# Mathematics

## Mathematical Operators

Base.:-Method
-(x)

Unary minus operator.

Examples

julia> -1
-1

julia> -(2)
-2

julia> -[1 2; 3 4]
2×2 Array{Int64,2}:
-1  -2
-3  -4
source
Base.:+Function
dt::Date + t::Time -> DateTime

The addition of a Date with a Time produces a DateTime. The hour, minute, second, and millisecond parts of the Time are used along with the year, month, and day of the Date to create the new DateTime. Non-zero microseconds or nanoseconds in the Time type will result in an InexactError being thrown.

source
+(x, y...)

Addition operator. x+y+z+... calls this function with all arguments, i.e. +(x, y, z, ...).

Examples

julia> 1 + 20 + 4
25

julia> +(1, 20, 4)
25
source
Base.:-Method
-(x, y)

Subtraction operator.

Examples

julia> 2 - 3
-1

julia> -(2, 4.5)
-2.5
source
Base.:*Method
*(x, y...)

Multiplication operator. x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...).

Examples

julia> 2 * 7 * 8
112

julia> *(2, 7, 8)
112
source
Base.:/Function
/(x, y)

Right division operator: multiplication of x by the inverse of y on the right. Gives floating-point results for integer arguments.

Examples

julia> 1/2
0.5

julia> 4/2
2.0

julia> 4.5/2
2.25
source
Base.:\Method
\(x, y)

Left division operator: multiplication of y by the inverse of x on the left. Gives floating-point results for integer arguments.

Examples

julia> 3 \ 6
2.0

julia> inv(3) * 6
2.0

julia> A = [4 3; 2 1]; x = [5, 6];

julia> A \ x
2-element Array{Float64,1}:
6.5
-7.0

julia> inv(A) * x
2-element Array{Float64,1}:
6.5
-7.0
source
Base.:^Method
^(x, y)

Exponentiation operator. If x is a matrix, computes matrix exponentiation.

If y is an Int literal (e.g. 2 in x^2 or -3 in x^-3), the Julia code x^y is transformed by the compiler to Base.literal_pow(^, x, Val(y)), to enable compile-time specialization on the value of the exponent. (As a default fallback we have Base.literal_pow(^, x, Val(y)) = ^(x,y), where usually ^ == Base.^ unless ^ has been defined in the calling namespace.)

julia> 3^5
243

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> A^3
2×2 Array{Int64,2}:
37   54
81  118
source
Base.fmaFunction
fma(x, y, z)

Computes x*y+z without rounding the intermediate result x*y. On some systems this is significantly more expensive than x*y+z. fma is used to improve accuracy in certain algorithms. See muladd.

source

Combined multiply-add: computes x*y+z, but allowing the add and multiply to be merged with each other or with surrounding operations for performance. For example, this may be implemented as an fma if the hardware supports it efficiently. The result can be different on different machines and can also be different on the same machine due to constant propagation or other optimizations. See fma.

Examples

7

julia> 3 * 2 + 1
7
source
Base.invMethod
inv(x)

Return the multiplicative inverse of x, such that x*inv(x) or inv(x)*x yields one(x) (the multiplicative identity) up to roundoff errors.

If x is a number, this is essentially the same as one(x)/x, but for some types inv(x) may be slightly more efficient.

Examples

julia> inv(2)
0.5

julia> inv(1 + 2im)
0.2 - 0.4im

julia> inv(1 + 2im) * (1 + 2im)
1.0 + 0.0im

julia> inv(2//3)
3//2
Julia 1.2

inv(::Missing) requires at least Julia 1.2.

source
Base.divFunction
div(x, y)
÷(x, y)

The quotient from Euclidean division. Computes x/y, truncated to an integer.

Examples

julia> 9 ÷ 4
2

julia> -5 ÷ 3
-1

julia> 5.0 ÷ 2
2.0
source
Base.fldFunction
fld(x, y)

Largest integer less than or equal to x/y. Equivalent to div(x, y, RoundDown).

Examples

julia> fld(7.3,5.5)
1.0
source
Base.cldFunction
cld(x, y)

Smallest integer larger than or equal to x/y. Equivalent to div(x, y, RoundUp).

Examples

julia> cld(5.5,2.2)
3.0
source
Base.modFunction
mod(x::Integer, r::AbstractUnitRange)

Find y in the range r such that $x ≡ y (mod n)$, where n = length(r), i.e. y = mod(x - first(r), n) + first(r).

