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GNU Emacs supports two numeric data types: integers and
floating point numbers. Integers are whole numbers such as
-3, 0, 7, 13, and 511. Their values are exact. Floating point
numbers are numbers with fractional parts, such as -4.5, 0.0, or
2.71828. They can also be expressed in an exponential notation as well:
thus, 1.5e2 equals 150; in this example, ``e2'` stands for ten to the
second power, and is multiplied by 1.5. Floating point values are not
exact; they have a fixed, limited amount of precision.

Support for floating point numbers is a new feature in Emacs 19, and it is controlled by a separate compilation option, so you may encounter a site where Emacs does not support them.

The range of values for an integer depends on the machine. The range is -8388608 to 8388607 (24 bits; i.e., to ) on most machines, but on others it is -16777216 to 16777215 (25 bits), or -33554432 to 33554431 (26 bits). All of the examples shown below assume an integer has 24 bits.

The Lisp reader reads numbers as a sequence of digits with an optional sign.

1 ; The integer 1. +1 ; Also the integer 1. -1 ; The integer -1. 16777217 ; Also the integer 1, due to overflow. 0 ; The number 0. -0 ; The number 0. 1. ; The integer 1.

To understand how various functions work on integers, especially the bitwise operators (see section Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.

In 24 bit binary, the decimal integer 5 looks like this:

0000 0000 0000 0000 0000 0101

(We have inserted spaces between groups of 4 bits, and two spaces between groups of 8 bits, to make the binary integer easier to read.)

The integer -1 looks like this:

1111 1111 1111 1111 1111 1111

-1 is represented as 24 ones. (This is called two's complement notation.)

The negative integer, -5, is creating by subtracting 4 from -1. In binary, the decimal integer 4 is 100. Consequently, -5 looks like this:

1111 1111 1111 1111 1111 1011

In this implementation, the largest 24 bit binary integer is the decimal integer 8,388,607. In binary, this number looks like this:

0111 1111 1111 1111 1111 1111

Since the arithmetic functions do not check whether integers go outside their range, when you add 1 to 8,388,607, the value is negative integer -8,388,608:

(+ 1 8388607) => -8388608 => 1000 0000 0000 0000 0000 0000

Many of the following functions accept markers for arguments as well
as integers. (See section Markers.) More precisely, the actual parameters
to such functions may be either integers or markers, which is why we
often give these parameters the name `int-or-marker`. When the
actual parameter is a marker, the position value of the marker is used
and the buffer of the marker is ignored.

Emacs version 19 supports floating point numbers, if compiled with the
macro `LISP_FLOAT_TYPE`

defined. The precise range of floating
point numbers is machine-specific; it is the same as the range of the C
data type `double`

on the machine in question.

The printed representation for floating point numbers requires either
a decimal point (with at least one digit following), an exponent, or
both. For example, ``1500.0'`, ``15e2'`, ``15.0e2'`,
``1.5e3'`, and ``.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
``-1.0'`.

You can use `logb`

to extract the binary exponent of a floating
point number (or estimate the logarithm of an integer):

This function returns the binary exponent of `number`. More
precisely, the value is the logarithm of `number` base 2, rounded
down to an integer.

The functions in this section test whether the argument is a number or
whether it is a certain sort of number. The functions `integerp`

and `floatp`

can take any type of Lisp object as argument (the
predicates would not be of much use otherwise); but the `zerop`

predicate requires a number as its argument. See also
`integer-or-marker-p`

and `number-or-marker-p`

, in
section Predicates on Markers.

This predicate tests whether its argument is a floating point
number and returns `t`

if so, `nil`

otherwise.

`floatp`

does not exist in Emacs versions 18 and earlier.

This predicate tests whether its argument is an integer, and returns
`t`

if so, `nil`

otherwise.

This predicate tests whether its argument is a number (either integer or
floating point), and returns `t`

if so, `nil`

otherwise.

The `natnump`

predicate (whose name comes from the phrase
"natural-number-p") tests to see whether its argument is a nonnegative
integer, and returns `t`

if so, `nil`

otherwise. 0 is
considered non-negative.

Markers are not converted to integers, hence `natnump`

of a marker
is always `nil`

.

People have pointed out that this function is misnamed, because the term "natural number" is usually understood as excluding zero. We are open to suggestions for a better name to use in a future version.

