Numbers

# Numbers

## Standard Numeric Types

### Abstract number types

`Core.Number` — Type.

    Number

Abstract supertype for all number types.

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`Core.Real` — Type.

    Real <: Number

Abstract supertype for all real numbers.

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`Core.AbstractFloat` — Type.

    AbstractFloat <: Real

Abstract supertype for all floating point numbers.

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`Core.Integer` — Type.

    Integer <: Real

Abstract supertype for all integers.

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`Core.Signed` — Type.

    Signed <: Integer

Abstract supertype for all signed integers.

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`Core.Unsigned` — Type.

    Unsigned <: Integer

Abstract supertype for all unsigned integers.

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### Concrete number types

`Core.Float16` — Type.

    Float16 <: AbstractFloat

16-bit floating point number type.

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`Core.Float32` — Type.

    Float32 <: AbstractFloat

32-bit floating point number type.

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`Core.Float64` — Type.

    Float64 <: AbstractFloat

64-bit floating point number type.

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`Base.MPFR.BigFloat` — Type.

    BigFloat <: AbstractFloat

Arbitrary precision floating point number type.

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`Core.Bool` — Type.

    Bool <: Integer

Boolean type.

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`Core.Int8` — Type.

    Int8 <: Signed

8-bit signed integer type.

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`Core.UInt8` — Type.

    UInt8 <: Unsigned

8-bit unsigned integer type.

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`Core.Int16` — Type.

    Int16 <: Signed

16-bit signed integer type.

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`Core.UInt16` — Type.

    UInt16 <: Unsigned

16-bit unsigned integer type.

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`Core.Int32` — Type.

    Int32 <: Signed

32-bit signed integer type.

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`Core.UInt32` — Type.

    UInt32 <: Unsigned

32-bit unsigned integer type.

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`Core.Int64` — Type.

    Int64 <: Signed

64-bit signed integer type.

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`Core.UInt64` — Type.

    UInt64 <: Unsigned

64-bit unsigned integer type.

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`Core.Int128` — Type.

    Int128 <: Signed

128-bit signed integer type.

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`Core.UInt128` — Type.

    UInt128 <: Unsigned

128-bit unsigned integer type.

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`Base.GMP.BigInt` — Type.

    BigInt <: Integer

Arbitrary precision integer type.

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`Base.Complex` — Type.

    Complex{T<:Real} <: Number

Complex number type with real and imaginary part of type `T`.

`Complex32`, `Complex64` and `Complex128` are aliases for `Complex{Float16}`, `Complex{Float32}` and `Complex{Float64}` respectively.

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`Base.Rational` — Type.

    Rational{T<:Integer} <: Real

Rational number type, with numerator and denominator of type `T`.

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`Base.Irrational` — Type.

    Irrational <: Real

Irrational number type.

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## Data Formats

`Base.bin` — Function.

    bin(n, pad::Int=1)

Convert an integer to a binary string, optionally specifying a number of digits to pad to.

    julia> bin(10,2)
    "1010"

    julia> bin(10,8)
    "00001010"

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`Base.hex` — Function.

    hex(n, pad::Int=1)

Convert an integer to a hexadecimal string, optionally specifying a number of digits to pad to.

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`Base.dec` — Function.

    dec(n, pad::Int=1)

Convert an integer to a decimal string, optionally specifying a number of digits to pad to.

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`Base.oct` — Function.

    oct(n, pad::Int=1)

Convert an integer to an octal string, optionally specifying a number of digits to pad to.

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`Base.base` — Function.

    base(base::Integer, n::Integer, pad::Integer=1)

Convert an integer `n` to a string in the given `base`, optionally specifying a number of digits to pad to.

    julia> base(13,5,4)
    "0005"

    julia> base(5,13,4)
    "0023"

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`Base.digits` — Function.

    digits([T<:Integer], n::Integer, base::T=10, pad::Integer=1)

Returns an array with element type `T` (default `Int`) of the digits of `n` in the given base, optionally padded with zeros to a specified size. More significant digits are at higher indexes, such that `n == sum([digits[k]*base^(k-1) for k=1:length(digits)])`.