Examples

julia> mod(0, Base.OneTo(3))
3

julia> mod(3, 0:2)
0
Julia 1.3

This method requires at least Julia 1.3.

source
mod(x, y)
rem(x, y, RoundDown)

The reduction of x modulo y, or equivalently, the remainder of x after floored division by y, i.e. x - y*fld(x,y) if computed without intermediate rounding.

The result will have the same sign as y, and magnitude less than abs(y) (with some exceptions, see note below).

Note

When used with floating point values, the exact result may not be representable by the type, and so rounding error may occur. In particular, if the exact result is very close to y, then it may be rounded to y.

julia> mod(8, 3)
2

julia> mod(9, 3)
0

julia> mod(8.9, 3)
2.9000000000000004

julia> mod(eps(), 3)
2.220446049250313e-16

julia> mod(-eps(), 3)
3.0
source
rem(x::Integer, T::Type{<:Integer}) -> T
mod(x::Integer, T::Type{<:Integer}) -> T
%(x::Integer, T::Type{<:Integer}) -> T

Find y::T such that xy (mod n), where n is the number of integers representable in T, and y is an integer in [typemin(T),typemax(T)]. If T can represent any integer (e.g. T == BigInt), then this operation corresponds to a conversion to T.

Examples

julia> 129 % Int8
-127
source
Base.remFunction
rem(x, y)
%(x, y)

Remainder from Euclidean division, returning a value of the same sign as x, and smaller in magnitude than y. This value is always exact.

Examples

julia> x = 15; y = 4;

julia> x % y
3

julia> x == div(x, y) * y + rem(x, y)
true
source
Base.Math.rem2piFunction
rem2pi(x, r::RoundingMode)

Compute the remainder of x after integer division by , with the quotient rounded according to the rounding mode r. In other words, the quantity

x - 2π*round(x/(2π),r)

without any intermediate rounding. This internally uses a high precision approximation of 2π, and so will give a more accurate result than rem(x,2π,r)

• if r == RoundNearest, then the result is in the interval $[-π, π]$. This will generally be the most accurate result. See also RoundNearest.

• if r == RoundToZero, then the result is in the interval $[0, 2π]$ if x is positive,. or $[-2π, 0]$ otherwise. See also RoundToZero.

• if r == RoundDown, then the result is in the interval $[0, 2π]$. See also RoundDown.

• if r == RoundUp, then the result is in the interval $[-2π, 0]$. See also RoundUp.

Examples

julia> rem2pi(7pi/4, RoundNearest)
-0.7853981633974485

julia> rem2pi(7pi/4, RoundDown)
5.497787143782138
source
Base.Math.mod2piFunction
mod2pi(x)

Modulus after division by , returning in the range $[0,2π)$.

This function computes a floating point representation of the modulus after division by numerically exact , and is therefore not exactly the same as mod(x,2π), which would compute the modulus of x relative to division by the floating-point number .

Note

Depending on the format of the input value, the closest representable value to 2π may be less than 2π. For example, the expression mod2pi(2π) will not return 0, because the intermediate value of 2*π is a Float64 and 2*Float64(π) < 2*big(π). See rem2pi for more refined control of this behavior.

Examples

julia> mod2pi(9*pi/4)
0.7853981633974481
source
Base.divremFunction
divrem(x, y, r::RoundingMode=RoundToZero)

The quotient and remainder from Euclidean division. Equivalent to (div(x,y,r), rem(x,y,r)). Equivalently, with the the default value of r, this call is equivalent to (x÷y, x%y).

Examples

julia> divrem(3,7)
(0, 3)

julia> divrem(7,3)
(2, 1)
source
Base.fldmodFunction
fldmod(x, y)

The floored quotient and modulus after division. A convenience wrapper for divrem(x, y, RoundDown). Equivalent to (fld(x,y), mod(x,y)).

source
Base.fld1Function
fld1(x, y)

Flooring division, returning a value consistent with mod1(x,y)

Examples

julia> x = 15; y = 4;

julia> fld1(x, y)
4

julia> x == fld(x, y) * y + mod(x, y)
true

julia> x == (fld1(x, y) - 1) * y + mod1(x, y)
true
source
Base.mod1Function
mod1(x, y)

Modulus after flooring division, returning a value r such that mod(r, y) == mod(x, y) in the range $(0, y]$ for positive y and in the range $[y,0)$ for negative y.

Examples

julia> mod1(4, 2)
2

julia> mod1(4, 3)
1
source
Base.://Function
//(num, den)

Divide two integers or rational numbers, giving a Rational result.

Examples

julia> 3 // 5
3//5

julia> (3 // 5) // (2 // 1)
3//10
source
Base.rationalizeFunction
rationalize([T<:Integer=Int,] x; tol::Real=eps(x))

Approximate floating point number x as a Rational number with components of the given integer type. The result will differ from x by no more than tol.