This predicate tests whether its argument is zero, and returns `t`

if so, `nil`

otherwise. The argument must be a number.

These two forms are equivalent: `(zerop x) == (= x 0)`

.

Floating point numbers in Emacs Lisp actually take up storage, and
there can be many distinct floating point number objects with the same
numeric value. If you use `eq`

to compare them, then you test
whether two values are the same *object*. If you want to compare
just the numeric values, use `=`

.

If you use `eq`

to compare two integers, it always returns
`t`

if they have the same value. This is sometimes useful, because
`eq`

accepts arguments of any type and never causes an error,
whereas `=`

signals an error if the arguments are not numbers or
markers. However, it is a good idea to use `=`

if you can, even
for comparing integers, just in case we change the representation of
integers in a future Emacs version.

There is another wrinkle: because floating point arithmetic is not exact, it is often a bad idea to check for equality of two floating point values. Usually it is better to test for approximate equality. Here's a function to do this:

(defvar fuzz-factor 1.0e-6) (defun approx-equal (x y) (< (/ (abs (- x y)) (max (abs x) (abs y))) fuzz-factor))

Common Lisp note:because of the way numbers are implemented in Common Lisp, you generally need to use`to test for equality between numbers of any kind.`=`

'

__Function:__ **=** *number-or-marker1 number-or-marker2*

This function tests whether its arguments are the same number, and
returns `t`

if so, `nil`

otherwise.

__Function:__ **/=** *number-or-marker1 number-or-marker2*

This function tests whether its arguments are not the same number, and
returns `t`

if so, `nil`

otherwise.

__Function:__ **<** *number-or-marker1 number-or-marker2*

This function tests whether its first argument is strictly less than
its second argument. It returns `t`

if so, `nil`

otherwise.

__Function:__ **<=** *number-or-marker1 number-or-marker2*

This function tests whether its first argument is less than or equal
to its second argument. It returns `t`

if so, `nil`

otherwise.

__Function:__ **>** *number-or-marker1 number-or-marker2*

This function tests whether its first argument is strictly greater
than its second argument. It returns `t`

if so, `nil`

otherwise.

__Function:__ **>=** *number-or-marker1 number-or-marker2*

This function tests whether its first argument is greater than or
equal to its second argument. It returns `t`

if so, `nil`

otherwise.

__Function:__ **max** *number-or-marker &rest numbers-or-markers*

This function returns the largest of its arguments.

(max 20) => 20 (max 1 2) => 2 (max 1 3 2) => 3

__Function:__ **min** *number-or-marker &rest numbers-or-markers*

This function returns the smallest of its arguments.

To convert an integer to floating point, use the function `float`

.

This returns `number` converted to floating point.
If `number` is already a floating point number, `float`

returns
it unchanged.

There are four functions to convert floating point numbers to integers; they differ in how they round. You can call these functions with an integer argument also; if you do, they return it without change.

This returns `number`, converted to an integer by rounding towards
zero.

__Function:__ **floor** *number &optional divisor*

This returns `number`, converted to an integer by rounding downward
(towards negative infinity).

If `divisor` is specified, `number` is divided by `divisor`
before the floor is taken; this is the division operation that
corresponds to `mod`

. An `arith-error`

results if
`divisor` is 0.

This returns `number`, converted to an integer by rounding upward
(towards positive infinity).

This returns `number`, converted to an integer by rounding towards the
nearest integer.

Emacs Lisp provides the traditional four arithmetic operations: addition, subtraction, multiplication, and division. Remainder and modulus functions supplement the division functions. The functions to add or subtract 1 are provided because they are traditional in Lisp and commonly used.

All of these functions except `%`

return a floating point value
if any argument is floating.

It is important to note that in GNU Emacs Lisp, arithmetic functions
do not check for overflow. Thus `(1+ 8388607)`

may equal
-8388608, depending on your hardware.

This function returns `number-or-marker` plus 1.
For example,

(setq foo 4) => 4 (1+ foo) => 5

This function is not analogous to the C operator `++`

---it does
not increment a variable. It just computes a sum. Thus,

foo => 4

If you want to increment the variable, you must use `setq`

,
like this:

(setq foo (1+ foo)) => 5

This function returns `number-or-marker` minus 1.

