# Mathematical Operations¶

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.

## Arithmetic and Bitwise Operators¶

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

`+x`

— unary plus is 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.

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

`~x`

— bitwise not.`x & y`

— bitwise and.`x | y`

— bitwise or.`x $ y`

— bitwise xor.`x >>> y`

— logical shift right.`x >> y`

— arithmetic shift right.`x << y`

— logical/arithmetic shift left.

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.)

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 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
```

Every binary arithmetic and bitwise operator also has an updating
version that assigns the result of the operation back into its left
operand. For example, the updating form of `+`

is the `+=`

operator.
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:

```
+= -= *= /= &= |= $= >>>= >>= <<=
```

## Numeric Comparisons¶

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

`==`

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

- finite numbers are ordered in the usual manner
`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, less than, or greater than anything, including itself.

The last point is potentially suprprising and thus worth noting:

```
julia> NaN == NaN
false
julia> NaN != NaN
true
julia> NaN < NaN
false
julia> NaN > NaN
false
```

For situations where one wants to compare floating-point values so that
`NaN`

equals `NaN`

, such as hash key comparisons, the function
`isequal`

is also provided, which considers `NaN`

s to be equal to
each other:

```
julia> isequal(NaN,NaN)
true
```

Mixed-type comparisons between signed integers, unsigned integers, and floats can be very tricky. A great deal of care has been taken to ensure that Julia does them correctly.

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 numeric comparisons use the `&`

operator, 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:

```
v(x) = (println(x); x)
julia> v(1) < v(2) <= v(3)
2
1
3
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 short-circuit `&&`

operator should
be used explicitly (see Short-Circuit Evaluation).

## Mathematical Functions¶

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, floating-point numbers, rationals, and complexes, wherever such definitions make sense.

`round(x)`

— round`x`

to the nearest integer.`iround(x)`

— round`x`

to the nearest integer, giving an integer-typed result.`floor(x)`

— round`x`

towards`-Inf`

.`ifloor(x)`

— round`x`

towards`-Inf`

, giving an integer-typed result.`ceil(x)`

— round`x`

towards`+Inf`

.`iceil(x)`

— round`x`

towards`+Inf`

, giving an integer-typed result.`trunc(x)`

— round`x`

towards zero.`itrunc(x)`

— round`x`

towards zero, giving an integer-typed result.`div(x,y)`

— truncated division; quotient rounded towards zero.`fld(x,y)`

— floored division; quotient rounded towards`-Inf`

.`rem(x,y)`

— remainder; satisfies`x == div(x,y)*y + rem(x,y)`

, implying that sign matches`x`

.`mod(x,y)`

— modulus; satisfies`x == fld(x,y)*y + mod(x,y)`

, implying that sign matches`y`

.`gcd(x,y...)`

— greatest common divisor of`x`

,`y`

... with sign matching`x`

.`lcm(x,y...)`

— least common multiple of`x`

,`y`

... with sign matching`x`

.`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 (1) or off (0).`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`

.`sqrt(x)`

— the square root of`x`

.`cbrt(x)`

— the cube root of`x`

.`hypot(x,y)`

— accurate`sqrt(x^2 + y^2)`

for all values of`x`

and`y`

.`exp(x)`

— the 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)`

— the natural logarithm of`x`

.`log(b,x)`

— the base`b`

logarithm of`x`

.`log2(x)`

— the base 2 logarithm of`x`

.`log10(x)`

— the base 10 logarithm of`x`

.`log1p(x)`

— accurate`log(1+x)`

for`x`

near zero.`logb(x)`

— returns the binary exponent of`x`

.`erf(x)`

— the error function at`x`

.`erfc(x)`

— accurate`1-erf(x)`

for large`x`

.`gamma(x)`

— the gamma function at`x`

.`lgamma(x)`

— accurate`log(gamma(x))`

for large`x`

.

For an overview of why functions like `hypot`

, `expm1`

, `log1p`

,
and `erfc`

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 functions are also defined:

```
sin cos tan cot sec csc
sinh cosh tanh coth sech csch
asin acos atan acot asec acsc
acoth asech acsch sinc cosc atan2
```

These are all single-argument functions, with the exception of
atan2, which gives the angle
in radians between the *x*-axis
and the point specified by its arguments, interpreted as *x* and *y*
coordinates. 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.

For notational convenience, the `rem`

functions has an operator form:

`x % y`

is equivalent to`rem(x,y)`

.

The spelled-out `rem`

operator is the “canonical” form, while the `%`

operator
form is retained for compatibility with other systems. Like arithmetic and bitwise
operators, `%`

and `^`

also have updating forms. As with other updating forms,
`x %= y`

means `x = x % y`

and `x ^= y`

means `x = x^y`

:

```
julia> x = 2; x ^= 5; x
32
julia> x = 7; x %= 4; x
3
```