Examples

julia> rationalize(5.6)
28//5

julia> a = rationalize(BigInt, 10.3)
103//10

julia> typeof(numerator(a))
BigInt
source
Base.numeratorFunction
numerator(x)

Numerator of the rational representation of x.

Examples

julia> numerator(2//3)
2

julia> numerator(4)
4
source
Base.denominatorFunction
denominator(x)

Denominator of the rational representation of x.

Examples

julia> denominator(2//3)
3

julia> denominator(4)
1
source
Base.:<<Function
<<(x, n)

Left bit shift operator, x << n. For n >= 0, the result is x shifted left by n bits, filling with 0s. This is equivalent to x * 2^n. For n < 0, this is equivalent to x >> -n.

Examples

julia> Int8(3) << 2
12

julia> bitstring(Int8(3))
"00000011"

julia> bitstring(Int8(12))
"00001100"

source
<<(B::BitVector, n) -> BitVector

Left bit shift operator, B << n. For n >= 0, the result is B with elements shifted n positions backwards, filling with false values. If n < 0, elements are shifted forwards. Equivalent to B >> -n.

Examples

julia> B = BitVector([true, false, true, false, false])
5-element BitArray{1}:
1
0
1
0
0

julia> B << 1
5-element BitArray{1}:
0
1
0
0
0

julia> B << -1
5-element BitArray{1}:
0
1
0
1
0
source
Base.:>>Function
>>(x, n)

Right bit shift operator, x >> n. For n >= 0, the result is x shifted right by n bits, where n >= 0, filling with 0s if x >= 0, 1s if x < 0, preserving the sign of x. This is equivalent to fld(x, 2^n). For n < 0, this is equivalent to x << -n.

Examples

julia> Int8(13) >> 2
3

julia> bitstring(Int8(13))
"00001101"

julia> bitstring(Int8(3))
"00000011"

julia> Int8(-14) >> 2
-4

julia> bitstring(Int8(-14))
"11110010"

julia> bitstring(Int8(-4))
"11111100"

source
>>(B::BitVector, n) -> BitVector

Right bit shift operator, B >> n. For n >= 0, the result is B with elements shifted n positions forward, filling with false values. If n < 0, elements are shifted backwards. Equivalent to B << -n.

Examples

julia> B = BitVector([true, false, true, false, false])
5-element BitArray{1}:
1
0
1
0
0

julia> B >> 1
5-element BitArray{1}:
0
1
0
1
0

julia> B >> -1
5-element BitArray{1}:
0
1
0
0
0
source
Base.:>>>Function
>>>(x, n)

Unsigned right bit shift operator, x >>> n. For n >= 0, the result is x shifted right by n bits, where n >= 0, filling with 0s. For n < 0, this is equivalent to x << -n.

For Unsigned integer types, this is equivalent to >>. For Signed integer types, this is equivalent to signed(unsigned(x) >> n).

Examples

julia> Int8(-14) >>> 2
60

julia> bitstring(Int8(-14))
"11110010"

julia> bitstring(Int8(60))
"00111100"

BigInts are treated as if having infinite size, so no filling is required and this is equivalent to >>.

source
>>>(B::BitVector, n) -> BitVector

Unsigned right bitshift operator, B >>> n. Equivalent to B >> n. See >> for details and examples.

source
Base.::Function
(:)(start, [step], stop)

Range operator. a:b constructs a range from a to b with a step size of 1 (a UnitRange) , and a:s:b is similar but uses a step size of s (a StepRange).

: is also used in indexing to select whole dimensions and for Symbol literals, as in e.g. :hello.

source
(:)(I::CartesianIndex, J::CartesianIndex)

Construct CartesianIndices from two CartesianIndex.

Julia 1.1

This method requires at least Julia 1.1.

Examples

julia> I = CartesianIndex(2,1);

julia> J = CartesianIndex(3,3);

julia> I:J
2×3 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
CartesianIndex(2, 1)  CartesianIndex(2, 2)  CartesianIndex(2, 3)
CartesianIndex(3, 1)  CartesianIndex(3, 2)  CartesianIndex(3, 3)
source
Base.rangeFunction
range(start[, stop]; length, stop, step=1)

Given a starting value, construct a range either by length or from start to stop, optionally with a given step (defaults to 1, a UnitRange). One of length or stop is required. If length, stop, and step are all specified, they must agree.

If length and stop are provided and step is not, the step size will be computed automatically such that there are length linearly spaced elements in the range (a LinRange).