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`Base.digits!` — Function.

    digits!(array, n::Integer, base::Integer=10)

Fills an array of the digits of `n` in the given base. More significant digits are at higher indexes. If the array length is insufficient, the least significant digits are filled up to the array length. If the array length is excessive, the excess portion is filled with zeros.

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`Base.bits` — Function.

    bits(n)

A string giving the literal bit representation of a number.

    julia> bits(4)
    "0000000000000000000000000000000000000000000000000000000000000100"

    julia> bits(2.2)
    "0100000000000001100110011001100110011001100110011001100110011010"

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`Base.parse` — Method.

    parse(type, str, [base])

Parse a string as a number. If the type is an integer type, then a base can be specified (the default is 10). If the type is a floating point type, the string is parsed as a decimal floating point number. If the string does not contain a valid number, an error is raised.

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`Base.tryparse` — Function.

    tryparse(type, str, [base])

Like `parse`, but returns a `Nullable` of the requested type. The result will be null if the string does not contain a valid number.

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`Base.big` — Function.

    big(x)

Convert a number to a maximum precision representation (typically `BigInt` or `BigFloat`). See `BigFloat` for information about some pitfalls with floating-point numbers.

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`Base.signed` — Function.

    signed(x)

Convert a number to a signed integer. If the argument is unsigned, it is reinterpreted as signed without checking for overflow.

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`Base.unsigned` — Function.

    unsigned(x) -> Unsigned

Convert a number to an unsigned integer. If the argument is signed, it is reinterpreted as unsigned without checking for negative values.

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`Base.float` — Method.

    float(x)

Convert a number or array to a floating point data type. When passed a string, this function is equivalent to `parse(Float64, x)`.

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`Base.Math.significand` — Function.

    significand(x)

Extract the `significand(s)` (a.k.a. mantissa), in binary representation, of a floating-point number. If `x` is a non-zero finite number, then the result will be a number of the same type on the interval $[1,2)$. Otherwise `x` is returned.

Examples

    julia> significand(15.2)/15.2
    0.125

    julia> significand(15.2)*8
    15.2

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`Base.Math.exponent` — Function.

    exponent(x) -> Int

Get the exponent of a normalized floating-point number.

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`Base.complex` — Method.

    complex(r, [i])

Convert real numbers or arrays to complex. `i` defaults to zero.

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`Base.bswap` — Function.

    bswap(n)

Byte-swap an integer. Flip the bits of its binary representation.

    julia> a = bswap(4)
    288230376151711744

    julia> bswap(a)
    4

    julia> bin(1)
    "1"

    julia> bin(bswap(1))
    "100000000000000000000000000000000000000000000000000000000"

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`Base.num2hex` — Function.

    num2hex(f)

Get a hexadecimal string of the binary representation of a floating point number.

    julia> num2hex(2.2)
    "400199999999999a"

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`Base.hex2num` — Function.

    hex2num(str)

Convert a hexadecimal string to the floating point number it represents.

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`Base.hex2bytes` — Function.

    hex2bytes(s::AbstractString)

Convert an arbitrarily long hexadecimal string to its binary representation. Returns an `Array{UInt8,1}`, i.e. an array of bytes.

    julia> a = hex(12345)
    "3039"

    julia> hex2bytes(a)
    2-element Array{UInt8,1}:
     0x30
     0x39

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`Base.bytes2hex` — Function.

    bytes2hex(bin_arr::Array{UInt8, 1}) -> String

Convert an array of bytes to its hexadecimal representation. All characters are in lower-case.

    julia> a = hex(12345)
    "3039"

    julia> b = hex2bytes(a)
    2-element Array{UInt8,1}:
     0x30
     0x39

    julia> bytes2hex(b)
    "3039"

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## General Number Functions and Constants

`Base.one` — Function.

    one(x)
    one(T::type)

Return a multiplicative identity for `x`: a value such that `one(x)*x == x*one(x) == x`. Alternatively `one(T)` can take a type `T`, in which case `one` returns a multiplicative identity for any `x` of type `T`.

