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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`Base.AbstractIrrational` — Type.

    AbstractIrrational <: Real

Number type representing an exact irrational value.

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

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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 Array{Int64,1}:
     0
     1

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

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

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

Examples

    julia> unsigned(-2)
    0xfffffffffffffffe

    julia> unsigned(2)
    0x0000000000000002

    julia> signed(unsigned(-2))
    -2

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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) -> 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.

Examples

    julia> complex(7)
    7 + 0im

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

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`Base.bswap` — Function.

    bswap(n)

Reverse the byte order of `n`.

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 Array{UInt8,1}:
     0x30
     0x39

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

    julia> hex2bytes(a)
    3-element Array{UInt8,1}:
     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 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.

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)

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 Array{Float64,2}:
     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 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.

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` is `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

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`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 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.

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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    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

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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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