Numbers

Data Formats

Base.bin — Function.

bin(n, pad::Int=1)

Convert an integer to a binary string, optionally specifying a number of digits to pad to.

julia> bin(10,2)
"1010"

julia> bin(10,8)
"00001010"

source

Base.hex — Function.

hex(n, pad::Int=1)

Convert an integer to a hexadecimal string, optionally specifying a number of digits to pad to.

julia> hex(20)
"14"

julia> hex(20, 3)
"014"

source

Base.dec — Function.

dec(n, pad::Int=1)

Convert an integer to a decimal string, optionally specifying a number of digits to pad to.

Examples

julia> dec(20)
"20"

julia> dec(20, 3)
"020"

source

Base.oct — Function.

oct(n, pad::Int=1)

Convert an integer to an octal string, optionally specifying a number of digits to pad to.

julia> oct(20)
"24"

julia> oct(20, 3)
"024"

source

Base.base — Function.

base(base::Integer, n::Integer, pad::Integer=1)

Convert an integer n to a string in the given base, optionally specifying a number of digits to pad to.

julia> base(13,5,4)
"0005"

julia> base(5,13,4)
"0023"

source

Base.digits — Function.

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

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

Examples

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

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

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

source

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 indexes. 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, 2)
4-element Array{Int64,1}:
 0
 1
 0
 1

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

source

Base.bits — Function.

bits(n)

A string giving the literal bit representation of a number.

Example

julia> bits(4)
"0000000000000000000000000000000000000000000000000000000000000100"

julia> bits(2.2)
"0100000000000001100110011001100110011001100110011001100110011010"

source

Base.parse — Method.

parse(type, str, [base])

Parse a string as a number. If the type is an integer type, then a base can be specified (the default is 10). If the type is a floating point type, the string is parsed as a decimal floating point number. If the string does not contain a valid number, an error is raised.

julia> parse(Int, "1234")
1234

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

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

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

source

Base.tryparse — Function.

tryparse(type, str, [base])

Like parse, but returns a Nullable of the requested type. The result will be null if the string does not contain a valid number.

source

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.

source

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.

source

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

source

Base.float — Method.

float(x)

Convert a number or array to a floating point data type. When passed a string, this function is equivalent to parse(Float64, x).

source

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

source

Base.Math.exponent — Function.

exponent(x) -> Int

Get the exponent of a normalized floating-point number.

source

Base.complex — Method.

complex(r, [i])

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

source

Base.bswap — Function.

bswap(n)

Byte-swap an integer. Flip the bits of its binary representation.

Examples

julia> a = bswap(4)
288230376151711744

julia> bswap(a)
4

julia> bin(1)
"1"

julia> bin(bswap(1))
"100000000000000000000000000000000000000000000000000000000"

source

Base.num2hex — Function.

num2hex(f)

Get a hexadecimal string of the binary representation of a floating point number.

Example

julia> num2hex(2.2)
"400199999999999a"

source

Base.hex2num — Function.

hex2num(str)

Convert a hexadecimal string to the floating point number it represents.

source

Base.hex2bytes — Function.

hex2bytes(s::AbstractString)

Convert an arbitrarily long hexadecimal string to its binary representation. Returns an Array{UInt8,1}, i.e. an array of bytes.

julia> a = hex(12345)
"3039"

julia> hex2bytes(a)
2-element Array{UInt8,1}:
 0x30
 0x39

source

Base.bytes2hex — Function.

bytes2hex(bin_arr::Array{UInt8, 1}) -> String

Convert an array of bytes to its hexadecimal representation. All characters are in lower-case.

julia> a = hex(12345)
"3039"

julia> b = hex2bytes(a)
2-element Array{UInt8,1}:
 0x30
 0x39

julia> bytes2hex(b)
"3039"

source

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.

julia> one(3.7)
1.0

julia> one(Int)
1

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

source

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).

julia> oneunit(3.7)
1.0

julia> oneunit(Dates.Day)
1 day

source

Base.zero — Function.

zero(x)

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

julia> zero(1)
0

julia> zero(big"2.0")
0.000000000000000000000000000000000000000000000000000000000000000000000000000000

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

source

Base.pi — Constant.

pi
π

The constant pi.

julia> pi
π = 3.1415926535897...

source

Base.im — Constant.

im

The imaginary unit.

source

Base.eu — Constant.

e
eu

The constant e.

julia> e
e = 2.7182818284590...

source

Base.catalan — Constant.

catalan

Catalan's constant.

julia> catalan
catalan = 0.9159655941772...

source

Base.eulergamma — Constant.

