"Factor attempts to preserve natural mathematical semantics for numbers. Multiplying two large integers never results in overflow, and dividing two integers yields an exact ratio. Floating point numbers are also supported, along with complex numbers."
"Floating point division never raises an error if the denominator is zero. This means that if at least one of the two inputs to " { $link / } ", " { $link /f } " or " { $link mod } " is a float, the result will be a floating point infinity or not a number value."
"The behavior of integer division is hardware specific. On x86 processors, " { $link /i } " and " { $link mod } " raise an error if both inputs are integers and the denominator is zero. On PowerPC, integer division by zero yields a result of zero."
"On the other hand, the " { $link / } " word, when given integer arguments, implements a much more expensive division algorithm which always yields an exact rational answer, and this word always tests for division by zero explicitly." ;
"Math operations obey certain numerical upgrade rules. If one of the inputs is a bignum and the other is a fixnum, the latter is first coerced to a bignum; if one of the inputs is a float, the other is coerced to a float."
"Real numbers (but not complex numbers) can be ordered:"
{ $subsection < }
{ $subsection <= }
{ $subsection > }
{ $subsection >= } ;
ARTICLE: "integers" "Integers"
{ $subsection integer }
"Integers come in two varieties -- fixnums and bignums. Fixnums fit in a machine word and are faster to manipulate; if the result of a fixnum operation is too large to fit in a fixnum, the result is upgraded to a bignum. Here is an example where two fixnums are multiplied yielding a bignum:"
"The " { $link . } " word prints numbers in decimal. A set of words in the " { $vocab-link "prettyprint" } " vocabulary is provided to print integers using another base."
"There are two ways of looking at an integer -- as an abstract mathematical entity, or as a string of bits. The latter representation motivates " { $emphasis "bitwise operations" } "."
"Some applications, such as binary communication protocols and assemblers, need to construct integers from elaborate bit field specifications. Hand-coding this using " { $link shift } " and " { $link bitor } " results in repetitive code. A higher-level facility exists to factor out this repetition:"
"When we add, subtract or multiply any two integers, the result is always an integer. However, dividing a numerator by a denominator that is not an integral divisor of the denominator yields a ratio:"
{ $example "1210 11 / ." "110" }
{ $example "100 330 / ." "10/33" }
"Ratios are printed and can be input literally in the form above. Ratios are always reduced to lowest terms by factoring out the greatest common divisor of the numerator and denominator. A ratio with a denominator of 1 becomes an integer. Division with a denominator of 0 throws an error."
"Rational numbers represent " { $emphasis "exact" } " quantities. On the other hand, a floating point number is an " { $emphasis "approximation" } ". While rationals can grow to any required precision, floating point numbers are fixed-width, and manipulating them is usually faster than manipulating ratios or bignums (but slower than manipulating fixnums). Floating point numbers are often used to represent irrational numbers, which have no exact representation as a ratio of two integers."
"Introducing a floating point number in a computation forces the result to be expressed in floating point."
{ $example "5/4 1/2 + ." "7/4" }
{ $example "5/4 0.5 + ." "1.75" }
"Integers and rationals can be converted to floats:"
{ $subsection >float }
"Two real numbers can be divided yielding a float result:"
{ $subsection /f }
"Floating point numbers are represented internally in IEEE 754 double-precision format. This internal representation can be accessed for advanced operations and input/output purposes."
"Complex numbers arise as solutions to quadratic equations whose graph does not intersect the " { $emphasis "x" } " axis. Their literal syntax is covered in " { $link "syntax-complex-numbers" } "."
"Unlike math, where all real numbers are also complex numbers, Factor only considers a number to be a complex number if its imaginary part is non-zero. However, complex number operations are fully supported for real numbers; they are treated as having an imaginary part of zero."
ARTICLE: "number-strings" "Converting between numbers and strings"
"These words only convert between real numbers and strings. Complex numbers are constructed by the parser (" { $link "parser" } ") and printed by the prettyprinter (" { $link "prettyprint" } ")."
"Note that only integers can be converted to and from strings using a representation other than base 10. Calling a word such as " { $link >oct } " on a float will give a result in base 10."
"Interval arithmetic is performed on ranges of real numbers, rather than exact values. It is used by the Factor compiler to convert arbitrary-precision arithmetic to machine arithmetic, by inferring bounds for integer calculations."
