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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