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Numbers

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.

Integer Basics

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.

Floating Point Basics

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):

Function: logb number

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

Type Predicates for Numbers

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.

Function: floatp object

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.

Function: integerp object

This predicate tests whether its argument is an integer, and returns t if so, nil otherwise.

Function: numberp object

This predicate tests whether its argument is a number (either integer or floating point), and returns t if so, nil otherwise.

Function: natnump object

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.

Function: zerop number

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).

Comparison of Numbers

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.

Numeric Conversions

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

Function: float number

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.

Function: truncate number

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.

Function: ceiling number

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

Function: round number

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

Arithmetic Operations

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.

Function: 1+ number-or-marker

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

Function: 1- number-or-marker

This function returns number-or-marker minus 1.

Function: abs number

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.

Function: % dividend divisor

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,

(+ (% dividend divisor)
   (* (/ dividend divisor) 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,

(+ (mod dividend divisor)
   (* (floor dividend divisor) divisor))

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

Bitwise Operations on Integers

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.

Function: lsh integer1 count

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.

Function: ash integer1 count

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 nth bit is set in the result if, and only if, the nth 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 nth bit is set in the result if, and only if, the nth 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 nth bit is set in the result if, and only if, the nth 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

Function: lognot integer

This function returns the logical complement of its argument: the nth bit is one in the result if, and only if, the nth 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

Transcendental Functions

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

Function: sin arg

Function: cos arg

Function: tan arg

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

Function: asin arg

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

Function: acos arg

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

Function: atan arg

The value of (atan arg) is a number between - pi / 2 and pi / 2 (exclusive) whose tangent is arg.

Function: exp 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.

Function: log10 arg

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

Function: expt x y

This function returns x raised to power y.

Function: sqrt arg

This returns the square root of arg.

Random Numbers

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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