@cindex numbers
GNU Emacs supports two numeric data types: @dfn{integers} and
-@dfn{floating point numbers}. Integers are whole numbers such as
-@minus{}3, 0, 7, 13, and 511. Their values are exact. Floating-point
-numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
-2.71828. They can also be expressed in exponential notation: 1.5e2
-equals 150; in this example, @samp{e2} stands for ten to the second
-power, and that is multiplied by 1.5. Floating point values are not
-exact; they have a fixed, limited amount of precision.
+@dfn{floating-point numbers}. Integers are whole numbers such as
+@minus{}3, 0, 7, 13, and 511. Floating-point numbers are numbers with
+fractional parts, such as @minus{}4.5, 0.0, and 2.71828. They can
+also be expressed in exponential notation: @samp{1.5e2} is the same as
+@samp{150.0}; here, @samp{e2} stands for ten to the second power, and
+that is multiplied by 1.5. Integer computations are exact, though
+they may overflow. Floating-point computations often involve rounding
+errors, as the numbers have a fixed amount of precision.
@menu
* Integer Basics:: Representation and range of integers.
@section Integer Basics
The range of values for an integer depends on the machine. The
-minimum range is @minus{}536870912 to 536870911 (30 bits; i.e.,
+minimum range is @minus{}536,870,912 to 536,870,911 (30 bits; i.e.,
@ifnottex
@minus{}2**29
@end ifnottex
1. ; @r{The integer 1.}
+1 ; @r{Also the integer 1.}
-1 ; @r{The integer @minus{}1.}
- 1073741825 ; @r{The floating point number 1073741825.0.}
+ 9000000000000000000
+ ; @r{The floating-point number 9e18.}
0 ; @r{The integer 0.}
-0 ; @r{The integer 0.}
@end example
@cindex largest Lisp integer
@cindex maximum Lisp integer
@defvar most-positive-fixnum
-The value of this variable is the largest integer that Emacs Lisp
-can handle.
+The value of this variable is the largest integer that Emacs Lisp can
+handle. Typical values are
+@ifnottex
+2**29 @minus{} 1
+@end ifnottex
+@tex
+@math{2^{29}-1}
+@end tex
+on 32-bit and
+@ifnottex
+2**61 @minus{} 1
+@end ifnottex
+@tex
+@math{2^{61}-1}
+@end tex
+on 64-bit platforms.
@end defvar
@cindex smallest Lisp integer
@cindex minimum Lisp integer
@defvar most-negative-fixnum
The value of this variable is the smallest integer that Emacs Lisp can
-handle. It is negative.
+handle. It is negative. Typical values are
+@ifnottex
+@minus{}2**29
+@end ifnottex
+@tex
+@math{-2^{29}}
+@end tex
+on 32-bit and
+@ifnottex
+@minus{}2**61
+@end ifnottex
+@tex
+@math{-2^{61}}
+@end tex
+on 64-bit platforms.
@end defvar
In Emacs Lisp, text characters are represented by integers. Any
@cindex @acronym{IEEE} floating point
Floating-point numbers are useful for representing numbers that are
-not integral. The precise range of floating-point numbers is
-machine-specific; it is the same as the range of the C data type
-@code{double} on the machine you are using. Emacs uses the
-@acronym{IEEE} floating-point standard, which is supported by all
-modern computers.
+not integral. The range of floating-point numbers is
+the same as the range of the C data type @code{double} on the machine
+you are using. On all computers currently supported by Emacs, this is
+double-precision @acronym{IEEE} floating point.
The read syntax for floating-point numbers requires either a decimal
-point (with at least one digit following), an exponent, or both. For
-example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and
-@samp{.15e4} are five ways of writing a floating-point number whose
-value is 1500. They are all equivalent. You can also use a minus
-sign to write negative floating-point numbers, as in @samp{-1.0}.
-
- Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero (with
-respect to @code{equal} and @code{=}), even though the two are
-distinguishable in the @acronym{IEEE} floating-point standard.
+point, an exponent, or both. Optional signs (@samp{+} or @samp{-})
+precede the number and its exponent. For example, @samp{1500.0},
+@samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
+five ways of writing a floating-point number whose value is 1500.
+They are all equivalent. Like Common Lisp, Emacs Lisp requires at
+least one digit after any decimal point in a floating-point number;
+@samp{1500.} is an integer, not a floating-point number.