γ
eulergamma

Euler's constant.

julia> eulergamma
γ = 0.5772156649015...

source

Base.golden — Constant.

φ
golden

The golden ratio.

julia> golden
φ = 1.6180339887498...

source

Base.Inf — Constant.

Inf

Positive infinity of type Float64.

source

Base.Inf32 — Constant.

Inf32

Positive infinity of type Float32.

source

Base.Inf16 — Constant.

Inf16

Positive infinity of type Float16.

source

Base.NaN — Constant.

NaN

A not-a-number value of type Float64.

source

Base.NaN32 — Constant.

NaN32

A not-a-number value of type Float32.

source

Base.NaN16 — Constant.

NaN16

A not-a-number value of type Float16.

source

Base.issubnormal — Function.

issubnormal(f) -> Bool

Test whether a floating point number is subnormal.

source

Base.isfinite — Function.

isfinite(f) -> Bool

Test whether a number is finite.

julia> isfinite(5)
true

julia> isfinite(NaN32)
false

source

Base.isinf — Function.

isinf(f) -> Bool

Test whether a number is infinite.

source

Base.isnan — Function.

isnan(f) -> Bool

Test whether a floating point number is not a number (NaN).

source

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.

source

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.

source

nextfloat(x::AbstractFloat)

Returns 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 returns x.

source

Base.prevfloat — Function.

prevfloat(x::AbstractFloat)

Returns 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 returns x.

source

Base.isinteger — Function.

isinteger(x) -> Bool

Test whether x is numerically equal to some integer.

julia> isinteger(4.0)
true

source

Base.isreal — Function.

isreal(x) -> Bool

Test whether x or all its elements are numerically equal to some real number.

julia> isreal(5.)
true

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

source

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.

source

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.

source

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.

julia> parse(BigInt, "42")
42

julia> big"313"
313

source

Base.MPFR.BigFloat — Method.

BigFloat(x)

Create an arbitrary precision floating point number. x may be an Integer, a Float64 or a BigInt. The usual mathematical operators are defined for this type, and results are promoted to a BigFloat.

Note that because decimal literals are converted to floating point numbers when parsed, BigFloat(2.1) may not yield what you expect. You may instead prefer to initialize constants from strings via parse, or using the big string literal.

julia> BigFloat(2.1)
2.100000000000000088817841970012523233890533447265625000000000000000000000000000

julia> big"2.1"
2.099999999999999999999999999999999999999999999999999999999999999999999999999986

source

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.

source

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 may affect other types, for instance changing the rounding mode of Float64 will change the rounding mode of Float32. See RoundingMode for available modes.

Warning

This feature is still experimental, and may give unexpected or incorrect values.

source

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.

Warning

This feature is still experimental, and may give unexpected or incorrect values. A known problem is the interaction with compiler optimisations, e.g.

julia> setrounding(Float64,RoundDown) do
           1.1 + 0.1
       end
1.2000000000000002

Here the compiler is constant folding, that is evaluating a known constant expression at compile time, however the rounding mode is only changed at runtime, so this is not reflected in the function result. This can be avoided by moving constants outside the expression, e.g.

julia> x = 1.1; y = 0.1;

julia> setrounding(Float64,RoundDown) do
           x + y
       end
1.2

source

Base.Rounding.get_zero_subnormals — Function.

get_zero_subnormals() -> Bool

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

source

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).

source

Integers

Base.count_ones — Function.

count_ones(x::Integer) -> Integer

Number of ones in the binary representation of x.

julia> count_ones(7)
3

source

Base.count_zeros — Function.

count_zeros(x::Integer) -> Integer

Number of zeros in the binary representation of x.