If possible, `one(x)` returns a value of the same type as `x`, and `one(T)` returns a value of type `T`. However, this may not be the case for types representing dimensionful quantities (e.g. time in days), since the multiplicative identity must be dimensionless. In that case, `one(x)` should return an identity value of the same precision (and shape, for matrices) as `x`.

If you want a quantity that is of the same type as `x`, or of type `T`, even if `x` is dimensionful, use `oneunit` instead.

    julia> one(3.7)
    1.0

    julia> one(Int)
    1

    julia> one(Dates.Day(1))
    1

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`Base.oneunit` — Function.

    oneunit(x::T)
    oneunit(T::Type)

Returns `T(one(x))`, where `T` is either the type of the argument or (if a type is passed) the argument. This differs from `one` for dimensionful quantities: `one` is dimensionless (a multiplicative identity) while `oneunit` is dimensionful (of the same type as `x`, or of type `T`).

    julia> oneunit(3.7)
    1.0

    julia> oneunit(Dates.Day)
    1 day

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`Base.zero` — Function.

    zero(x)

Get the additive identity element for the type of `x` (`x` can also specify the type itself).

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`Base.pi` — Constant.

    pi
    π

The constant pi.

    julia> pi
    π = 3.1415926535897...

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`Base.im` — Constant.

    im

The imaginary unit.

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`Base.eu` — Constant.

    e
    eu

The constant e.

    julia> e
    e = 2.7182818284590...

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`Base.catalan` — Constant.

    catalan

Catalan's constant.

    julia> catalan
    catalan = 0.9159655941772...

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`Base.eulergamma` — Constant.

    γ
    eulergamma

Euler's constant.

    julia> eulergamma
    γ = 0.5772156649015...

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`Base.golden` — Constant.

    φ
    golden

The golden ratio.

    julia> golden
    φ = 1.6180339887498...

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`Base.Inf` — Constant.

    Inf

Positive infinity of type `Float64`.

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`Base.Inf32` — Constant.

    Inf32

Positive infinity of type `Float32`.

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`Base.Inf16` — Constant.

    Inf16

Positive infinity of type `Float16`.

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`Base.NaN` — Constant.

    NaN

A not-a-number value of type `Float64`.

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`Base.NaN32` — Constant.

    NaN32

A not-a-number value of type `Float32`.

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`Base.NaN16` — Constant.

    NaN16

A not-a-number value of type `Float16`.

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`Base.issubnormal` — Function.

    issubnormal(f) -> Bool

Test whether a floating point number is subnormal.

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`Base.isfinite` — Function.

    isfinite(f) -> Bool

Test whether a number is finite.

    julia> isfinite(5)
    true

    julia> isfinite(NaN32)
    false

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`Base.isinf` — Function.

    isinf(f) -> Bool

Test whether a number is infinite.

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`Base.isnan` — Function.

    isnan(f) -> Bool

Test whether a floating point number is not a number (NaN).

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`Base.iszero` — Function.

    iszero(x)

Return `true` if `x == zero(x)`; if `x` is an array, this checks whether all of the elements of `x` are zero.

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`Base.nextfloat` — Function.

    nextfloat(x::AbstractFloat, n::Integer)

The result of `n` iterative applications of `nextfloat` to `x` if `n >= 0`, or `-n` applications of `prevfloat` if `n < 0`.

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    nextfloat(x::AbstractFloat)

Returns the smallest floating point number `y` of the same type as `x` such `x < y`. If no such `y` exists (e.g. if `x` is `Inf` or `NaN`), then returns `x`.

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`Base.prevfloat` — Function.

    prevfloat(x::AbstractFloat)

Returns the largest floating point number `y` of the same type as `x` such `y < x`. If no such `y` exists (e.g. if `x` is `-Inf` or `NaN`), then returns `x`.

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`Base.isinteger` — Function.

    isinteger(x) -> Bool

Test whether `x` is numerically equal to some integer.

    julia> isinteger(4.0)
    true

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`Base.isreal` — Function.

    isreal(x) -> Bool

Test whether `x` or all its elements are numerically equal to some real number.

    julia> isreal(5.)
    true

    julia> isreal([4.; complex(0,1)])
    false

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`Core.Float32` — Method.