This returns the absolute value of `number`.

__Function:__ **+** *&rest numbers-or-markers*

This function adds its arguments together. When given no arguments,
`+`

returns 0. It does not check for overflow.

(+) => 0 (+ 1) => 1 (+ 1 2 3 4) => 10

__Function:__ **-** *&optional number-or-marker &rest other-numbers-or-markers*

The `-`

function serves two purposes: negation and subtraction.
When `-`

has a single argument, the value is the negative of the
argument. When there are multiple arguments, each of the
`other-numbers-or-markers` is subtracted from `number-or-marker`,
cumulatively. If there are no arguments, the result is 0. This
function does not check for overflow.

(- 10 1 2 3 4) => 0 (- 10) => -10 (-) => 0

__Function:__ ***** *&rest numbers-or-markers*

This function multiplies its arguments together, and returns the
product. When given no arguments, `*`

returns 1. It does
not check for overflow.

(*) => 1 (* 1) => 1 (* 1 2 3 4) => 24

__Function:__ **/** *dividend divisor &rest divisors*

This function divides `dividend` by `divisors` and returns the
quotient. If there are additional arguments `divisors`, then
`dividend` is divided by each divisor in turn. Each argument may be
a number or a marker.

If all the arguments are integers, then the result is an integer too.
This means the result has to be rounded. On most machines, the result
is rounded towards zero after each division, but some machines may round
differently with negative arguments. This is because the Lisp function
`/`

is implemented using the C division operator, which has the
same possibility for machine-dependent rounding. As a practical matter,
all known machines round in the standard fashion.

If you divide by 0, an `arith-error`

error is signaled.
(See section Errors.)

(/ 6 2) => 3 (/ 5 2) => 2 (/ 25 3 2) => 4 (/ -17 6) => -2

Since the division operator in Emacs Lisp is implemented using the
division operator in C, the result of dividing negative numbers may in
principle vary from machine to machine, depending on how they round the
result. Thus, the result of `(/ -17 6)`

could be -3 on some
machines. In practice, all known machines round the quotient towards
0.

This function returns the integer remainder after division of `dividend`
by `divisor`. The arguments must be integers or markers.

For negative arguments, the value is in principle machine-dependent since the quotient is; but in practice, all known machines behave alike.

An `arith-error`

results if `divisor` is 0.

(% 9 4) => 1 (% -9 4) => -1 (% 9 -4) => 1 (% -9 -4) => -1

For any two integers `dividend` and `divisor`,

(+ (%dividenddivisor) (* (/dividenddivisor)divisor))

always equals `dividend`.

__Function:__ **mod** *dividend divisor*

This function returns the value of `dividend` modulo `divisor`;
in other words, the remainder after division of `dividend`
by `divisor`, but with the same sign as `divisor`.
The arguments must be numbers or markers.

Unlike `%`

, the result is well-defined for negative arguments.
Also, floating point arguments are permitted.

An `arith-error`

results if `divisor` is 0.

(mod 9 4) => 1 (mod -9 4) => 3 (mod 9 -4) => -3 (mod -9 -4) => -1

For any two numbers `dividend` and `divisor`,

(+ (moddividenddivisor) (* (floordividenddivisor)divisor))

always equals `dividend`, subject to rounding error if
either argument is floating point.

In a computer, an integer is represented as a binary number, a sequence of bits (digits which are either zero or one). A bitwise operation acts on the individual bits of such a sequence. For example, shifting moves the whole sequence left or right one or more places, reproducing the same pattern "moved over".

The bitwise operations in Emacs Lisp apply only to integers.

`lsh`

, which is an abbreviation for logical shift, shifts the
bits in `integer1` to the left `count` places, or to the
right if `count` is negative. If `count` is negative,
`lsh`

shifts zeros into the most-significant bit, producing a
positive result even if `integer1` is negative. Contrast this with
`ash`

, below.

Thus, the decimal number 5 is the binary number 00000101. Shifted once to the left, with a zero put in the one's place, the number becomes 00001010, decimal 10.

Here are two examples of shifting the pattern of bits one place to the
left. Since the contents of the rightmost place has been moved one
place to the left, a value has to be inserted into the rightmost place.
With `lsh`

, a zero is placed into the rightmost place. (These
examples show only the low-order eight bits of the binary pattern; the
rest are all zero.)