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

source

Base.leading_zeros — Function.

leading_zeros(x::Integer) -> Integer

Number of zeros leading the binary representation of x.

julia> leading_zeros(Int32(1))
31

source

Base.leading_ones — Function.

leading_ones(x::Integer) -> Integer

Number of ones leading the binary representation of x.

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

source

Base.trailing_zeros — Function.

trailing_zeros(x::Integer) -> Integer

Number of zeros trailing the binary representation of x.

julia> trailing_zeros(2)
1

source

Base.trailing_ones — Function.

trailing_ones(x::Integer) -> Integer

Number of ones trailing the binary representation of x.

julia> trailing_ones(3)
2

source

Base.isodd — Function.

isodd(x::Integer) -> Bool

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

julia> isodd(9)
true

julia> isodd(10)
false

source

Base.iseven — Function.

iseven(x::Integer) -> Bool

Returns true is x is even (that is, divisible by 2), and false otherwise.

julia> iseven(9)
false

julia> iseven(10)
true

source

BigFloats

The BigFloat type implements arbitrary-precision floating-point arithmetic using the GNU MPFR library.

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.

source

Base.precision — Method.

precision(BigFloat)

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

source

Base.MPFR.setprecision — Function.

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

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

source

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

Change the T arithmetic precision (in bits) for the duration of f. It is logically equivalent to:

old = precision(BigFloat)
setprecision(BigFloat, precision)
f()
setprecision(BigFloat, old)

Often used as setprecision(T, precision) do ... end

source

Base.MPFR.BigFloat — Method.

BigFloat(x, prec::Int)

Create a representation of x as a BigFloat with precision prec.

source

Base.MPFR.BigFloat — Method.

BigFloat(x, rounding::RoundingMode)

Create a representation of x as a BigFloat with the current global precision and rounding mode rounding.

source

Base.MPFR.BigFloat — Method.

BigFloat(x, prec::Int, rounding::RoundingMode)

Create a representation of x as a BigFloat with precision prec and rounding mode rounding.

source

Base.MPFR.BigFloat — Method.

BigFloat(x::String)

Create a representation of the string x as a BigFloat.

source

Random Numbers

Random number generation in Julia uses the Mersenne Twister library via MersenneTwister objects. Julia has a global RNG, which is used by default. Other RNG types can be plugged in by inheriting the AbstractRNG type; they can then be used to have multiple streams of random numbers. Besides MersenneTwister, Julia also provides the RandomDevice RNG type, which is a wrapper over the OS provided entropy.

Most functions related to random generation accept an optional AbstractRNG as the first argument, rng , which defaults to the global one if not provided. Morever, some of them accept optionally dimension specifications dims... (which can be given as a tuple) to generate arrays of random values.

A MersenneTwister or RandomDevice RNG can generate random numbers of the following types: Float16, Float32, Float64, Bool, Int8, UInt8, Int16, UInt16, Int32, UInt32, Int64, UInt64, Int128, UInt128, BigInt (or complex numbers of those types). Random floating point numbers are generated uniformly in $[0, 1)$. As BigInt represents unbounded integers, the interval must be specified (e.g. rand(big(1:6))).

Base.Random.srand — Function.

srand([rng=GLOBAL_RNG], [seed]) -> rng
srand([rng=GLOBAL_RNG], filename, n=4) -> rng

Reseed the random number generator. If a seed is provided, the RNG will give a reproducible sequence of numbers, otherwise Julia will get entropy from the system. For MersenneTwister, the seed may be a non-negative integer, a vector of UInt32 integers or a filename, in which case the seed is read from a file (4n bytes are read from the file, where n is an optional argument). RandomDevice does not support seeding.

source

Base.Random.MersenneTwister — Type.

MersenneTwister(seed)

Create a MersenneTwister RNG object. Different RNG objects can have their own seeds, which may be useful for generating different streams of random numbers.