If step and stop are provided and length is not, the overall range length will be computed automatically such that the elements are step spaced (a StepRange).

stop may be specified as either a positional or keyword argument.

Julia 1.1

stop as a positional argument requires at least Julia 1.1.

Examples

julia> range(1, length=100)
1:100

julia> range(1, stop=100)
1:100

julia> range(1, step=5, length=100)
1:5:496

julia> range(1, step=5, stop=100)
1:5:96

julia> range(1, 10, length=101)
1.0:0.09:10.0

julia> range(1, 100, step=5)
1:5:96
source
Base.OneToType
Base.OneTo(n)

Define an AbstractUnitRange that behaves like 1:n, with the added distinction that the lower limit is guaranteed (by the type system) to be 1.

source
Base.StepRangeLenType
StepRangeLen{T,R,S}(ref::R, step::S, len, [offset=1]) where {T,R,S}
StepRangeLen(       ref::R, step::S, len, [offset=1]) where {  R,S}

A range r where r[i] produces values of type T (in the second form, T is deduced automatically), parameterized by a reference value, a step, and the length. By default ref is the starting value r[1], but alternatively you can supply it as the value of r[offset] for some other index 1 <= offset <= len. In conjunction with TwicePrecision this can be used to implement ranges that are free of roundoff error.

source
Base.:==Function
==(x, y)

Generic equality operator. Falls back to ===. Should be implemented for all types with a notion of equality, based on the abstract value that an instance represents. For example, all numeric types are compared by numeric value, ignoring type. Strings are compared as sequences of characters, ignoring encoding. For collections, == is generally called recursively on all contents, though other properties (like the shape for arrays) may also be taken into account.

This operator follows IEEE semantics for floating-point numbers: 0.0 == -0.0 and NaN != NaN.

The result is of type Bool, except when one of the operands is missing, in which case missing is returned (three-valued logic). For collections, missing is returned if at least one of the operands contains a missing value and all non-missing values are equal. Use isequal or === to always get a Bool result.

Implementation

New numeric types should implement this function for two arguments of the new type, and handle comparison to other types via promotion rules where possible.

isequal falls back to ==, so new methods of == will be used by the Dict type to compare keys. If your type will be used as a dictionary key, it should therefore also implement hash.

source
==(x)

Create a function that compares its argument to x using ==, i.e. a function equivalent to y -> y == x.

The returned function is of type Base.Fix2{typeof(==)}, which can be used to implement specialized methods.

source
==(a::AbstractString, b::AbstractString) -> Bool

Test whether two strings are equal character by character (technically, Unicode code point by code point).

Examples

julia> "abc" == "abc"
true

julia> "abc" == "αβγ"
false
source
Base.:!=Function
!=(x, y)
≠(x,y)

Not-equals comparison operator. Always gives the opposite answer as ==.

Implementation

New types should generally not implement this, and rely on the fallback definition !=(x,y) = !(x==y) instead.

Examples

julia> 3 != 2
true

julia> "foo" ≠ "foo"
false
source
!=(x)

Create a function that compares its argument to x using !=, i.e. a function equivalent to y -> y != x. The returned function is of type Base.Fix2{typeof(!=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:!==Function
!==(x, y)
≢(x,y)

Always gives the opposite answer as ===.

Examples

julia> a = [1, 2]; b = [1, 2];

julia> a ≢ b
true

julia> a ≢ a
false
source
Base.:<Function
<(x, y)

Less-than comparison operator. Falls back to isless. Because of the behavior of floating-point NaN values, this operator implements a partial order.

Implementation

New numeric types with a canonical partial order should implement this function for two arguments of the new type. Types with a canonical total order should implement isless instead. (x < y) | (x == y)

Examples

julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false
source
<(x)

Create a function that compares its argument to x using <, i.e. a function equivalent to y -> y < x. The returned function is of type Base.Fix2{typeof(<)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:<=Function
<=(x, y)
≤(x,y)

Less-than-or-equals comparison operator. Falls back to (x < y) | (x == y).

Examples

julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false
source
<=(x)

Create a function that compares its argument to x using <=, i.e. a function equivalent to y -> y <= x. The returned function is of type Base.Fix2{typeof(<=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:>Function
>(x, y)

Greater-than comparison operator. Falls back to y < x.

Implementation

Generally, new types should implement < instead of this function, and rely on the fallback definition >(x, y) = y < x.

Examples

julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true
source
>(x)

Create a function that compares its argument to x using >, i.e. a function equivalent to y -> y > x. The returned function is of type Base.Fix2{typeof(>)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.:>=Function
>=(x, y)
≥(x,y)

Greater-than-or-equals comparison operator. Falls back to y <= x.