    Float32(x [, mode::RoundingMode])

Create a Float32 from `x`. If `x` is not exactly representable then `mode` determines how `x` is rounded.

    julia> Float32(1/3, RoundDown)
    0.3333333f0

    julia> Float32(1/3, RoundUp)
    0.33333334f0

See `RoundingMode` for available rounding modes.

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`Core.Float64` — Method.

    Float64(x [, mode::RoundingMode])

Create a Float64 from `x`. If `x` is not exactly representable then `mode` determines how `x` is rounded.

    julia> Float64(pi, RoundDown)
    3.141592653589793

    julia> Float64(pi, RoundUp)
    3.1415926535897936

See `RoundingMode` for available rounding modes.

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`Base.GMP.BigInt` — Method.

    BigInt(x)

Create an arbitrary precision integer. `x` may be an `Int` (or anything that can be converted to an `Int`). The usual mathematical operators are defined for this type, and results are promoted to a `BigInt`.

Instances can be constructed from strings via `parse`, or using the `big` string literal.

    julia> parse(BigInt, "42")
    42

    julia> big"313"
    313

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`Base.MPFR.BigFloat` — Method.

    BigFloat(x)

Create an arbitrary precision floating point number. `x` may be an `Integer`, a `Float64` or a `BigInt`. The usual mathematical operators are defined for this type, and results are promoted to a `BigFloat`.

Note that because decimal literals are converted to floating point numbers when parsed, `BigFloat(2.1)` may not yield what you expect. You may instead prefer to initialize constants from strings via `parse`, or using the `big` string literal.

    julia> BigFloat(2.1)
    2.100000000000000088817841970012523233890533447265625000000000000000000000000000

    julia> big"2.1"
    2.099999999999999999999999999999999999999999999999999999999999999999999999999986

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`Base.Rounding.rounding` — Function.

    rounding(T)

Get the current floating point rounding mode for type `T`, controlling the rounding of basic arithmetic functions (`+`, `-`, `*`, `/` and `sqrt`) and type conversion.

See `RoundingMode` for available modes.

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`Base.Rounding.setrounding` — Method.

    setrounding(T, mode)

Set the rounding mode of floating point type `T`, controlling the rounding of basic arithmetic functions (`+`, `-`, `*`, `/` and `sqrt`) and type conversion. Other numerical functions may give incorrect or invalid values when using rounding modes other than the default `RoundNearest`.

Note that this may affect other types, for instance changing the rounding mode of `Float64` will change the rounding mode of `Float32`. See `RoundingMode` for available modes.

Warning

This feature is still experimental, and may give unexpected or incorrect values.

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`Base.Rounding.setrounding` — Method.

    setrounding(f::Function, T, mode)

Change the rounding mode of floating point type `T` for the duration of `f`. It is logically equivalent to:

    old = rounding(T)
    setrounding(T, mode)
    f()
    setrounding(T, old)

See `RoundingMode` for available rounding modes.

Warning

This feature is still experimental, and may give unexpected or incorrect values. A known problem is the interaction with compiler optimisations, e.g.

    julia> setrounding(Float64,RoundDown) do
               1.1 + 0.1
           end
    1.2000000000000002

Here the compiler is constant folding, that is evaluating a known constant expression at compile time, however the rounding mode is only changed at runtime, so this is not reflected in the function result. This can be avoided by moving constants outside the expression, e.g.

    julia> x = 1.1; y = 0.1;

    julia> setrounding(Float64,RoundDown) do
               x + y
           end
    1.2

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`Base.Rounding.get_zero_subnormals` — Function.

    get_zero_subnormals() -> Bool

Returns `false` if operations on subnormal floating-point values ("denormals") obey rules for IEEE arithmetic, and `true` if they might be converted to zeros.

