# Module `Big_int`

Operations on arbitrary-precision integers.

Big integers (type `big_int`) are signed integers of arbitrary size.

`type big_int`

The type of big integers.

`val zero_big_int : big_int`

The big integer `0`.

`val unit_big_int : big_int`

The big integer `1`.

###### Arithmetic operations
`val minus_big_int : big_int -> big_int`

Unary negation.

`val abs_big_int : big_int -> big_int`

Absolute value.

`val add_big_int : big_int -> big_int -> big_int`

`val succ_big_int : big_int -> big_int`

`val add_int_big_int : int -> big_int -> big_int`

Addition of a small integer to a big integer.

`val sub_big_int : big_int -> big_int -> big_int`

Subtraction.

`val pred_big_int : big_int -> big_int`

Predecessor (subtract 1).

`val mult_big_int : big_int -> big_int -> big_int`

Multiplication of two big integers.

`val mult_int_big_int : int -> big_int -> big_int`

Multiplication of a big integer by a small integer

`val square_big_int : big_int -> big_int`

Return the square of the given big integer

`val sqrt_big_int : big_int -> big_int`

`sqrt_big_int a` returns the integer square root of `a`, that is, the largest big integer `r` such that `r * r <= a`. Raise `Invalid_argument` if `a` is negative.

`val quomod_big_int : big_int -> big_int -> big_int * big_int`

Euclidean division of two big integers. The first part of the result is the quotient, the second part is the remainder. Writing `(q,r) = quomod_big_int a b`, we have `a = q * b + r` and `0 <= r < |b|`. Raise `Division_by_zero` if the divisor is zero.

`val div_big_int : big_int -> big_int -> big_int`

Euclidean quotient of two big integers. This is the first result `q` of `quomod_big_int` (see above).

`val mod_big_int : big_int -> big_int -> big_int`

Euclidean modulus of two big integers. This is the second result `r` of `quomod_big_int` (see above).

`val gcd_big_int : big_int -> big_int -> big_int`

Greatest common divisor of two big integers.

`val power_int_positive_int : int -> int -> big_int`
`val power_big_int_positive_int : big_int -> int -> big_int`
`val power_int_positive_big_int : int -> big_int -> big_int`
`val power_big_int_positive_big_int : big_int -> big_int -> big_int`

Exponentiation functions. Return the big integer representing the first argument `a` raised to the power `b` (the second argument). Depending on the function, `a` and `b` can be either small integers or big integers. Raise `Invalid_argument` if `b` is negative.

###### Comparisons and tests
`val sign_big_int : big_int -> int`

Return `0` if the given big integer is zero, `1` if it is positive, and `-1` if it is negative.

`val compare_big_int : big_int -> big_int -> int`

`compare_big_int a b` returns `0` if `a` and `b` are equal, `1` if `a` is greater than `b`, and `-1` if `a` is smaller than `b`.

`val eq_big_int : big_int -> big_int -> bool`
`val le_big_int : big_int -> big_int -> bool`
`val ge_big_int : big_int -> big_int -> bool`
`val lt_big_int : big_int -> big_int -> bool`
`val gt_big_int : big_int -> big_int -> bool`

Usual boolean comparisons between two big integers.

`val max_big_int : big_int -> big_int -> big_int`

Return the greater of its two arguments.

`val min_big_int : big_int -> big_int -> big_int`

Return the smaller of its two arguments.

`val num_digits_big_int : big_int -> int`

Return the number of machine words used to store the given big integer.

`val num_bits_big_int : big_int -> int`

Return the number of significant bits in the absolute value of the given big integer. `num_bits_big_int a` returns 0 if `a` is 0; otherwise it returns a positive integer `n` such that `2^(n-1) <= |a| < 2^n`.

since
4.03.0
###### Conversions to and from strings
`val string_of_big_int : big_int -> string`

Return the string representation of the given big integer, in decimal (base 10).