Examples

julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true
source
>=(x)

Create a function that compares its argument to x using >=, i.e. a function equivalent to y -> y >= x. The returned function is of type Base.Fix2{typeof(>=)}, which can be used to implement specialized methods.

Julia 1.2

This functionality requires at least Julia 1.2.

source
Base.cmpFunction
cmp(x,y)

Return -1, 0, or 1 depending on whether x is less than, equal to, or greater than y, respectively. Uses the total order implemented by isless.

Examples

julia> cmp(1, 2)
-1

julia> cmp(2, 1)
1

julia> cmp(2+im, 3-im)
ERROR: MethodError: no method matching isless(::Complex{Int64}, ::Complex{Int64})
[...]
source
cmp(<, x, y)

Return -1, 0, or 1 depending on whether x is less than, equal to, or greater than y, respectively. The first argument specifies a less-than comparison function to use.

source
cmp(a::AbstractString, b::AbstractString) -> Int

Compare two strings. Return 0 if both strings have the same length and the character at each index is the same in both strings. Return -1 if a is a prefix of b, or if a comes before b in alphabetical order. Return 1 if b is a prefix of a, or if b comes before a in alphabetical order (technically, lexicographical order by Unicode code points).

Examples

julia> cmp("abc", "abc")
0

julia> cmp("ab", "abc")
-1

julia> cmp("abc", "ab")
1

julia> cmp("ab", "ac")
-1

julia> cmp("ac", "ab")
1

julia> cmp("α", "a")
1

julia> cmp("b", "β")
-1
source
Base.:~Function
~(x)

Bitwise not.

Examples

julia> ~4
-5

julia> ~10
-11

julia> ~true
false
source
Base.:&Function
&(x, y)

Bitwise and. Implements three-valued logic, returning missing if one operand is missing and the other is true.

Examples

julia> 4 & 10
0

julia> 4 & 12
4

julia> true & missing
missing

julia> false & missing
false
source
Base.:|Function
|(x, y)

Bitwise or. Implements three-valued logic, returning missing if one operand is missing and the other is false.

Examples

julia> 4 | 10
14

julia> 4 | 1
5

julia> true | missing
true

julia> false | missing
missing
source
Base.xorFunction
xor(x, y)
⊻(x, y)

Bitwise exclusive or of x and y. Implements three-valued logic, returning missing if one of the arguments is missing.

The infix operation a ⊻ b is a synonym for xor(a,b), and can be typed by tab-completing \xor or \veebar in the Julia REPL.

Examples

julia> xor(true, false)
true

julia> xor(true, true)
false

julia> xor(true, missing)
missing

julia> false ⊻ false
false

julia> [true; true; false] .⊻ [true; false; false]
3-element BitArray{1}:
0
1
0
source
Base.:!Function
!(x)

Boolean not. Implements three-valued logic, returning missing if x is missing.

Examples

julia> !true
false

julia> !false
true

julia> !missing
missing

julia> .![true false true]
1×3 BitArray{2}:
0  1  0
source
!f::Function

Predicate function negation: when the argument of ! is a function, it returns a function which computes the boolean negation of f.

Examples

julia> str = "∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"
"∀ ε > 0, ∃ δ > 0: |x-y| < δ ⇒ |f(x)-f(y)| < ε"

julia> filter(isletter, str)
"εδxyδfxfyε"

julia> filter(!isletter, str)
"∀  > 0, ∃  > 0: |-| <  ⇒ |()-()| < "
source
&&Keyword
x && y

Short-circuiting boolean AND.

source
||Keyword
x || y

Short-circuiting boolean OR.

source

## Mathematical Functions

Base.isapproxFunction
isapprox(x, y; rtol::Real=atol>0 ? 0 : √eps, atol::Real=0, nans::Bool=false, norm::Function)

Inexact equality comparison: true if norm(x-y) <= max(atol, rtol*max(norm(x), norm(y))). The default atol is zero and the default rtol depends on the types of x and y. The keyword argument nans determines whether or not NaN values are considered equal (defaults to false).

For real or complex floating-point values, if an atol > 0 is not specified, rtol defaults to the square root of eps of the type of x or y, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an atol > 0 is supplied, rtol defaults to zero.

x and y may also be arrays of numbers, in which case norm defaults to the usual norm function in LinearAlgebra, but may be changed by passing a norm::Function keyword argument. (For numbers, norm is the same thing as abs.) When x and y are arrays, if norm(x-y) is not finite (i.e. ±Inf or NaN), the comparison falls back to checking whether all elements of x and y are approximately equal component-wise.

