upper and lower case letters as though they were the same character, but string=? treats upper and lower case as distinct characters.
These procedures are the lexicographic extensions to strings of the corresponding orderings on characters. For example, string<? is the lexicographic ordering on strings induced by the ordering char<? on characters. If two strings differ in length but are the same up to the length of the shorter string, the shorter string is considered to be lexicographically less than the longer string.
Implementations may generalize these and the string=? and string-ci=? procedures to take more than two arguments, as with the corresponding numerical predicates.
String must be a string, and start and end must be exact integers satisfying
| 0 < start < end < (string-length string). |
Substring returns a newly allocated string formed from the characters of string beginning with index start (inclusive) and ending with index end (exclusive).
Returns a newly allocated string whose characters form the concatenation of the given strings.
String->list returns a newly allocated list of the characters that make up the given string. List->string returns a newly allocated string formed from the characters in the list list, which must be a list of characters. String->list and list->string are inverses so far as equal? is concerned.
Returns a newly allocated copy of the given string.
Stores char in every element of the given string and returns an unspecified value.
Vectors are heterogenous structures whose elements are indexed by integers. A vector typically occupies less space than a list of the same length, and the average time required to access a randomly chosen element is typically less for the vector than for the list.
The length of a vector is the number of elements that it contains. This number is a non-negative integer that is fixed when the vector is created. The valid indexes of a vector are the exact non-negative integers less than the length of the vector. The first element in a vector is indexed by zero, and the last element is indexed by one less than the length of the vector.
Vectors are written using the notation #(obj ...). For example, a vector of length 3 containing the number zero in element 0, the list (2 2 2 2) in element 1, and the string "Anna" in element 2 can be written as following:
#(0 (2 2 2 2) "Anna")
Note that this is the external representation of a vector, not an expression evaluating to a vector. Like list constants, vector constants must be quoted:
'#(0 (2 2 2 2) "Anna")
===> #(0 (2 2 2 2) "Anna")
Returns #t if obj is a vector, otherwise returns #f.
Returns a newly allocated vector of k elements. If a second argument is given, then each element is initialized to fill. Otherwise the initial contents of each element is unspecified.
Returns a newly allocated vector whose elements contain the given arguments. Analogous to list.
(vector 'a 'b 'c) ===> #(a b c)
Returns the number of elements in vector as an exact integer.
k must be a valid index of vector. Vector-ref returns the contents of element k of vector.
(vector-ref '#(1 1 2 3 5 8 13 21)
5)
===> 8
(vector-ref '#(1 1 2 3 5 8 13 21)
(let ((i (round (* 2 (acos -1)))))
(if (inexact? i)
(inexact->exact i)
i)))
===> 13
k must be a valid index of vector.
Vector-set! stores obj in element k of vector.
The value returned by vector-set! is unspecified.
(let ((vec (vector 0 '(2 2 2 2) "Anna")))
(vector-set! vec 1 '("Sue" "Sue"))
vec)
===> #(0 ("Sue" "Sue") "Anna")
(vector-set! '#(0 1 2) 1 "doe")
===> error ; constant vector
Vector->list returns a newly allocated list of the objects contained in the elements of vector. List->vector returns a newly created vector initialized to the elemen