+
+ Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
+with respect to @code{equal} and @code{=}. This follows the
+@acronym{IEEE} floating-point standard, which says @code{-0.0} and
+@code{0.0} are numerically equal even though other operations can
+distinguish them.
@cindex positive infinity
@cindex negative infinity
infinity and negative infinity as floating-point values. It also
provides for a class of values called NaN or ``not-a-number'';
numerical functions return such values in cases where there is no
-correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@. (NaN
-values can also carry a sign, but for practical purposes there's no
-significant difference between different NaN values in Emacs Lisp.)
-
-When a function is documented to return a NaN, it returns an
-implementation-defined value when Emacs is running on one of the
-now-rare platforms that do not use @acronym{IEEE} floating point. For
-example, @code{(log -1.0)} typically returns a NaN, but on
-non-@acronym{IEEE} platforms it returns an implementation-defined
-value.
+correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@.
+Although NaN values carry a sign, for practical purposes there is no other
+significant difference between different NaN values in Emacs Lisp.
-Here are the read syntaxes for these special floating-point values:
+Here are read syntaxes for these special floating-point values:
@table @asis
-@item positive infinity
-@samp{1.0e+INF}
-@item negative infinity
-@samp{-1.0e+INF}
-@item Not-a-number
-@samp{0.0e+NaN} or @samp{-0.0e+NaN}.
+@item infinity
+@samp{1.0e+INF} and @samp{-1.0e+INF}
+@item not-a-number
+@samp{0.0e+NaN} and @samp{-0.0e+NaN}
@end table
-@defun isnan number
-This predicate tests whether its argument is NaN, and returns @code{t}
-if so, @code{nil} otherwise. The argument must be a number.
-@end defun
-
- The following functions are specialized for handling floating point
+ The following functions are specialized for handling floating-point
numbers:
-@defun frexp x
-This function returns a cons cell @code{(@var{sig} . @var{exp})},
-where @var{sig} and @var{exp} are respectively the significand and
-exponent of the floating point number @var{x}:
+@defun isnan x
+This predicate returns @code{t} if its floating-point argument is a NaN,
+@code{nil} otherwise.
+@end defun
-@smallexample
-@var{x} = @var{sig} * 2^@var{exp}
-@end smallexample
+@defun frexp x
+This function returns a cons cell @code{(@var{s} . @var{e})},
+where @var{s} and @var{e} are respectively the significand and
+exponent of the floating-point number @var{x}.
-@var{sig} is a floating point number between 0.5 (inclusive) and 1.0
-(exclusive). If @var{x} is zero, the return value is @code{(0 . 0)}.
+If @var{x} is finite, @var{s} is a floating-point number between 0.5
+(inclusive) and 1.0 (exclusive), @var{e} is an integer, and
+@ifnottex
+@var{x} = @var{s} * 2**@var{e}.
+@end ifnottex
+@tex
+@math{x = s 2^e}.
+@end tex
+If @var{x} is zero or infinity, @var{s} is the same as @var{x}.
+If @var{x} is a NaN, @var{s} is also a NaN.
+If @var{x} is zero, @var{e} is 0.
@end defun
@defun ldexp sig &optional exp
-This function returns a floating point number corresponding to the
+This function returns a floating-point number corresponding to the
significand @var{sig} and exponent @var{exp}.
@end defun
@defun copysign x1 x2
This function copies the sign of @var{x2} to the value of @var{x1},
-and returns the result. @var{x1} and @var{x2} must be floating point
-numbers.
+and returns the result. @var{x1} and @var{x2} must be floating point.
@end defun
@defun logb number
returns @code{t} if so, @code{nil} otherwise. 0 is considered
non-negative.
-@findex wholenump number
-This is a synonym for @code{natnump}.
+@findex wholenump
+@code{wholenump} is a synonym for @code{natnump}.
@end defun
@defun zerop number
floating-point arguments, and returns a floating-point number if any
argument is floating point.
- It is important to note that in Emacs Lisp, arithmetic functions
-do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to
+ Emacs Lisp arithmetic functions do not check for integer overflow.
+Thus @code{(1+ 536870911)} may evaluate to
@minus{}536870912, depending on your hardware.
@defun 1+ number-or-marker
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