The binary operator is equivalent to isapprox with the default arguments, and x ≉ y is equivalent to !isapprox(x,y).

Note that x ≈ 0 (i.e., comparing to zero with the default tolerances) is equivalent to x == 0 since the default atol is 0. In such cases, you should either supply an appropriate atol (or use norm(x) ≤ atol) or rearrange your code (e.g. use x ≈ y rather than x - y ≈ 0). It is not possible to pick a nonzero atol automatically because it depends on the overall scaling (the "units") of your problem: for example, in x - y ≈ 0, atol=1e-9 is an absurdly small tolerance if x is the radius of the Earth in meters, but an absurdly large tolerance if x is the radius of a Hydrogen atom in meters.

Examples

julia> 0.1 ≈ (0.1 - 1e-10)
true

julia> isapprox(10, 11; atol = 2)
true

julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])
true

julia> 1e-10 ≈ 0
false

julia> isapprox(1e-10, 0, atol=1e-8)
true
source
Base.sinMethod
sin(x)

Compute sine of x, where x is in radians.

source
Base.cosMethod
cos(x)

Compute cosine of x, where x is in radians.

source
Base.Math.sincosMethod
sincos(x)

Simultaneously compute the sine and cosine of x, where the x is in radians.

source
Base.tanMethod
tan(x)

Compute tangent of x, where x is in radians.

source
Base.Math.sinpiFunction
sinpi(x)

Compute $\sin(\pi x)$ more accurately than sin(pi*x), especially for large x.

source
Base.Math.cospiFunction
cospi(x)

Compute $\cos(\pi x)$ more accurately than cos(pi*x), especially for large x.

source
Base.asinMethod
asin(x)

Compute the inverse sine of x, where the output is in radians.

source
Base.acosMethod
acos(x)

Compute the inverse cosine of x, where the output is in radians

source
Base.atanMethod
atan(y)
atan(y, x)

Compute the inverse tangent of y or y/x, respectively.

For one argument, this is the angle in radians between the positive x-axis and the point (1, y), returning a value in the interval $[-\pi/2, \pi/2]$.

For two arguments, this is the angle in radians between the positive x-axis and the point (x, y), returning a value in the interval $[-\pi, \pi]$. This corresponds to a standard atan2 function.

source
Base.Math.asindFunction
asind(x)

Compute the inverse sine of x, where the output is in degrees.

source
Base.Math.acosdFunction
acosd(x)

Compute the inverse cosine of x, where the output is in degrees.

source
Base.Math.atandFunction
atand(y)
atand(y,x)

Compute the inverse tangent of y or y/x, respectively, where the output is in degrees.

source
Base.Math.asecMethod
asec(x)

Compute the inverse secant of x, where the output is in radians.

source
Base.Math.acscMethod
acsc(x)

Compute the inverse cosecant of x, where the output is in radians.

source
Base.Math.acotMethod
acot(x)

Compute the inverse cotangent of x, where the output is in radians.

source
Base.Math.asecdFunction
asecd(x)

Compute the inverse secant of x, where the output is in degrees.

source
Base.Math.acscdFunction
acscd(x)

Compute the inverse cosecant of x, where the output is in degrees.

source
Base.Math.acotdFunction
acotd(x)

Compute the inverse cotangent of x, where the output is in degrees.

source
Base.Math.sincFunction
sinc(x)

Compute $\sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.

source
Base.Math.coscFunction
cosc(x)

Compute $\cos(\pi x) / x - \sin(\pi x) / (\pi x^2)$ if $x \neq 0$, and $0$ if $x = 0$. This is the derivative of sinc(x).

source

Convert x from degrees to radians.

Examples

1.5707963267948966
source
Base.Math.hypotFunction
hypot(x, y)

Compute the hypotenuse $\sqrt{|x|^2+|y|^2}$ avoiding overflow and underflow.

This code is an implementation of the algorithm described in: An Improved Algorithm for hypot(a,b) by Carlos F. Borges The article is available online at ArXiv at the link https://arxiv.org/abs/1904.09481

Examples

julia> a = Int64(10)^10;

julia> hypot(a, a)
1.4142135623730951e10

julia> √(a^2 + a^2) # a^2 overflows
ERROR: DomainError with -2.914184810805068e18:
sqrt will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
Stacktrace:
[...]

julia> hypot(3, 4im)
5.0
source
hypot(x...)

Compute the hypotenuse $\sqrt{\sum |x_i|^2}$ avoiding overflow and underflow.

