# 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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`Base.AbstractIrrational` — Type

    AbstractIrrational <: Real

Number type representing an exact irrational value, which is automatically rounded to the correct precision in arithmetic operations with other numeric quantities.

Subtypes `MyIrrational <: AbstractIrrational` should implement at least `==(::MyIrrational, ::MyIrrational)`, `hash(x::MyIrrational, h::UInt)`, and `convert(::Type{F}, x::MyIrrational) where {F <: Union{BigFloat,Float32,Float64}}`.

If a subtype is used to represent values that may occasionally be rational (e.g. a square-root type that represents `√n` for integers `n` will give a rational result when `n` is a perfect square), then it should also implement `isinteger`, `iszero`, `isone`, and `==` with `Real` values (since all of these default to `false` for `AbstractIrrational` types), as well as defining `hash` to equal that of the corresponding `Rational`.

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### Concrete number types

`Core.Float16` — Type

    Float16 <: AbstractFloat

16-bit floating point number type (IEEE 754 standard).

Binary format: 1 sign, 5 exponent, 10 fraction bits.

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`Core.Float32` — Type

    Float32 <: AbstractFloat

32-bit floating point number type (IEEE 754 standard).

Binary format: 1 sign, 8 exponent, 23 fraction bits.

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`Core.Float64` — Type

    Float64 <: AbstractFloat

64-bit floating point number type (IEEE 754 standard).

Binary format: 1 sign, 11 exponent, 52 fraction bits.

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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, containing the values `true` and `false`.

`Bool` is a kind of number: `false` is numerically equal to `0` and `true` is numerically equal to `1`. Moreover, `false` acts as a multiplicative "strong zero":

    julia> false == 0
    true

    julia> true == 1
    true

    julia> 0 * NaN
    NaN

    julia> false * NaN
    0.0

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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 <: Signed

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`.

`ComplexF16`, `ComplexF32` and `ComplexF64` 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`. Rationals are checked for overflow.

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`Base.Irrational` — Type

    Irrational{sym} <: AbstractIrrational

Number type representing an exact irrational value denoted by the symbol `sym`.

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## Data Formats

`Base.digits` — Function

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

Return 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 indices, such that `n == sum(digits[k]*base^(k-1) for k=1:length(digits))`.

Examples

    julia> digits(10, base = 10)
    2-element Vector{Int64}:
     0
     1

    julia> digits(10, base = 2)
    4-element Vector{Int64}:
     0
     1
     0
     1

    julia> digits(10, base = 2, pad = 6)
    6-element Vector{Int64}:
     0
     1
     0
     1
     0
     0

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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 indices. 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.

Examples

    julia> digits!([2,2,2,2], 10, base = 2)
    4-element Vector{Int64}:
     0
     1
     0
     1

    julia> digits!([2,2,2,2,2,2], 10, base = 2)
    6-element Vector{Int64}:
     0
     1
     0
     1
     0
     0

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`Base.bitstring` — Function

    bitstring(n)

A string giving the literal bit representation of a number.

Examples

    julia> bitstring(4)
    "0000000000000000000000000000000000000000000000000000000000000100"

    julia> bitstring(2.2)
    "0100000000000001100110011001100110011001100110011001100110011010"

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`Base.parse` — Function

    parse(::Type{Platform}, triplet::AbstractString)

Parses a string platform triplet back into a `Platform` object.

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    parse(type, str; base)

Parse a string as a number. For `Integer` types, a base can be specified (the default is 10). For floating-point types, the string is parsed as a decimal floating-point number. `Complex` types are parsed from decimal strings of the form `"R±Iim"` as a `Complex(R,I)` of the requested type; `"i"` or `"j"` can also be used instead of `"im"`, and `"R"` or `"Iim"` are also permitted. If the string does not contain a valid number, an error is raised.

Julia 1.1

`parse(Bool, str)` requires at least Julia 1.1.

Examples

    julia> parse(Int, "1234")
    1234

    julia> parse(Int, "1234", base = 5)
    194

    julia> parse(Int, "afc", base = 16)
    2812

    julia> parse(Float64, "1.2e-3")
    0.0012

    julia> parse(Complex{Float64}, "3.2e-1 + 4.5im")
    0.32 + 4.5im

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`Base.tryparse` — Function

    tryparse(type, str; base)

Like `parse`, but returns either a value of the requested type, or `nothing` 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(T::Integer)

Convert an integer bitstype to the signed type of the same size.

Examples

    julia> signed(UInt16)
    Int16
    julia> signed(UInt64)
    Int64

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    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(T::Integer)

Convert an integer bitstype to the unsigned type of the same size.

Examples

    julia> unsigned(Int16)
    UInt16
    julia> unsigned(UInt64)
    UInt64

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`Base.float` — Method

    float(x)

Convert a number or array to a floating point data type.

