Numbers

Standard Numeric Types

Abstract number types

Core.NumberType

Number

Abstract supertype for all number types.

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Core.RealType

Real <: Number

Abstract supertype for all real numbers.

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Core.AbstractFloatType

AbstractFloat <: Real

Abstract supertype for all floating point numbers.

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Core.IntegerType

Integer <: Real

Abstract supertype for all integers.

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Core.SignedType

Signed <: Integer

Abstract supertype for all signed integers.

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Core.UnsignedType

Unsigned <: Integer

Abstract supertype for all unsigned integers.

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Base.AbstractIrrationalType

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.Float16Type

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.Float32Type

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.Float64Type

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.BigFloatType

BigFloat <: AbstractFloat

Arbitrary precision floating point number type.

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Core.BoolType

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.Int8Type

Int8 <: Signed

8-bit signed integer type.

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Core.UInt8Type

UInt8 <: Unsigned

8-bit unsigned integer type.

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Core.Int16Type

Int16 <: Signed

16-bit signed integer type.

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Core.UInt16Type

UInt16 <: Unsigned

16-bit unsigned integer type.

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Core.Int32Type

Int32 <: Signed

32-bit signed integer type.

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Core.UInt32Type

UInt32 <: Unsigned

32-bit unsigned integer type.

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Core.Int64Type

Int64 <: Signed

64-bit signed integer type.

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Core.UInt64Type

UInt64 <: Unsigned

64-bit unsigned integer type.

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Core.Int128Type

Int128 <: Signed

128-bit signed integer type.

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Core.UInt128Type

UInt128 <: Unsigned

128-bit unsigned integer type.

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Base.GMP.BigIntType

BigInt <: Signed

Arbitrary precision integer type.

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Base.ComplexType

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.RationalType

Rational{T<:Integer} <: Real

Rational number type, with numerator and denominator of type T. Rationals are checked for overflow.

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Base.IrrationalType

Irrational{sym} <: AbstractIrrational

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

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Data Formats

Base.digitsFunction

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.bitstringFunction

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.parseFunction

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.

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.tryparseFunction

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.bigFunction

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.signedFunction

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.unsignedFunction

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.floatMethod

float(x)

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

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Base.Math.significandFunction

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.exponentFunction

exponent(x) -> Int

Get the exponent of a normalized floating-point number.

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Base.complexMethod

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.bswapFunction

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.hex2bytesFunction

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.bytes2hexFunction

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.oneFunction

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.oneunitFunction

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.zeroFunction

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.imConstant

im

The imaginary unit.

Examples

julia> im * im
-1 + 0im
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Base.MathConstants.piConstant

π
pi

The constant pi.

Examples

julia> pi
π = 3.1415926535897...
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Base.MathConstants.ℯConstant

ℯ
e

The constant ℯ.

Examples

julia> ℯ
ℯ = 2.7182818284590...
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Base.MathConstants.catalanConstant

catalan

Catalan's constant.

Examples

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

γ
eulergamma

Euler's constant.

Examples

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

φ
golden

The golden ratio.

Examples

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

Inf, Inf64

Positive infinity of type Float64.

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Base.Inf32Constant

Inf32

Positive infinity of type Float32.

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Base.Inf16Constant

Inf16

Positive infinity of type Float16.

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Base.NaNConstant

NaN, NaN64

A not-a-number value of type Float64.

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Base.NaN32Constant

NaN32

A not-a-number value of type Float32.

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Base.NaN16Constant

NaN16

A not-a-number value of type Float16.

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Base.issubnormalFunction

issubnormal(f) -> Bool

Test whether a floating point number is subnormal.

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Base.isfiniteFunction

isfinite(f) -> Bool

Test whether a number is finite.

Examples

julia> isfinite(5)
true

julia> isfinite(NaN32)
false
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Base.isinfFunction

isinf(f) -> Bool

Test whether a number is infinite.

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Base.isnanFunction

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.iszeroFunction

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.isoneFunction

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.nextfloatFunction

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.prevfloatFunction

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.isintegerFunction

isinteger(x) -> Bool

Test whether x is numerically equal to some integer.

Examples

julia> isinteger(4.0)
true
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Base.isrealFunction

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.Float32Method

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.Float64Method

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.roundingFunction

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.setroundingMethod

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.

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.setroundingMethod

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_subnormalsFunction

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.

This function only affects the current thread.

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Base.Rounding.set_zero_subnormalsFunction

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).

This function only affects the current thread.

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Integers

Base.count_onesFunction

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_zerosFunction

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_zerosFunction

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_onesFunction

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_zerosFunction

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_onesFunction

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.isoddFunction

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.isevenFunction

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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[email protected]_strMacro

@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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[email protected]_strMacro

@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.BigFloatMethod

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.

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

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Base.precisionFunction

precision(num::AbstractFloat)

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

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Base.precisionMethod

precision(BigFloat)

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

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Base.MPFR.setprecisionFunction

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

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

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.BigIntMethod

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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[email protected]_strMacro

@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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© 2009–2020 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
https://docs.julialang.org/en/v1.5.3/base/numbers/