Examples

julia> hypot(-5.7)
5.7

julia> hypot(3, 4im, 12.0)
13.0
source
Base.logMethod
log(x)

Compute the natural logarithm of x. Throws DomainError for negative Real arguments. Use complex negative arguments to obtain complex results.

Examples

julia> log(2)
0.6931471805599453

julia> log(-3)
ERROR: DomainError with -3.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
source
Base.logMethod
log(b,x)

Compute the base b logarithm of x. Throws DomainError for negative Real arguments.

Examples

julia> log(4,8)
1.5

julia> log(4,2)
0.5

julia> log(-2, 3)
ERROR: DomainError with -2.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]

julia> log(2, -3)
ERROR: DomainError with -3.0:
log will only return a complex result if called with a complex argument. Try log(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
Note

If b is a power of 2 or 10, log2 or log10 should be used, as these will typically be faster and more accurate. For example,

julia> log(100,1000000)
2.9999999999999996

julia> log10(1000000)/2
3.0
source
Base.log2Function
log2(x)

Compute the logarithm of x to base 2. Throws DomainError for negative Real arguments.

Examples

julia> log2(4)
2.0

julia> log2(10)
3.321928094887362

julia> log2(-2)
ERROR: DomainError with -2.0:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
source
Base.log10Function
log10(x)

Compute the logarithm of x to base 10. Throws DomainError for negative Real arguments.

Examples

julia> log10(100)
2.0

julia> log10(2)
0.3010299956639812

julia> log10(-2)
ERROR: DomainError with -2.0:
NaN result for non-NaN input.
Stacktrace:
[1] nan_dom_err at ./math.jl:325 [inlined]
[...]
source
Base.log1pFunction
log1p(x)

Accurate natural logarithm of 1+x. Throws DomainError for Real arguments less than -1.

Examples

julia> log1p(-0.5)
-0.6931471805599453

julia> log1p(0)
0.0

julia> log1p(-2)
ERROR: DomainError with -2.0:
log1p will only return a complex result if called with a complex argument. Try log1p(Complex(x)).
Stacktrace:
[1] throw_complex_domainerror(::Symbol, ::Float64) at ./math.jl:31
[...]
source
Base.Math.frexpFunction
frexp(val)

Return (x,exp) such that x has a magnitude in the interval $[1/2, 1)$ or 0, and val is equal to $x \times 2^{exp}$.

source
Base.expMethod
exp(x)

Compute the natural base exponential of x, in other words $e^x$.

Examples

julia> exp(1.0)
2.718281828459045
source
Base.exp2Function
exp2(x)

Compute the base 2 exponential of x, in other words $2^x$.

Examples

julia> exp2(5)
32.0
source
Base.exp10Function
exp10(x)

Compute the base 10 exponential of x, in other words $10^x$.

Examples

julia> exp10(2)
100.0
source
exp10(x)

Compute $10^x$.

Examples

julia> exp10(2)
100.0

julia> exp10(0.2)
1.5848931924611136
source
Base.Math.modfFunction
modf(x)

Return a tuple (fpart, ipart) of the fractional and integral parts of a number. Both parts have the same sign as the argument.

Examples

julia> modf(3.5)
(0.5, 3.0)

julia> modf(-3.5)
(-0.5, -3.0)
source
Base.roundMethod
round([T,] x, [r::RoundingMode])
round(x, [r::RoundingMode]; digits::Integer=0, base = 10)
round(x, [r::RoundingMode]; sigdigits::Integer, base = 10)

Rounds the number x.

Without keyword arguments, x is rounded to an integer value, returning a value of type T, or of the same type of x if no T is provided. An InexactError will be thrown if the value is not representable by T, similar to convert.

If the digits keyword argument is provided, it rounds to the specified number of digits after the decimal place (or before if negative), in base base.

If the sigdigits keyword argument is provided, it rounds to the specified number of significant digits, in base base.

The RoundingModer controls the direction of the rounding; the default is RoundNearest, which rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer. Note that round may give incorrect results if the global rounding mode is changed (see rounding).

Examples

julia> round(1.7)
2.0

julia> round(Int, 1.7)
2

julia> round(1.5)
2.0

julia> round(2.5)
2.0

julia> round(pi; digits=2)
3.14

julia> round(pi; digits=3, base=2)
3.125

julia> round(123.456; sigdigits=2)
120.0

julia> round(357.913; sigdigits=4, base=2)
352.0
Note

Rounding to specified digits in bases other than 2 can be inexact when operating on binary floating point numbers. For example, the Float64 value represented by 1.15 is actually less than 1.15, yet will be rounded to 1.2.

