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Integer Basics

The Lisp reader reads an integer as a nonempty sequence of decimal digits with optional initial sign and optional final period.

 1               ; The integer 1.
 1.              ; The integer 1.
+1               ; Also the integer 1.
-1               ; The integer -1.
 0               ; The integer 0.
-0               ; The integer 0.

The syntax for integers in bases other than 10 consists of ‘#’ followed by a radix indication followed by one or more digits. The radix indications are ‘b’ for binary, ‘o’ for octal, ‘x’ for hex, and ‘radixr’ for radix radix. Thus, ‘#binteger’ reads integer in binary, and ‘#radixrinteger’ reads integer in radix radix. Allowed values of radix run from 2 to 36, and allowed digits are the first radix characters taken from ‘0’–‘9’, ‘A’–‘Z’. Letter case is ignored and there is no initial sign or final period. For example:

#b101100 ⇒ 44
#o54 ⇒ 44
#x2c ⇒ 44
#24r1k ⇒ 44

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

In binary, the decimal integer 5 looks like this:

…000101

(The ellipsis ‘…’ stands for a conceptually infinite number of bits that match the leading bit; here, an infinite number of 0 bits. Later examples also use this ‘…’ notation.)

The integer -1 looks like this:

…111111

-1 is represented as all ones. (This is called two’s complement notation.)

Subtracting 4 from -1 returns the negative integer -5. In binary, the decimal integer 4 is 100. Consequently, -5 looks like this:

…111011

Many of the functions described in this chapter accept markers for arguments in place of numbers. (See Markers.) Since the actual arguments to such functions may be either numbers or markers, we often give these arguments the name number-or-marker. When the argument value is a marker, its position value is used and its buffer is ignored.

In Emacs Lisp, text characters are represented by integers. Any integer between zero and the value of (max-char), inclusive, is considered to be valid as a character. See Character Codes.

Integers in Emacs Lisp are not limited to the machine word size. Under the hood, though, there are two kinds of integers: smaller ones, called fixnums, and larger ones, called bignums. Although Emacs Lisp code ordinarily should not depend on whether an integer is a fixnum or a bignum, older Emacs versions support only fixnums, some functions in Emacs still accept only fixnums, and older Emacs Lisp code may have trouble when given bignums. For example, while older Emacs Lisp code could safely compare integers for numeric equality with eq, the presence of bignums means that equality predicates like eql and = should now be used to compare integers.

The range of values for bignums is limited by the amount of main memory, by machine characteristics such as the size of the word used to represent a bignum’s exponent, and by the integer-width variable. These limits are typically much more generous than the limits for fixnums. A bignum is never numerically equal to a fixnum; Emacs always represents an integer in fixnum range as a fixnum, not a bignum.

The range of values for a fixnum depends on the machine. The minimum range is -536,870,912 to 536,870,911 (30 bits; i.e., -2**29 to 2**29 - 1), but many machines provide a wider range.

Variable: most-positive-fixnum

The value of this variable is the greatest “small” integer that Emacs Lisp can handle. Typical values are 2**29 - 1 on 32-bit and 2**61 - 1 on 64-bit platforms.

Variable: most-negative-fixnum

The value of this variable is the numerically least “small” integer that Emacs Lisp can handle. It is negative. Typical values are -2**29 on 32-bit and -2**61 on 64-bit platforms.

Variable: integer-width

The value of this variable is a nonnegative integer that controls whether Emacs signals a range error when a large integer would be calculated. Integers with absolute values less than 2**n, where n is this variable’s value, do not signal a range error. Attempts to create larger integers typically signal a range error, although there might be no signal if a larger integer can be created cheaply. Setting this variable to a large number can be costly if a computation creates huge integers.