Numbers

Numbers

Standard Numeric Types

A type tree for all subtypes of Number in Base is shown below. Abstract types have been marked, the rest are concrete types.

Number  (Abstract Type)
├─ Complex
└─ Real  (Abstract Type)
   ├─ AbstractFloat  (Abstract Type)
   │  ├─ Float16
   │  ├─ Float32
   │  ├─ Float64
   │  └─ BigFloat
   ├─ Integer  (Abstract Type)
   │  ├─ Bool
   │  ├─ Signed  (Abstract Type)
   │  │  ├─ Int8
   │  │  ├─ Int16
   │  │  ├─ Int32
   │  │  ├─ Int64
   │  │  ├─ Int128
   │  │  └─ BigInt
   │  └─ Unsigned  (Abstract Type)
   │     ├─ UInt8
   │     ├─ UInt16
   │     ├─ UInt32
   │     ├─ UInt64
   │     └─ UInt128
   ├─ Rational
   └─ AbstractIrrational  (Abstract Type)
      └─ Irrational

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, 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.Float16 — Type

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.Float32 — Type

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.Float64 — Type

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.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.

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

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)).

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

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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 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

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Base.bitstring — Function

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"

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Base.parse — Function

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

Parses a string platform triplet back into a Platform object.

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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(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.

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.

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

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Base.oneunit — Function

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

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Base.zero — Function

zero(x)
zero(::Type)

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

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
julia> count_ones(Int32(-1))
32

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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
julia> count_zeros(-1)
0

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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::Number) -> Bool

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

Julia 1.7

Non-Integer arguments require Julia 1.7 or later.

Examples

julia> isodd(9)
true
julia> isodd(10)
false

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Base.iseven — Function

iseven(x::Number) -> Bool

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

Julia 1.7

Non-Integer arguments require Julia 1.7 or later.

Examples

julia> iseven(9)
false
julia> iseven(10)
true

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Core.@int128_str — Macro

@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"
[...]

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Core.@uint128_str — Macro

@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"
[...]

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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.