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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::AbstractFloat) -> Int

Get the exponent of a normalized floating-point number. Returns the largest integer `y` such that `2^y ≤ abs(x)`.

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`Base.complex` — Method

    complex(r, [i])

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

Examples

    julia> complex(7)
    7 + 0im

    julia> complex([1, 2, 3])
    3-element Vector{Complex{Int64}}:
     1 + 0im
     2 + 0im
     3 + 0im

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`Base.bswap` — Function

    bswap(n)

Reverse the byte order of `n`.

(See also `ntoh` and `hton` to convert between the current native byte order and big-endian order.)

Examples

    julia> a = bswap(0x10203040)
    0x40302010

    julia> bswap(a)
    0x10203040

    julia> string(1, base = 2)
    "1"

    julia> string(bswap(1), base = 2)
    "100000000000000000000000000000000000000000000000000000000"

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`Base.hex2bytes` — Function

    hex2bytes(s::Union{AbstractString,AbstractVector{UInt8}})

Given a string or array `s` of ASCII codes for a sequence of hexadecimal digits, returns a `Vector{UInt8}` of bytes corresponding to the binary representation: each successive pair of hexadecimal digits in `s` gives the value of one byte in the return vector.

The length of `s` must be even, and the returned array has half of the length of `s`. See also `hex2bytes!` for an in-place version, and `bytes2hex` for the inverse.

Examples

    julia> s = string(12345, base = 16)
    "3039"

    julia> hex2bytes(s)
    2-element Vector{UInt8}:
     0x30
     0x39

    julia> a = b"01abEF"
    6-element Base.CodeUnits{UInt8, String}:
     0x30
     0x31
     0x61
     0x62
     0x45
     0x46

    julia> hex2bytes(a)
    3-element Vector{UInt8}:
     0x01
     0xab
     0xef

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`Base.hex2bytes!` — Function

    hex2bytes!(d::AbstractVector{UInt8}, s::Union{String,AbstractVector{UInt8}})

Convert an array `s` of bytes representing a hexadecimal string to its binary representation, similar to `hex2bytes` except that the output is written in-place in `d`. The length of `s` must be exactly twice the length of `d`.

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`Base.bytes2hex` — Function

    bytes2hex(a::AbstractArray{UInt8}) -> String
    bytes2hex(io::IO, a::AbstractArray{UInt8})

Convert an array `a` of bytes to its hexadecimal string representation, either returning a `String` via `bytes2hex(a)` or writing the string to an `io` stream via `bytes2hex(io, a)`. The hexadecimal characters are all lowercase.

Examples

    julia> a = string(12345, base = 16)
    "3039"

    julia> b = hex2bytes(a)
    2-element Vector{UInt8}:
     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.

Examples

    julia> one(3.7)
    1.0

    julia> one(Int)
    1

    julia> import Dates; 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`).

Examples

    julia> oneunit(3.7)
    1.0

    julia> import Dates; oneunit(Dates.Day)
    1 day

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`Base.zero` — Function

    zero(x)
    zero(::Type)

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

Examples

    julia> zero(1)
    0

    julia> zero(big"2.0")
    0.0

    julia> zero(rand(2,2))
    2×2 Matrix{Float64}:
     0.0  0.0
     0.0  0.0

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`Base.im` — Constant

    im

The imaginary unit.

Examples

    julia> im * im
    -1 + 0im

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`Base.MathConstants.pi` — Constant

    π
    pi

The constant pi.

Examples

    julia> pi
    π = 3.1415926535897...

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`Base.MathConstants.ℯ` — Constant

    ℯ
    e

The constant ℯ.

Examples

    julia> ℯ
    ℯ = 2.7182818284590...

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`Base.MathConstants.catalan` — Constant

    catalan

Catalan's constant.

Examples

    julia> Base.MathConstants.catalan
    catalan = 0.9159655941772...

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`Base.MathConstants.eulergamma` — Constant

    γ
    eulergamma

Euler's constant.

Examples

    julia> Base.MathConstants.eulergamma
    γ = 0.5772156649015...

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`Base.MathConstants.golden` — Constant

    φ
    golden

The golden ratio.

Examples

    julia> Base.MathConstants.golden
    φ = 1.6180339887498...

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`Base.Inf` — Constant

    Inf, Inf64

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, NaN64

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.

Examples

    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 number value is a NaN, an indeterminate value which is neither an infinity nor a finite number ("not a number").

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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.

Examples

    julia> iszero(0.0)
    true

    julia> iszero([1, 9, 0])
    false

    julia> iszero([false, 0, 0])
    true

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`Base.isone` — Function

    isone(x)

Return `true` if `x == one(x)`; if `x` is an array, this checks whether `x` is an identity matrix.

Examples

    julia> isone(1.0)
    true

    julia> isone([1 0; 0 2])
    false

    julia> isone([1 0; 0 true])
    true

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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)

Return 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 return `x`.

