Julia provides a complete collection of basic arithmetic and bitwise operators across all of its numeric primitive types, as well as providing portable, efficient implementations of a comprehensive collection of standard mathematical functions.
The following arithmetic operators are supported on all primitive numeric types:
Expression  Name  Description 

+x 
unary plus  the identity operation 
x 
unary minus  maps values to their additive inverses 
x + y 
binary plus  performs addition 
x  y 
binary minus  performs subtraction 
x * y 
times  performs multiplication 
x / y 
divide  performs division 
x ÷ y 
integer divide  x / y, truncated to an integer 
x \ y 
inverse divide  equivalent to y / x

x ^ y 
power  raises x to the y th power 
x % y 
remainder  equivalent to rem(x,y)

as well as the negation on Bool
types:
Expression  Name  Description 

!x 
negation  changes true to false and vice versa 
Julia's promotion system makes arithmetic operations on mixtures of argument types "just work" naturally and automatically. See Conversion and Promotion for details of the promotion system.
Here are some simple examples using arithmetic operators:
julia> 1 + 2 + 3 6 julia> 1  2 1 julia> 3*2/12 0.5
(By convention, we tend to space operators more tightly if they get applied before other nearby operators. For instance, we would generally write x + 2
to reflect that first x
gets negated, and then 2
is added to that result.)
The following bitwise operators are supported on all primitive integer types:
Expression  Name 

~x 
bitwise not 
x & y 
bitwise and 
x  y 
bitwise or 
x ⊻ y 
bitwise xor (exclusive or) 
x >>> y 
logical shift right 
x >> y 
arithmetic shift right 
x << y 
logical/arithmetic shift left 
Here are some examples with bitwise operators:
julia> ~123 124 julia> 123 & 234 106 julia> 123  234 251 julia> 123 ⊻ 234 145 julia> xor(123, 234) 145 julia> ~UInt32(123) 0xffffff84 julia> ~UInt8(123) 0x84
Every binary arithmetic and bitwise operator also has an updating version that assigns the result of the operation back into its left operand. The updating version of the binary operator is formed by placing a =
immediately after the operator. For example, writing x += 3
is equivalent to writing x = x + 3
:
julia> x = 1 1 julia> x += 3 4 julia> x 4
The updating versions of all the binary arithmetic and bitwise operators are:
+= = *= /= \= ÷= %= ^= &= = ⊻= >>>= >>= <<=
An updating operator rebinds the variable on the lefthand side. As a result, the type of the variable may change.
julia> x = 0x01; typeof(x) UInt8 julia> x *= 2 # Same as x = x * 2 2 julia> typeof(x) Int64
For every binary operation like ^
, there is a corresponding "dot" operation .^
that is automatically defined to perform ^
elementbyelement on arrays. For example, [1,2,3] ^ 3
is not defined, since there is no standard mathematical meaning to "cubing" a (nonsquare) array, but [1,2,3] .^ 3
is defined as computing the elementwise (or "vectorized") result [1^3, 2^3, 3^3]
. Similarly for unary operators like !
or √
, there is a corresponding .√
that applies the operator elementwise.
julia> [1,2,3] .^ 3 3element Array{Int64,1}: 1 8 27
More specifically, a .^ b
is parsed as the "dot" call (^).(a,b)
, which performs a broadcast operation: it can combine arrays and scalars, arrays of the same size (performing the operation elementwise), and even arrays of different shapes (e.g. combining row and column vectors to produce a matrix). Moreover, like all vectorized "dot calls," these "dot operators" are fusing. For example, if you compute 2 .* A.^2 .+ sin.(A)
(or equivalently @. 2A^2 + sin(A)
, using the @.
macro) for an array A
, it performs a single loop over A
, computing 2a^2 + sin(a)
for each element of A
. In particular, nested dot calls like f.(g.(x))
are fused, and "adjacent" binary operators like x .+ 3 .* x.^2
are equivalent to nested dot calls (+).(x, (*).(3, (^).(x, 2)))
.
Furthermore, "dotted" updating operators like a .+= b
(or @. a += b
) are parsed as a .= a .+ b
, where .=
is a fused inplace assignment operation (see the dot syntax documentation).
Note the dot syntax is also applicable to userdefined operators. For example, if you define ⊗(A,B) = kron(A,B)
to give a convenient infix syntax A ⊗ B
for Kronecker products (kron
), then [A,B] .⊗ [C,D]
will compute [A⊗C, B⊗D]
with no additional coding.
Combining dot operators with numeric literals can be ambiguous. For example, it is not clear whether 1.+x
means 1. + x
or 1 .+ x
. Therefore this syntax is disallowed, and spaces must be used around the operator in such cases.
Standard comparison operations are defined for all the primitive numeric types:
Operator  Name 

