Mathematical Operations and Elementary Functions¶

Arithmetic Operators¶

The following arithmetic operators are supported on all primitive numeric types:

as well as the negation on Bool types:

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 less tightly binding operators less tightly, but there are no syntactic constraints.)

Bitwise Operators¶

The following bitwise operators are supported on all primitive integer types:

Here are some examples with bitwise operators:

julia> ~123
-124

julia> 123 & 234
106

julia> 123 | 234
251

julia> 123 $ 234
145

julia> ~UInt32(123)
0xffffff84

julia> ~UInt8(123)
0x84

Updating operators¶

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:

+=  -=  *=  /=  \=  ÷=  %=  ^=  &=  |=  $=  >>>=  >>=  <<=

julia> x = 0x01; typeof(x)
UInt8

julia> x *= 2 #Same as x = x * 2
2

julia> isa(x, Int)
true

Numeric Comparisons¶

Standard comparison operations are defined for all the primitive numeric types:

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. Floating-point numbers are compared according to the IEEE 754 standard:

• Finite numbers are ordered in the usual manner.
• Positive zero is equal but not greater than negative zero.
• 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:

isequal() considers NaNs 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

Mixed-type 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).

julia> 1 < 2 <= 2 < 3 == 3 > 2 >= 1 == 1 < 3 != 5
true

v(x) = (println(x); x)

julia> v(1) < v(2) <= v(3)
2
1
3
true

julia> v(1) > v(2) <= v(3)
2
1
false

Numerical Conversions¶

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.
  • If T is a floating-point type, the result is the nearest representable value, which could be positive or negative infinity.
  • If 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()
 in call at essentials.jl:56

julia> Int8(127.0)
127

julia> Int8(3.14)
ERROR: InexactError()
 in call at essentials.jl:56

julia> Int8(128.0)
ERROR: InexactError()
 in call at essentials.jl:56

julia> 127 % Int8
127

julia> 128 % Int8
-128

julia> round(Int8,127.4)
127

julia> round(Int8,127.6)
ERROR: InexactError()
 in trunc at float.jl:374
 in round at float.jl:181

See Conversion and Promotion for how to define your own conversions and promotions.

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   atan2

sind   cosd   tand   cotd   secd   cscd
asind  acosd  atand  acotd  asecd  acscd