Example

julia> rng = MersenneTwister(1234);

source

Base.Random.RandomDevice — Type.

RandomDevice()

Create a RandomDevice RNG object. Two such objects will always generate different streams of random numbers.

source

Base.Random.rand — Function.

rand([rng=GLOBAL_RNG], [S], [dims...])

Pick a random element or array of random elements from the set of values specified by S; S can be

an indexable collection (for example 1:n or ['x','y','z']), or
a type: the set of values to pick from is then equivalent to typemin(S):typemax(S) for integers (this is not applicable to BigInt), and to $[0, 1)$ for floating point numbers;

S defaults to Float64.

source

Base.Random.rand! — Function.

rand!([rng=GLOBAL_RNG], A, [coll])

Populate the array A with random values. If the indexable collection coll is specified, the values are picked randomly from coll. This is equivalent to copy!(A, rand(rng, coll, size(A))) or copy!(A, rand(rng, eltype(A), size(A))) but without allocating a new array.

Example

julia> rng = MersenneTwister(1234);

julia> rand!(rng, zeros(5))
5-element Array{Float64,1}:
 0.590845
 0.766797
 0.566237
 0.460085
 0.794026

source

Base.Random.bitrand — Function.

bitrand([rng=GLOBAL_RNG], [dims...])

Generate a BitArray of random boolean values.

Example

julia> rng = MersenneTwister(1234);

julia> bitrand(rng, 10)
10-element BitArray{1}:
  true
  true
  true
 false
  true
 false
 false
  true
 false
  true

source

Base.Random.randn — Function.

randn([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a normally-distributed random number of type T with mean 0 and standard deviation 1. Optionally generate an array of normally-distributed random numbers. The Base module currently provides an implementation for the types Float16, Float32, and Float64 (the default).

Examples

julia> rng = MersenneTwister(1234);

julia> randn(rng, Float64)
0.8673472019512456

julia> randn(rng, Float32, (2, 4))
2×4 Array{Float32,2}:
 -0.901744  -0.902914  2.21188   -0.271735
 -0.494479   0.864401  0.532813   0.502334

source

Base.Random.randn! — Function.

randn!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array A with normally-distributed (mean 0, standard deviation 1) random numbers. Also see the rand function.

Example

julia> rng = MersenneTwister(1234);

julia> randn!(rng, zeros(5))
5-element Array{Float64,1}:
  0.867347
 -0.901744
 -0.494479
 -0.902914
  0.864401

source

Base.Random.randexp — Function.

randexp([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a random number of type T according to the exponential distribution with scale 1. Optionally generate an array of such random numbers. The Base module currently provides an implementation for the types Float16, Float32, and Float64 (the default).

Examples

julia> rng = MersenneTwister(1234);

julia> randexp(rng, Float32)
2.4835055f0

julia> randexp(rng, 3, 3)
3×3 Array{Float64,2}:
 1.5167    1.30652   0.344435
 0.604436  2.78029   0.418516
 0.695867  0.693292  0.643644

source

Base.Random.randexp! — Function.

randexp!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array A with random numbers following the exponential distribution (with scale 1).

Example

julia> rng = MersenneTwister(1234);

julia> randexp!(rng, zeros(5))
5-element Array{Float64,1}:
 2.48351
 1.5167
 0.604436
 0.695867
 1.30652

source

Base.Random.randjump — Function.

randjump(r::MersenneTwister, jumps::Integer, [jumppoly::AbstractString=dSFMT.JPOLY1e21]) -> Vector{MersenneTwister}

Create an array of the size jumps of initialized MersenneTwister RNG objects. The first RNG object given as a parameter and following MersenneTwister RNGs in the array are initialized such that a state of the RNG object in the array would be moved forward (without generating numbers) from a previous RNG object array element on a particular number of steps encoded by the jump polynomial jumppoly.

Default jump polynomial moves forward MersenneTwister RNG state by 10^20 steps.

source

Numbers

Standard Numeric Types

Abstract number types

Concrete number types

Data Formats

General Number Functions and Constants

Integers

BigFloats

Random Numbers