== 
equality 
!= , ≠

inequality 
< 
less than 
<= , ≤

less than or equal to 
> 
greater than 
>= , ≥

greater than or equal to 
Here are some simple examples:
julia> 1 == 1 true julia> 1 == 2 false julia> 1 != 2 true julia> 1 == 1.0 true julia> 1 < 2 true julia> 1.0 > 3 false julia> 1 >= 1.0 true julia> 1 <= 1 true julia> 1 <= 1 true julia> 1 <= 2 false julia> 3 < 0.5 false
Integers are compared in the standard manner – by comparison of bits. Floatingpoint numbers are compared according to the IEEE 754 standard:
Inf
is equal to itself and greater than everything else except NaN
.Inf
is equal to itself and less then everything else except NaN
.NaN
is not equal to, not less than, and not greater than anything, including itself.The last point is potentially surprising and thus worth noting:
julia> NaN == NaN false julia> NaN != NaN true julia> NaN < NaN false julia> NaN > NaN false
and can cause especial headaches with arrays:
julia> [1 NaN] == [1 NaN] false
Julia provides additional functions to test numbers for special values, which can be useful in situations like hash key comparisons:
Function  Tests if 

isequal(x, y) 
x and y are identical 
isfinite(x) 
x is a finite number 
isinf(x) 
x is infinite 
isnan(x) 
x is not a number 
isequal
considers NaN
s equal to each other:
julia> isequal(NaN, NaN) true julia> isequal([1 NaN], [1 NaN]) true julia> isequal(NaN, NaN32) true
isequal
can also be used to distinguish signed zeros:
julia> 0.0 == 0.0 true julia> isequal(0.0, 0.0) false
Mixedtype comparisons between signed integers, unsigned integers, and floats can be tricky. A great deal of care has been taken to ensure that Julia does them correctly.
For other types, isequal
defaults to calling ==
, so if you want to define equality for your own types then you only need to add a ==
method. If you define your own equality function, you should probably define a corresponding hash
method to ensure that isequal(x,y)
implies hash(x) == hash(y)
.
Unlike most languages, with the notable exception of Python, comparisons can be arbitrarily chained:
julia> 1 < 2 <= 2 < 3 == 3 > 2 >= 1 == 1 < 3 != 5 true
Chaining comparisons is often quite convenient in numerical code. Chained comparisons use the &&
operator for scalar comparisons, and the &
operator for elementwise comparisons, which allows them to work on arrays. For example, 0 .< A .< 1
gives a boolean array whose entries are true where the corresponding elements of A
are between 0 and 1.
Note the evaluation behavior of chained comparisons:
julia> v(x) = (println(x); x) v (generic function with 1 method) julia> v(1) < v(2) <= v(3) 2 1 3 true julia> v(1) > v(2) <= v(3) 2 1 false
The middle expression is only evaluated once, rather than twice as it would be if the expression were written as v(1) < v(2) && v(2) <= v(3)
. However, the order of evaluations in a chained comparison is undefined. It is strongly recommended not to use expressions with side effects (such as printing) in chained comparisons. If side effects are required, the shortcircuit &&
operator should be used explicitly (see ShortCircuit Evaluation).
Julia provides a comprehensive collection of mathematical functions and operators. These mathematical operations are defined over as broad a class of numerical values as permit sensible definitions, including integers, floatingpoint numbers, rationals, and complex numbers, wherever such definitions make sense.
Moreover, these functions (like any Julia function) can be applied in "vectorized" fashion to arrays and other collections with the dot syntax f.(A)
, e.g. sin.(A)
will compute the sine of each element of an array A
.
Julia applies the following order and associativity of operations, from highest precedence to lowest:
Category  Operators  Associativity 