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`Base.Rounding.set_zero_subnormals` — Function.

    set_zero_subnormals(yes::Bool) -> Bool

If `yes` is `false`, subsequent floating-point operations follow rules for IEEE arithmetic on subnormal values ("denormals"). Otherwise, floating-point operations are permitted (but not required) to convert subnormal inputs or outputs to zero. Returns `true` unless `yes==true` but the hardware does not support zeroing of subnormal numbers.

`set_zero_subnormals(true)` can speed up some computations on some hardware. However, it can break identities such as `(x-y==0) == (x==y)`.

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### Integers

`Base.count_ones` — Function.

    count_ones(x::Integer) -> Integer

Number of ones in the binary representation of `x`.

    julia> count_ones(7)
    3

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`Base.count_zeros` — Function.

    count_zeros(x::Integer) -> Integer

Number of zeros in the binary representation of `x`.

    julia> count_zeros(Int32(2 ^ 16 - 1))
    16

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`Base.leading_zeros` — Function.

    leading_zeros(x::Integer) -> Integer

Number of zeros leading the binary representation of `x`.

    julia> leading_zeros(Int32(1))
    31

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`Base.leading_ones` — Function.

    leading_ones(x::Integer) -> Integer

Number of ones leading the binary representation of `x`.

    julia> leading_ones(UInt32(2 ^ 32 - 2))
    31

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`Base.trailing_zeros` — Function.

    trailing_zeros(x::Integer) -> Integer

Number of zeros trailing the binary representation of `x`.

    julia> trailing_zeros(2)
    1

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`Base.trailing_ones` — Function.

    trailing_ones(x::Integer) -> Integer

Number of ones trailing the binary representation of `x`.

    julia> trailing_ones(3)
    2

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`Base.isodd` — Function.

    isodd(x::Integer) -> Bool

Returns `true` if `x` is odd (that is, not divisible by 2), and `false` otherwise.

    julia> isodd(9)
    true

    julia> isodd(10)
    false

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`Base.iseven` — Function.

    iseven(x::Integer) -> Bool

Returns `true` is `x` is even (that is, divisible by 2), and `false` otherwise.

    julia> iseven(9)
    false

    julia> iseven(10)
    true

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## BigFloats

The `BigFloat` type implements arbitrary-precision floating-point arithmetic using the GNU MPFR library.

`Base.precision` — Function.

    precision(num::AbstractFloat)

Get the precision of a floating point number, as defined by the effective number of bits in the mantissa.

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`Base.precision` — Method.

    precision(BigFloat)

Get the precision (in bits) currently used for `BigFloat` arithmetic.

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`Base.MPFR.setprecision` — Function.

    setprecision([T=BigFloat,] precision::Int)

Set the precision (in bits) to be used for `T` arithmetic.

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    setprecision(f::Function, [T=BigFloat,] precision::Integer)

Change the `T` arithmetic precision (in bits) for the duration of `f`. It is logically equivalent to:

    old = precision(BigFloat)
    setprecision(BigFloat, precision)
    f()
    setprecision(BigFloat, old)

Often used as `setprecision(T, precision) do ... end`

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`Base.MPFR.BigFloat` — Method.

    BigFloat(x, prec::Int)

Create a representation of `x` as a `BigFloat` with precision `prec`.

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`Base.MPFR.BigFloat` — Method.

    BigFloat(x, rounding::RoundingMode)

Create a representation of `x` as a `BigFloat` with the current global precision and rounding mode `rounding`.

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`Base.MPFR.BigFloat` — Method.

    BigFloat(x, prec::Int, rounding::RoundingMode)

Create a representation of `x` as a `BigFloat` with precision `prec` and rounding mode `rounding`.

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`Base.MPFR.BigFloat` — Method.

    BigFloat(x::String)

Create a representation of the string `x` as a `BigFloat`.

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## Random Numbers

Random number generation in Julia uses the Mersenne Twister library via `MersenneTwister` objects. Julia has a global RNG, which is used by default. Other RNG types can be plugged in by inheriting the `AbstractRNG` type; they can then be used to have multiple streams of random numbers. Besides `MersenneTwister`, Julia also provides the `RandomDevice` RNG type, which is a wrapper over the OS provided entropy.