`val big_int_of_string : string -> big_int`

Convert a string to a big integer, in decimal. The string consists of an optional `-` or `+` sign, followed by one or several decimal digits.

`val big_int_of_string_opt : string -> big_int option`

Convert a string to a big integer, in decimal. The string consists of an optional `-` or `+` sign, followed by one or several decimal digits. Other the function returns `None`.

since
4.05
###### Conversions to and from other numerical types
`val big_int_of_int : int -> big_int`

Convert a small integer to a big integer.

`val is_int_big_int : big_int -> bool`

Test whether the given big integer is small enough to be representable as a small integer (type `int`) without loss of precision. On a 32-bit platform, `is_int_big_int a` returns `true` if and only if `a` is between 230 and 230-1. On a 64-bit platform, `is_int_big_int a` returns `true` if and only if `a` is between -262 and 262-1.

`val int_of_big_int : big_int -> int`

Convert a big integer to a small integer (type `int`). Raises `Failure "int_of_big_int"` if the big integer is not representable as a small integer.

`val int_of_big_int_opt : big_int -> int option`

Convert a big integer to a small integer (type `int`). Return `None` if the big integer is not representable as a small integer.

since
4.05
`val big_int_of_int32 : int32 -> big_int`

Convert a 32-bit integer to a big integer.

`val big_int_of_nativeint : nativeint -> big_int`

Convert a native integer to a big integer.

`val big_int_of_int64 : int64 -> big_int`

Convert a 64-bit integer to a big integer.

`val int32_of_big_int : big_int -> int32`

Convert a big integer to a 32-bit integer. Raises `Failure` if the big integer is outside the range [-231, 231-1].

`val int32_of_big_int_opt : big_int -> int32 option`

Convert a big integer to a 32-bit integer. Return `None` if the big integer is outside the range [-231, 231-1].

since
4.05
`val nativeint_of_big_int : big_int -> nativeint`

Convert a big integer to a native integer. Raises `Failure` if the big integer is outside the range `[Nativeint.min_int, Nativeint.max_int]`.

`val nativeint_of_big_int_opt : big_int -> nativeint option`

Convert a big integer to a native integer. Return `None` if the big integer is outside the range ```[Nativeint.min_int, Nativeint.max_int]```;

since
4.05
`val int64_of_big_int : big_int -> int64`

Convert a big integer to a 64-bit integer. Raises `Failure` if the big integer is outside the range [-263, 263-1].

`val int64_of_big_int_opt : big_int -> int64 option`

Convert a big integer to a 64-bit integer. Return `None` if the big integer is outside the range [-263, 263-1].

since
4.05
`val float_of_big_int : big_int -> float`

Returns a floating-point number approximating the given big integer.

###### Bit-oriented operations
`val and_big_int : big_int -> big_int -> big_int`

Bitwise logical 'and'. The arguments must be positive or zero.

`val or_big_int : big_int -> big_int -> big_int`

Bitwise logical 'or'. The arguments must be positive or zero.

`val xor_big_int : big_int -> big_int -> big_int`

Bitwise logical 'exclusive or'. The arguments must be positive or zero.

`val shift_left_big_int : big_int -> int -> big_int`

`shift_left_big_int b n` returns `b` shifted left by `n` bits. Equivalent to multiplication by 2^n.

`val shift_right_big_int : big_int -> int -> big_int`

`shift_right_big_int b n` returns `b` shifted right by `n` bits. Equivalent to division by 2^n with the result being rounded towards minus infinity.

`val shift_right_towards_zero_big_int : big_int -> int -> big_int`

`shift_right_towards_zero_big_int b n` returns `b` shifted right by `n` bits. The shift is performed on the absolute value of `b`, and the result has the same sign as `b`. Equivalent to division by 2^n with the result being rounded towards zero.

`val extract_big_int : big_int -> int -> int -> big_int`

`extract_big_int bi ofs n` returns a nonnegative number corresponding to bits `ofs` to `ofs + n - 1` of the binary representation of `bi`. If `bi` is negative, a two's complement representation is used.