(lsh 5 1) => 10 ;; Decimal 5 becomes decimal 10. 00000101 => 00001010 (lsh 7 1) => 14 ;; Decimal 7 becomes decimal 14. 00000111 => 00001110

As the examples illustrate, shifting the pattern of bits one place to the left produces a number that is twice the value of the previous number.

Note, however that functions do not check for overflow, and a returned value may be negative (and in any case, no more than a 24 bit value) when an integer is sufficiently left shifted.

For example, left shifting 8,388,607 produces -2:

(lsh 8388607 1) ; left shift => -2

In binary, in the 24 bit implementation, the numbers looks like this:

;; Decimal 8,388,607 0111 1111 1111 1111 1111 1111

which becomes the following when left shifted:

;; Decimal -2 1111 1111 1111 1111 1111 1110

Shifting the pattern of bits two places to the left produces results like this (with 8-bit binary numbers):

(lsh 3 2) => 12 ;; Decimal 3 becomes decimal 12. 00000011 => 00001100

On the other hand, shifting the pattern of bits one place to the right looks like this:

(lsh 6 -1) => 3 ;; Decimal 6 becomes decimal 3. 00000110 => 00000011 (lsh 5 -1) => 2 ;; Decimal 5 becomes decimal 2. 00000101 => 00000010

As the example illustrates, shifting the pattern of bits one place to the right divides the value of the binary number by two, rounding downward.

`ash`

(arithmetic shift) shifts the bits in `integer1`
to the left `count` places, or to the right if `count`
is negative.

`ash`

gives the same results as `lsh`

except when
`integer1` and `count` are both negative. In that case,
`ash`

puts a one in the leftmost position, while `lsh`

puts
a zero in the leftmost position.

Thus, with `ash`

, shifting the pattern of bits one place to the right
looks like this:

(ash -6 -1) => -3 ;; Decimal -6 ;; becomes decimal -3. 1111 1111 1111 1111 1111 1010 => 1111 1111 1111 1111 1111 1101

In contrast, shifting the pattern of bits one place to the right with
`lsh`

looks like this:

(lsh -6 -1) => 8388605 ;; Decimal -6 ;; becomes decimal 8,388,605. 1111 1111 1111 1111 1111 1010 => 0111 1111 1111 1111 1111 1101

In this case, the 1 in the leftmost position is shifted one place to the right, and a zero is shifted into the leftmost position.

Here are other examples:

; 24-bit binary values (lsh 5 2) ; 5 = 0000 0000 0000 0000 0000 0101 => 20 ; 20 = 0000 0000 0000 0000 0001 0100 (ash 5 2) => 20 (lsh -5 2) ; -5 = 1111 1111 1111 1111 1111 1011 => -20 ; -20 = 1111 1111 1111 1111 1110 1100 (ash -5 2) => -20 (lsh 5 -2) ; 5 = 0000 0000 0000 0000 0000 0101 => 1 ; 1 = 0000 0000 0000 0000 0000 0001 (ash 5 -2) => 1 (lsh -5 -2) ; -5 = 1111 1111 1111 1111 1111 1011 => 4194302 ; 0011 1111 1111 1111 1111 1110 (ash -5 -2) ; -5 = 1111 1111 1111 1111 1111 1011 => -2 ; -2 = 1111 1111 1111 1111 1111 1110

__Function:__ **logand** *&rest ints-or-markers*

This function returns the "logical and" of the arguments: the
`n`th bit is set in the result if, and only if, the `n`th bit is
set in all the arguments. ("Set" means that the value of the bit is 1
rather than 0.)

For example, using 4-bit binary numbers, the "logical and" of 13 and 12 is 12: 1101 combined with 1100 produces 1100.

In both the binary numbers, the leftmost two bits are set (i.e., they are 1's), so the leftmost two bits of the returned value are set. However, for the rightmost two bits, each is zero in at least one of the arguments, so the rightmost two bits of the returned value are 0's.

Therefore,

(logand 13 12) => 12

If `logand`

is not passed any argument, it returns a value of
-1. This number is an identity element for `logand`

because its binary representation consists entirely of ones. If
`logand`

is passed just one argument, it returns that argument.