Syntax 
. followed by ::

Left 
Exponentiation  ^ 
Right 
Unary  +  √ 
Right[1] 
Bitshifts  << >> >>> 
Left 
Fractions  // 
Left 
Multiplication  * / % & \ ÷ 
Left[2] 
Addition  +   ⊻ 
Left[2] 
Syntax  : .. 
Left 
Syntax  > 
Left 
Syntax  < 
Right 
Comparisons  > < >= <= == === != !== <: 
Nonassociative 
Control flow 
&& followed by  followed by ?

Right 
Pair  => 
Right 
Assignments  = += = *= /= //= \= ^= ÷= %= = &= ⊻= <<= >>= >>>= 
Right 
The unary operators +
and 
require explicit parentheses around their argument to disambiguate them from the operator ++
, etc. Other compositions of unary operators are parsed with rightassociativity, e. g., √√a
as √(√(a))
.
The operators +
, ++
and *
are nonassociative. a + b + c
is parsed as +(a, b, c)
not +(+(a, b), c)
. However, the fallback methods for +(a, b, c, d...)
and *(a, b, c, d...)
both default to leftassociative evaluation.
For a complete list of every Julia operator's precedence, see the top of this file: src/juliaparser.scm
You can also find the numerical precedence for any given operator via the builtin function Base.operator_precedence
, where higher numbers take precedence:
julia> Base.operator_precedence(:+), Base.operator_precedence(:*), Base.operator_precedence(:.) (11, 13, 17) julia> Base.operator_precedence(:sin), Base.operator_precedence(:+=), Base.operator_precedence(:(=)) # (Note the necessary parens on `:(=)`) (0, 1, 1)
A symbol representing the operator associativity can also be found by calling the builtin function Base.operator_associativity
:
julia> Base.operator_associativity(:), Base.operator_associativity(:+), Base.operator_associativity(:^) (:left, :none, :right) julia> Base.operator_associativity(:⊗), Base.operator_associativity(:sin), Base.operator_associativity(:→) (:left, :none, :right)
Note that symbols such as :sin
return precedence 0
. This value represents invalid operators and not operators of lowest precedence. Similarly, such operators are assigned associativity :none
.
Julia supports three forms of numerical conversion, which differ in their handling of inexact conversions.
The notation T(x)
or convert(T,x)
converts x
to a value of type T
.
T
is a floatingpoint type, the result is the nearest representable value, which could be positive or negative infinity.T
is an integer type, an InexactError
is raised if x
is not representable by T
.x % T
converts an integer x
to a value of integer type T
congruent to x
modulo 2^n
, where n
is the number of bits in T
. In other words, the binary representation is truncated to fit.
The Rounding functions take a type T
as an optional argument. For example, round(Int,x)
is a shorthand for Int(round(x))
.
The following examples show the different forms.
julia> Int8(127) 127 julia> Int8(128) ERROR: InexactError: trunc(Int8, 128) Stacktrace: [...] julia> Int8(127.0) 127 julia> Int8(3.14) ERROR: InexactError: Int8(3.14) Stacktrace: [...] julia> Int8(128.0) ERROR: InexactError: Int8(128.0) Stacktrace: [...] julia> 127 % Int8 127 julia> 128 % Int8 128 julia> round(Int8,127.4) 127 julia> round(Int8,127.6) ERROR: InexactError: trunc(Int8, 128.0) Stacktrace: [...]
See Conversion and Promotion for how to define your own conversions and promotions.
Function  Description  Return type 

