Numbers

Number

Abstract supertype for all number types.

Real <: Number

Abstract supertype for all real numbers.

AbstractFloat <: Real

Abstract supertype for all floating point numbers.

Integer <: Real

Abstract supertype for all integers.

Signed <: Integer

Abstract supertype for all signed integers.

Unsigned <: Integer

Abstract supertype for all unsigned integers.

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.

Float16 <: AbstractFloat

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

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

Float32 <: AbstractFloat

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

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

Float64 <: AbstractFloat

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

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

BigFloat <: AbstractFloat

Arbitrary precision floating point number 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

See also: digits, iszero, NaN.

Int8 <: Signed

8-bit signed integer type.

UInt8 <: Unsigned

8-bit unsigned integer type.

Int16 <: Signed

16-bit signed integer type.

UInt16 <: Unsigned

16-bit unsigned integer type.

Int32 <: Signed

32-bit signed integer type.

UInt32 <: Unsigned

32-bit unsigned integer type.

Int64 <: Signed

64-bit signed integer type.

UInt64 <: Unsigned

64-bit unsigned integer type.

Int128 <: Signed

128-bit signed integer type.

UInt128 <: Unsigned

128-bit unsigned integer type.

BigInt <: Signed

Arbitrary precision integer 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.

See also: Real, complex, real.

Rational{T<:Integer} <: Real

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

Irrational{sym} <: AbstractIrrational

Number type representing an exact irrational value denoted by the symbol sym, such as π, ℯ and γ.

See also AbstractIrrational.

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)).

See also ndigits, digits!, and for base 2 also bitstring, count_ones.

Examples

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

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

julia> digits(-256, base = 10, pad = 5)
5-element Vector{Int64}:
 -6
 -5
 -2
  0
  0

julia> n = rand(-999:999);

julia> n == evalpoly(13, digits(n, base = 13))
true

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

bitstring(n)

A string giving the literal bit representation of a primitive type.

See also count_ones, count_zeros, digits.

Examples

julia> bitstring(Int32(4))
"00000000000000000000000000000100"

julia> bitstring(2.2)
"0100000000000001100110011001100110011001100110011001100110011010"

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.

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

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

Parses a string platform triplet back into a Platform object.

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.

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.

signed(T::Integer)

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

Examples

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

signed(x)

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

See also: unsigned, sign, signbit.

unsigned(T::Integer)

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

Examples

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

float(x)

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

See also: complex, oftype, convert.

Examples

julia> float(1:1000)
1.0:1.0:1000.0

julia> float(typemax(Int32))
2.147483647e9

significand(x)

Extract the significand (a.k.a. mantissa) of a floating-point number. If x is a non-zero finite number, then the result will be a number of the same type and sign as x, and whose absolute value is on the interval $[1,2)$. Otherwise x is returned.

Examples

julia> significand(15.2)
1.9

julia> significand(-15.2)
-1.9

julia> significand(-15.2) * 2^3
-15.2

julia> significand(-Inf), significand(Inf), significand(NaN)
(-Inf, Inf, NaN)

exponent(x) -> Int

Returns the largest integer y such that 2^y ≤ abs(x). For a normalized floating-point number x, this corresponds to the exponent of x.

Examples

julia> exponent(8)
3

julia> exponent(64//1)
6

julia> exponent(6.5)
2

julia> exponent(16.0)
4

julia> exponent(3.142e-4)
-12

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

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"

hex2bytes(itr)

Given an iterable itr 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 itr gives the value of one byte in the return vector.

The length of itr must be even, and the returned array has half of the length of itr. 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

hex2bytes!(dest::AbstractVector{UInt8}, itr)

Convert an iterable itr of bytes representing a hexadecimal string to its binary representation, similar to hex2bytes except that the output is written in-place to dest. The length of dest must be half the length of itr.

bytes2hex(itr) -> String
bytes2hex(io::IO, itr)

Convert an iterator itr of bytes to its hexadecimal string representation, either returning a String via bytes2hex(itr) or writing the string to an io stream via bytes2hex(io, itr). 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"

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.

See also the identity function, and I in LinearAlgebra for the identity matrix.

Examples

julia> one(3.7)
1.0

julia> one(Int)
1

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

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

Return 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

zero(x)
zero(::Type)

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

See also iszero, one, oneunit, oftype.

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

im

The imaginary unit.

See also: imag, angle, complex.

Examples

julia> im * im
-1 + 0im

julia> (2.0 + 3im)^2
-5.0 + 12.0im

π
pi

The constant pi.

Unicode π can be typed by writing \pi then pressing tab in the Julia REPL, and in many editors.

See also: sinpi, sincospi, deg2rad.

Examples

julia> pi
π = 3.1415926535897...

julia> 1/2pi
0.15915494309189535

ℯ
e

The constant ℯ.

Unicode ℯ can be typed by writing \euler and pressing tab in the Julia REPL, and in many editors.

See also: exp, cis, cispi.

Examples

julia> ℯ
ℯ = 2.7182818284590...

julia> log(ℯ)
1

julia> ℯ^(im)π ≈ -1
true

catalan

Catalan's constant.

Examples

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

julia> sum(log(x)/(1+x^2) for x in 1:0.01:10^6) * 0.01
0.9159466120554123

γ
eulergamma

Euler's constant.