; 24-bit binary values (logand 14 13) ; 14 = 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 1101 => 12 ; 12 = 0000 0000 0000 0000 0000 1100 (logand 14 13 4) ; 14 = 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 1101 ; 4 = 0000 0000 0000 0000 0000 0100 => 4 ; 4 = 0000 0000 0000 0000 0000 0100 (logand) => -1 ; -1 = 1111 1111 1111 1111 1111 1111

__Function:__ **logior** *&rest ints-or-markers*

This function returns the "inclusive or" of its arguments: the `n`th bit
is set in the result if, and only if, the `n`th bit is set in at least
one of the arguments. If there are no arguments, the result is zero,
which is an identity element for this operation. If `logior`

is
passed just one argument, it returns that argument.

; 24-bit binary values (logior 12 5) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 => 13 ; 13 = 0000 0000 0000 0000 0000 1101 (logior 12 5 7) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0111 => 15 ; 15 = 0000 0000 0000 0000 0000 1111

__Function:__ **logxor** *&rest ints-or-markers*

This function returns the "exclusive or" of its arguments: the
`n`th bit is set in the result if, and only if, the `n`th bit
is set in an odd number of the arguments. If there are no arguments,
the result is 0. If `logxor`

is passed just one argument, it returns
that argument.

; 24-bit binary values (logxor 12 5) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 => 9 ; 9 = 0000 0000 0000 0000 0000 1001 (logxor 12 5 7) ; 12 = 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0111 => 14 ; 14 = 0000 0000 0000 0000 0000 1110

This function returns the logical complement of its argument: the `n`th
bit is one in the result if, and only if, the `n`th bit is zero in
`integer`, and vice-versa.

;; 5 = 0000 0000 0000 0000 0000 0101 ;; becomes ;; -6 = 1111 1111 1111 1111 1111 1010 (lognot 5) => -6

These mathematical functions are available if floating point is supported. They allow integers as well as floating point numbers as arguments.

These are the ordinary trigonometric functions, with argument measured in radians.

The value of `(asin `

is a number between - pi / 2
and pi / 2 (inclusive) whose sine is `arg`)`arg`; if, however, `arg`
is out of range (outside [-1, 1]), then the result is a NaN.

The value of `(acos `

is a number between 0 and pi
(inclusive) whose cosine is `arg`)`arg`; if, however, `arg`
is out of range (outside [-1, 1]), then the result is a NaN.

The value of `(atan `

is a number between - pi / 2
and pi / 2 (exclusive) whose tangent is `arg`)`arg`.

This is the exponential function; it returns *e* to the power
`arg`.

__Function:__ **log** *arg &optional base*

This function returns the logarithm of `arg`, with base `base`.
If you don't specify `base`, the base `e` is used. If `arg`
is negative, the result is a NaN.

This function returns the logarithm of `arg`, with base 10. If
`arg` is negative, the result is a NaN.

This function returns `x` raised to power `y`.

This returns the square root of `arg`.

In a computer, a series of pseudo-random numbers is generated in a deterministic fashion. The numbers are not truly random, but they have certain properties that mimic a random series. For example, all possible values occur equally often in a pseudo-random series.

In Emacs, pseudo-random numbers are generated from a "seed" number.
Starting from any given seed, the `random`

function always
generates the same sequence of numbers. Emacs always starts with the
same seed value, so the sequence of values of `random`

is actually
the same in each Emacs run! For example, in one operating system, the
first call to `(random)`

after you start Emacs always returns
-1457731, and the second one always returns -7692030. This is helpful
for debugging.

If you want truly unpredictable random numbers, execute ```
(random
t)
```

. This chooses a new seed based on the current time of day and on
Emacs' process ID number.

__Function:__ **random** *&optional limit*

This function returns a pseudo-random integer. When called more than once, it returns a series of pseudo-random integers.

If `limit` is `nil`

, then the value may in principle be any
integer. If `limit` is a positive integer, the value is chosen to
be nonnegative and less than `limit` (only in Emacs 19).

If `limit` is `t`

, it means to choose a new seed based on the
current time of day and on Emacs's process ID number.

On some machines, any integer representable in Lisp may be the result
of `random`

. On other machines, the result can never be larger
than a certain maximum or less than a certain (negative) minimum.

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