round(x) 
round x to the nearest integer 
typeof(x) 
round(T, x) 
round x to the nearest integer 
T 
floor(x) 
round x towards Inf

typeof(x) 
floor(T, x) 
round x towards Inf

T 
ceil(x) 
round x towards +Inf

typeof(x) 
ceil(T, x) 
round x towards +Inf

T 
trunc(x) 
round x towards zero 
typeof(x) 
trunc(T, x) 
round x towards zero 
T 
Function  Description 

div(x,y) , x÷y

truncated division; quotient rounded towards zero 
fld(x,y) 
floored division; quotient rounded towards Inf

cld(x,y) 
ceiling division; quotient rounded towards +Inf

rem(x,y) 
remainder; satisfies x == div(x,y)*y + rem(x,y) ; sign matches x

mod(x,y) 
modulus; satisfies x == fld(x,y)*y + mod(x,y) ; sign matches y

mod1(x,y) 
mod with offset 1; returns r∈(0,y] for y>0 or r∈[y,0) for y<0 , where mod(r, y) == mod(x, y)

mod2pi(x) 
modulus with respect to 2pi; 0 <= mod2pi(x) < 2pi

divrem(x,y) 
returns (div(x,y),rem(x,y))

fldmod(x,y) 
returns (fld(x,y),mod(x,y))

gcd(x,y...) 
greatest positive common divisor of x , y ,... 
lcm(x,y...) 
least positive common multiple of x , y ,... 
Function  Description 

abs(x) 
a positive value with the magnitude of x

abs2(x) 
the squared magnitude of x

sign(x) 
indicates the sign of x , returning 1, 0, or +1 
signbit(x) 
indicates whether the sign bit is on (true) or off (false) 
copysign(x,y) 
a value with the magnitude of x and the sign of y

flipsign(x,y) 
a value with the magnitude of x and the sign of x*y

Function  Description 

sqrt(x) , √x

square root of x

cbrt(x) , ∛x

cube root of x

hypot(x,y) 
hypotenuse of rightangled triangle with other sides of length x and y

exp(x) 
natural exponential function at x

expm1(x) 
accurate exp(x)1 for x near zero 
ldexp(x,n) 
x*2^n computed efficiently for integer values of n

log(x) 
natural logarithm of x

log(b,x) 
base b logarithm of x

log2(x) 
base 2 logarithm of x

log10(x) 
base 10 logarithm of x

log1p(x) 
accurate log(1+x) for x near zero 
exponent(x) 
binary exponent of x

significand(x) 
binary significand (a.k.a. mantissa) of a floatingpoint number x

For an overview of why functions like hypot
, expm1
, and log1p
are necessary and useful, see John D. Cook's excellent pair of blog posts on the subject: expm1, log1p, erfc, and hypot.
All the standard trigonometric and hyperbolic functions are also defined:
sin cos tan cot sec csc sinh cosh tanh coth sech csch asin acos atan acot asec acsc asinh acosh atanh acoth asech acsch sinc cosc
These are all singleargument functions, with atan
also accepting two arguments corresponding to a traditional atan2
function.
Additionally, sinpi(x)
and cospi(x)
are provided for more accurate computations of sin(pi*x)
and cos(pi*x)
respectively.
In order to compute trigonometric functions with degrees instead of radians, suffix the function with d
. For example, sind(x)
computes the sine of x
where x
is specified in degrees. The complete list of trigonometric functions with degree variants is:
sind cosd tand cotd secd cscd asind acosd atand acotd asecd acscd
Many other special mathematical functions are provided by the package SpecialFunctions.jl.
© 2009–2019 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
https://docs.julialang.org/en/v1.2.0/manual/mathematicaloperations/