Numerical operations

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 a 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)).
The numbers +inf.0, -inf.0, and +nan.0 are real but not rational.
Note: The behavior of these type predicates on inexact numbers is unreliable, since any inaccuracy might affect the result.
Note: In many implementations the complex? procedure will be the same as number?, but unusual implementations may 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.

Returns #t if z is both exact and an integer; otherwise returns #f.

The finite? procedure returns #t on all real numbers except +inf.0, -inf.0, and +nan.0, and on complex numbers if their real and imaginary parts are both finite. Otherwise it returns #f.

The infinite? procedure returns #t on the real numbers +inf.0 and -inf.0, and on complex numbers if their real or imaginary parts or both are infinite. Otherwise it returns #f.

The nan? procedure returns #t on +nan.0, and on complex numbers if their real or imaginary parts or both are +nan.0. Otherwise it returns #f.

These procedures return #t if their arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically non-decreasing, or monotonically non-increasing, and #f otherwise. If any of the arguments are +nan.0, all the predicates return #f. They do not distinguish between inexact zero and inexact negative zero.
These predicates are transitive.
Note: While it is not an error to compare inexact numbers using these predicates, the results are unreliable because a small inaccuracy can 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.

These procedures return the maximum or minimum of their arguments.
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.

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.
It is an error if any argument of / other than the first is an exact zero. If the first argument is an exact zero, an implementation may return an exact zero unless one of the other arguments is a NaN.

The abs procedure returns the absolute value of its argument.

These procedures implement number-theoretic (integer) division. It is an error if n2 is zero. The procedures ending in / return two integers; the other procedures return an integer. All the procedures compute a quotient nq and remainder nr such that n1 = n2nq + nr. For each of the division operators, there are three procedures defined as follows: Examples:

The quotient and remainder procedures are equivalent to truncate-quotient and truncate-remainder, respectively, and modulo is equivalent to floor-remainder.

These procedures return the greatest common divisor or least common multiple of their arguments. The result is always non-negative.

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.

These procedures return integers. The floor procedure returns the largest integer not larger than x. The ceiling procedure 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, and round returns the closest integer to x, rounding to even when x is halfway between two integers.
Rationale: The round procedure rounds to even for consistency with the default rounding mode specified by the IEEE 754 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 can be passed to the exact procedure. If the argu- ment is infinite or a NaN, then it is returned.

The rationalize procedure returns the simplest rational number differing from x by no more than y.

These procedures compute the usual transcendental functions.
The log procedure computes the natural logarithm of z (not the base ten logarithm) if a single argument is given, or the base-z2 logarithm of z1 if two arguments are given.

Returns the square of z. This is equivalent to (* z z).

Returns the principal square root of z. The result will have either a positive real part, or a zero real part and a non-negative imaginary part.

Returns two non-negative exact integers s and r where k = s2 + r and k < (s + 1)2.

Returns z1 raised to the power z2.

Let x1, x2, x3, and x4 be real numbers and z be a complex number such that z = x1 + x2i = x3 * e ^ (ix4) Then all of are true.
The make-polar procedure may return an inexact complex number even if its arguments are exact. The real-part and imag-part procedures may return exact real numbers when applied to an inexact complex number if the corresponding argument passed to make-rectangular was exact.
Rationale: The magnitude procedure is the same as abs for a real argument, but abs is in the base library, whereas magnitude is in the optional complex library.

The procedure inexact returns an inexact representation of z. The value returned is the inexact number that is numerically closest to the argument. For inexact arguments, the result is the same as the argument. For exact complex numbers, the result is a complex number whose real and imaginary parts are the result of applying inexact to the real and imaginary parts of the argument, respectively. If an exact argument has no reasonably close inexact equivalent (in the sense of ==>, then a violation of an implementation restriction may be reported.
The procedure exact returns an exact representation of z. The value returned is the exact number that is numerically closest to the argument. For exact arguments, the result is the same as the argument. For inexact nonintegral real arguments, the implementation may return a rational approximation, or may report an implementation violation. For inexact complex arguments, the result is a complex number whose real and imaginary parts are the result of applying exact to the real and imaginary parts of the argument, respectively. If an inexact argument has no reasonably close exact equivalent, (in the sense of ==>, 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.
Note: These procedures were known in R5RS as exact->inexact and inexact->exact, respectively, but they have always accepted arguments of any exactness. The new names are clearer and shorter, as well as being compatible with R6RS.