Examples

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

julia> dx = 10^-6;

julia> sum(-exp(-x) * log(x) for x in dx:dx:100) * dx
0.5772078382499133

φ
golden

The golden ratio.

Examples

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

julia> (2ans - 1)^2 ≈ 5
true

Inf, Inf64

Positive infinity of type Float64.

See also: isfinite, typemax, NaN, Inf32.

Examples

julia> π/0
Inf

julia> +1.0 / -0.0
-Inf

julia> ℯ^-Inf
0.0

Inf, Inf64

Positive infinity of type Float64.

See also: isfinite, typemax, NaN, Inf32.

Examples

julia> π/0
Inf

julia> +1.0 / -0.0
-Inf

julia> ℯ^-Inf
0.0

Inf32

Positive infinity of type Float32.

Inf16

Positive infinity of type Float16.

NaN, NaN64

A not-a-number value of type Float64.

See also: isnan, missing, NaN32, Inf.

Examples

julia> 0/0
NaN

julia> Inf - Inf
NaN

julia> NaN == NaN, isequal(NaN, NaN), NaN === NaN
(false, true, true)

NaN, NaN64

A not-a-number value of type Float64.

See also: isnan, missing, NaN32, Inf.

Examples

julia> 0/0
NaN

julia> Inf - Inf
NaN

julia> NaN == NaN, isequal(NaN, NaN), NaN === NaN
(false, true, true)

NaN32

A not-a-number value of type Float32.

NaN16

A not-a-number value of type Float16.

issubnormal(f) -> Bool

Test whether a floating point number is subnormal.

An IEEE floating point number is subnormal when its exponent bits are zero and its significand is not zero.

Examples

```jldoctest julia> floatmin(Float32) 1.1754944f-38

julia> issubnormal(1.0f-37) false

julia> issubnormal(1.0f-38) true

isfinite(f) -> Bool

Test whether a number is finite.

Examples

julia> isfinite(5)
true

julia> isfinite(NaN32)
false

isinf(f) -> Bool

Test whether a number is infinite.

See also: Inf, iszero, isfinite, isnan.

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").

See also: iszero, isone, isinf, ismissing.

iszero(x)

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

See also: isone, isinteger, isfinite, isnan.

Examples

julia> iszero(0.0)
true

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

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

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

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.

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.

See also: prevfloat, eps, issubnormal.

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.

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.

isinteger(x) -> Bool

Test whether x is numerically equal to some integer.

Examples

julia> isinteger(4.0)
true

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(1 - 3im)
false

julia> isreal(Inf + 0im)
true

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

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.

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.

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.

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.

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.

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.

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).

count_ones(x::Integer) -> Integer

Number of ones in the binary representation of x.

Examples

julia> count_ones(7)
3

julia> count_ones(Int32(-1))
32

count_zeros(x::Integer) -> Integer

Number of zeros in the binary representation of x.

Examples

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

julia> count_zeros(-1)
0

leading_zeros(x::Integer) -> Integer

Number of zeros leading the binary representation of x.

Examples

julia> leading_zeros(Int32(1))
31

leading_ones(x::Integer) -> Integer

Number of ones leading the binary representation of x.

Examples

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

trailing_zeros(x::Integer) -> Integer

Number of zeros trailing the binary representation of x.

Examples

julia> trailing_zeros(2)
1

trailing_ones(x::Integer) -> Integer

Number of ones trailing the binary representation of x.

Examples

julia> trailing_ones(3)
2

isodd(x::Number) -> Bool

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

Examples

julia> isodd(9)
true

julia> isodd(10)
false

iseven(x::Number) -> Bool

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

Examples

julia> iseven(9)
false

julia> iseven(10)
true

@int128_str str

Parse str as an Int128. Throw an ArgumentError if the string is not a valid integer.

Examples

julia> int128"123456789123"
123456789123

julia> int128"123456789123.4"
ERROR: LoadError: ArgumentError: invalid base 10 digit '.' in "123456789123.4"
[...]

@uint128_str str

Parse str as an UInt128. Throw an ArgumentError if the string is not a valid integer.

Examples

julia> uint128"123456789123"
0x00000000000000000000001cbe991a83

julia> uint128"-123456789123"
ERROR: LoadError: ArgumentError: invalid base 10 digit '-' in "-123456789123"
[...]

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.

See also:

• @big_str
• rounding and setrounding
• precision and setprecision

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

precision(num::AbstractFloat; base::Integer=2)
precision(T::Type; base::Integer=2)

Get the precision of a floating point number, as defined by the effective number of bits in the significand, or the precision of a floating-point type T (its current default, if T is a variable-precision type like BigFloat).

If base is specified, then it returns the maximum corresponding number of significand digits in that base.

setprecision([T=BigFloat,] precision::Int; base=2)

Set the precision (in bits, by default) to be used for T arithmetic. If base is specified, then the precision is the minimum required to give at least precision digits in the given base.

setprecision(f::Function, [T=BigFloat,] precision::Integer; base=2)

Change the T arithmetic precision (in the given base) 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.

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

@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

julia> big"_"
ERROR: ArgumentError: invalid number format _ for BigInt or BigFloat
[...]