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`Base.prevfloat` — Function

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

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

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    prevfloat(x::AbstractFloat)

Return 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 return `x`.

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`Base.isinteger` — Function

    isinteger(x) -> Bool

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

Examples

    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 including infinities and NaNs. `isreal(x)` is true if `isequal(x, real(x))` is true.

Examples

    julia> isreal(5.)
    true

    julia> isreal(Inf + 0im)
    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.

Examples

    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.

Examples

    julia> Float64(pi, RoundDown)
    3.141592653589793

    julia> Float64(pi, RoundUp)
    3.1415926535897936

See `RoundingMode` for available rounding modes.

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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 is currently only supported for `T == BigFloat`.

Warning

This function is not thread-safe. It will affect code running on all threads, but its behavior is undefined if called concurrently with computations that use the setting.

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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.

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`Base.Rounding.get_zero_subnormals` — Function

    get_zero_subnormals() -> Bool

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

Warning

This function only affects the current thread.

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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)`.

Warning

This function only affects the current thread.

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### Integers

`Base.count_ones` — Function

    count_ones(x::Integer) -> Integer

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

Examples

    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`.

Examples

    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`.

Examples

    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`.

Examples

    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`.

Examples

    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`.

Examples

    julia> trailing_ones(3)
    2

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`Base.isodd` — Function

    isodd(x::Integer) -> Bool

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

Examples

    julia> isodd(9)
    true

    julia> isodd(10)
    false

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`Base.iseven` — Function

    iseven(x::Integer) -> Bool

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

Examples

    julia> iseven(9)
    false

    julia> iseven(10)
    true

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`Core.@int128_str` — Macro

    @int128_str str
    @int128_str(str)

`@int128_str` parses a string into a Int128 Throws an `ArgumentError` if the string is not a valid integer

source
`Core.@uint128_str` — Macro

    @uint128_str str
    @uint128_str(str)

`@uint128_str` parses a string into a UInt128 Throws an `ArgumentError` if the string is not a valid integer

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## BigFloats and BigInts

The `BigFloat` and `BigInt` types implements arbitrary-precision floating point and integer arithmetic, respectively. For `BigFloat` the GNU MPFR library is used, and for `BigInt` the GNU Multiple Precision Arithmetic Library (GMP) is used.

`Base.MPFR.BigFloat` — Method

    BigFloat(x::Union{Real, AbstractString} [, rounding::RoundingMode=rounding(BigFloat)]; [precision::Integer=precision(BigFloat)])

Create an arbitrary precision floating point number from `x`, with precision `precision`. The `rounding` argument specifies the direction in which the result should be rounded if the conversion cannot be done exactly. If not provided, these are set by the current global values.

`BigFloat(x::Real)` is the same as `convert(BigFloat,x)`, except if `x` itself is already `BigFloat`, in which case it will return a value with the precision set to the current global precision; `convert` will always return `x`.

`BigFloat(x::AbstractString)` is identical to `parse`. This is provided for convenience since decimal literals are converted to `Float64` when parsed, so `BigFloat(2.1)` may not yield what you expect.

Julia 1.1

`precision` as a keyword argument requires at least Julia 1.1. In Julia 1.0 `precision` is the second positional argument (`BigFloat(x, precision)`).

Examples

    julia> BigFloat(2.1) # 2.1 here is a Float64
    2.100000000000000088817841970012523233890533447265625

    julia> BigFloat("2.1") # the closest BigFloat to 2.1
    2.099999999999999999999999999999999999999999999999999999999999999999999999999986

    julia> BigFloat("2.1", RoundUp)
    2.100000000000000000000000000000000000000000000000000000000000000000000000000021

    julia> BigFloat("2.1", RoundUp, precision=128)
    2.100000000000000000000000000000000000007

See also

- `@big_str`
- `rounding` and `setrounding`
- `precision` and `setprecision`

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`Base.precision` — Function

    precision(num::AbstractFloat)

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

source
`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.

Warning

This function is not thread-safe. It will affect code running on all threads, but its behavior is undefined if called concurrently with computations that use the setting.

source

    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`

Note: `nextfloat()`, `prevfloat()` do not use the precision mentioned by `setprecision`

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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.

Examples

    julia> parse(BigInt, "42")
    42

    julia> big"313"
    313

    julia> BigInt(10)^19
    10000000000000000000

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`Core.@big_str` — Macro

    @big_str str
    @big_str(str)

Parse a string into a `BigInt` or `BigFloat`, and throw an `ArgumentError` if the string is not a valid number. For integers `_` is allowed in the string as a separator.

Examples

    julia> big"123_456"
    123456

    julia> big"7891.5"
    7891.5

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Theme

This document was generated with Documenter.jl on Tuesday 16 March 2021. Using Julia version 1.6.0-rc3.