Most functions related to random generation accept an optional `AbstractRNG` as the first argument, `rng` , which defaults to the global one if not provided. Morever, some of them accept optionally dimension specifications `dims...` (which can be given as a tuple) to generate arrays of random values.

A `MersenneTwister` or `RandomDevice` RNG can generate random numbers of the following types: `Float16`, `Float32`, `Float64`, `Bool`, `Int8`, `UInt8`, `Int16`, `UInt16`, `Int32`, `UInt32`, `Int64`, `UInt64`, `Int128`, `UInt128`, `BigInt` (or complex numbers of those types). Random floating point numbers are generated uniformly in $[0, 1)$. As `BigInt` represents unbounded integers, the interval must be specified (e.g. `rand(big(1:6))`).

`Base.Random.srand` — Function.

    srand([rng=GLOBAL_RNG], [seed]) -> rng
    srand([rng=GLOBAL_RNG], filename, n=4) -> rng

Reseed the random number generator. If a `seed` is provided, the RNG will give a reproducible sequence of numbers, otherwise Julia will get entropy from the system. For `MersenneTwister`, the `seed` may be a non-negative integer, a vector of `UInt32` integers or a filename, in which case the seed is read from a file (`4n` bytes are read from the file, where `n` is an optional argument). `RandomDevice` does not support seeding.

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`Base.Random.MersenneTwister` — Type.

    MersenneTwister(seed)

Create a `MersenneTwister` RNG object. Different RNG objects can have their own seeds, which may be useful for generating different streams of random numbers.

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`Base.Random.RandomDevice` — Type.

    RandomDevice()

Create a `RandomDevice` RNG object. Two such objects will always generate different streams of random numbers.

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`Base.Random.rand` — Function.

    rand([rng=GLOBAL_RNG], [S], [dims...])

Pick a random element or array of random elements from the set of values specified by `S`; `S` can be

- an indexable collection (for example `1:n` or `['x','y','z']`), or

- a type: the set of values to pick from is then equivalent to `typemin(S):typemax(S)` for integers (this is not applicable to `BigInt`), and to $[0, 1)$ for floating point numbers;

`S` defaults to `Float64`.

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`Base.Random.rand!` — Function.

    rand!([rng=GLOBAL_RNG], A, [coll])

Populate the array `A` with random values. If the indexable collection `coll` is specified, the values are picked randomly from `coll`. This is equivalent to `copy!(A, rand(rng, coll, size(A)))` or `copy!(A, rand(rng, eltype(A), size(A)))` but without allocating a new array.

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`Base.Random.bitrand` — Function.

    bitrand([rng=GLOBAL_RNG], [dims...])

Generate a `BitArray` of random boolean values.

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`Base.Random.randn` — Function.

    randn([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a normally-distributed random number of type `T` with mean 0 and standard deviation 1. Optionally generate an array of normally-distributed random numbers. The `Base` module currently provides an implementation for the types `Float16`, `Float32`, and `Float64` (the default).

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`Base.Random.randn!` — Function.

    randn!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array `A` with normally-distributed (mean 0, standard deviation 1) random numbers. Also see the `rand` function.

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`Base.Random.randexp` — Function.

    randexp([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a random number of type `T` according to the exponential distribution with scale 1. Optionally generate an array of such random numbers. The `Base` module currently provides an implementation for the types `Float16`, `Float32`, and `Float64` (the default).

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`Base.Random.randexp!` — Function.

    randexp!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array `A` with random numbers following the exponential distribution (with scale 1).

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`Base.Random.randjump` — Function.

    randjump(r::MersenneTwister, jumps::Integer, [jumppoly::AbstractString=dSFMT.JPOLY1e21]) -> Vector{MersenneTwister}

Create an array of the size `jumps` of initialized `MersenneTwister` RNG objects. The first RNG object given as a parameter and following `MersenneTwister` RNGs in the array are initialized such that a state of the RNG object in the array would be moved forward (without generating numbers) from a previous RNG object array element on a particular number of steps encoded by the jump polynomial `jumppoly`.

Default jump polynomial moves forward `MersenneTwister` RNG state by `10^20` steps.

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