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See section Entry Format, 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.(3)
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.
These numerical predicates provide tests for the exactness of a quantity. For any Scheme number, precisely one of these predicates is true.
procedure+: exact-integer? object
procedure+: exact-nonnegative-integer? object
procedure+: exact-rational? object
These procedures test for some very common types of numbers. These tests could be written in terms of simpler predicates, but are more efficient.
These procedures return #t if their arguments are (respectively):
equal, monotonically increasing, monotonically decreasing, monotonically
nondecreasing, or monotonically nonincreasing.
These predicates are transitive. Note that 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 regarding inexact
numbers.
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.(4)
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
(1+ z) is equivalent to (+ z 1); (-1+ z) is
equivalent to (- z 1).
abs returns the magnitude of its argument.
(abs -7) => 7
These procedures implement number-theoretic (integer) division: for positive integers n1 and n2, if n3 and n4 are integers such that then
(quotient n1 n2) => n3 (remainder n1 n2) => n4 (modulo n1 n2) => n4
For integers n1 and n2 with n2 not equal to 0,
(= n1
(+ (* n2 (quotient n1 n2))
(remainder n1 n2)))
=> #t
provided all numbers involved in that computation are exact.
The value returned by quotient always has the sign of the product
of its arguments. remainder and modulo differ on negative
arguments -- the remainder always has the sign of the dividend,
the modulo always has the sign of the divisor:
(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
Note that quotient is the same as integer-truncate.
procedure+: integer-floor n1 n2
procedure+: integer-ceiling n1 n2
procedure+: integer-truncate n1 n2
procedure+: integer-round n1 n2
These procedures combine integer division with rounding. For example, the following are equivalent:
(integer-floor n1 n2) (floor (/ n1 n2))
However, the former is faster and does not produce an intermediate result.
Note that integer-truncate is the same as quotient.
procedure+: integer-divide n1 n2
procedure+: integer-divide-quotient qr
procedure+: integer-divide-remainder qr
integer-divide is equivalent to performing both quotient
and remainder at once. The result of integer-divide is an
object with two components; the procedures
integer-divide-quotient and integer-divide-remainder
select those components. These procedures are useful when both the
quotient and remainder are needed; often computing both of these numbers
simultaneously is much faster than computing them separately.
For example, the following are equivalent:
(lambda (n d)
(cons (quotient n d)
(remainder n d)))
(lambda (n d)
(let ((qr (integer-divide n d)))
(cons (integer-divide-quotient qr)
(integer-divide-remainder qr))))
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 rounding
modes required 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 (or use one of
the procedures below).
(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
These procedures are similar to the preceding procedures except that they always return an exact result. For example, the following are equivalent
(floor->exact x) (inexact->exact (floor x))
except that the former is faster and has fewer range restrictions.
procedure+: rationalize->exact x y
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 (both
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
rationalize->exact is similar to rationalize except that
it always returns an exact result.
procedure+: simplest-rational x y
procedure+: simplest-exact-rational x y
simplest-rational returns the simplest rational number between
x and y inclusive; simplest-exact-rational is similar
except that it always returns an exact result.
These procedures implement the same functionality as rationalize
and rationalize->exact, except that they specify the input range
by its endpoints; rationalize specifies the range by its center
point and its (half-) width.
These procedures compute the usual transcendental functions. log
computes the natural logarithm of z (not the base ten logarithm).
asin, acos, and atan compute arcsine, arccosine,
and arctangent, respectively. The two-argument variant of atan
computes (angle (make-rectangular x y)) (see
below).
In general, the mathematical functions log, arcsine, arccosine, and arctangent are multiply defined. For nonzero real x, the value of log x is defined to be the one whose imaginary part lies in the range minus pi (exclusive) to pi (inclusive). log 0 is undefined. The value of log z when z is complex is defined according to the formula With log defined this way, the values of arcsine, arccosine, and arctangent are according to the following formulae: The above specification follows Common Lisp: the Language, which in turn cites Principal Values and Branch Cuts in Complex APL; 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:
procedure: make-rectangular x1 x2
Suppose x1, x2, x3, and x4 are real numbers and
z is a complex number such that
Then make-rectangular and make-polar return z,
real-part returns x1, imag-part returns x2,
magnitude returns x3, and angle returns x4.
In the case of angle, whose value is not uniquely determined by
the preceding rule, the value returned will be the one in the range
minus pi (exclusive) to pi (inclusive).
conjugate returns the complex conjugate of z.
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; MIT Scheme signals an error of type
condition-type:bad-range-argument in this case.
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; in MIT Scheme this case does not occur because all inexact
numbers are representable as exact numbers.
These procedures implement the natural one-to-one correspondence between exact and inexact integers throughout an implementation-dependent range. See section Implementation restrictions.
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