# Examples

julia> x = 1.15
1.15

julia> @sprintf "%.20f" x
"1.14999999999999991118"

julia> x < 115//100
true

julia> round(x, digits=1)
1.2

Extensions

To extend round to new numeric types, it is typically sufficient to define Base.round(x::NewType, r::RoundingMode).

source
Base.Rounding.RoundingModeType
RoundingMode

A type used for controlling the rounding mode of floating point operations (via rounding/setrounding functions), or as optional arguments for rounding to the nearest integer (via the round function).

Currently supported rounding modes are:

source
Base.Rounding.RoundNearestConstant
RoundNearest

The default rounding mode. Rounds to the nearest integer, with ties (fractional values of 0.5) being rounded to the nearest even integer.

source
Base.Rounding.RoundFromZeroConstant
RoundFromZero

Rounds away from zero. This rounding mode may only be used with T == BigFloat inputs to round.

Examples

julia> BigFloat("1.0000000000000001", 5, RoundFromZero)
1.06
source
Base.roundMethod
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]])
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; digits=, base=10)
round(z::Complex[, RoundingModeReal, [RoundingModeImaginary]]; sigdigits=, base=10)

Return the nearest integral value of the same type as the complex-valued z to z, breaking ties using the specified RoundingModes. The first RoundingMode is used for rounding the real components while the second is used for rounding the imaginary components.

Example

julia> round(3.14 + 4.5im)
3.0 + 4.0im
source
Base.ceilFunction
ceil([T,] x)
ceil(x; digits::Integer= [, base = 10])
ceil(x; sigdigits::Integer= [, base = 10])

ceil(x) returns the nearest integral value of the same type as x that is greater than or equal to x.

ceil(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for round.

source
Base.floorFunction
floor([T,] x)
floor(x; digits::Integer= [, base = 10])
floor(x; sigdigits::Integer= [, base = 10])

floor(x) returns the nearest integral value of the same type as x that is less than or equal to x.

floor(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for round.

source
Base.truncFunction
trunc([T,] x)
trunc(x; digits::Integer= [, base = 10])
trunc(x; sigdigits::Integer= [, base = 10])

trunc(x) returns the nearest integral value of the same type as x whose absolute value is less than or equal to x.

trunc(T, x) converts the result to type T, throwing an InexactError if the value is not representable.

digits, sigdigits and base work as for round.

source
Base.unsafe_truncFunction
unsafe_trunc(T, x)

Return the nearest integral value of type T whose absolute value is less than or equal to x. If the value is not representable by T, an arbitrary value will be returned.

source
Base.minFunction
min(x, y, ...)

Return the minimum of the arguments. See also the minimum function to take the minimum element from a collection.

Examples

julia> min(2, 5, 1)
1
source
Base.maxFunction
max(x, y, ...)

Return the maximum of the arguments. See also the maximum function to take the maximum element from a collection.

Examples

julia> max(2, 5, 1)
5
source
Base.minmaxFunction
minmax(x, y)

Examples

julia> minmax('c','b')
('b', 'c')
source
Base.Math.clampFunction
clamp(x, lo, hi)

Return x if lo <= x <= hi. If x > hi, return hi. If x < lo, return lo. Arguments are promoted to a common type.

Examples

julia> clamp.([pi, 1.0, big(10.)], 2., 9.)
3-element Array{BigFloat,1}:
3.141592653589793238462643383279502884197169399375105820974944592307816406286198
2.0
9.0

julia> clamp.([11,8,5],10,6) # an example where lo > hi
3-element Array{Int64,1}:
6
6
10
source
Base.Math.clamp!Function
clamp!(array::AbstractArray, lo, hi)

Restrict values in array to the specified range, in-place. See also clamp.

source
Base.absFunction
abs(x)

The absolute value of x.

When abs is applied to signed integers, overflow may occur, resulting in the return of a negative value. This overflow occurs only when abs is applied to the minimum representable value of a signed integer. That is, when x == typemin(typeof(x)), abs(x) == x < 0, not -x as might be expected.

Examples

julia> abs(-3)
3

julia> abs(1 + im)
1.4142135623730951

julia> abs(typemin(Int64))
-9223372036854775808
source
Base.Checked.checked_absFunction
Base.checked_abs(x)

Calculates abs(x), checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. Int) cannot represent abs(typemin(Int)), thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_negFunction
Base.checked_neg(x)

Calculates -x, checking for overflow errors where applicable. For example, standard two's complement signed integers (e.g. Int) cannot represent -typemin(Int), thus leading to an overflow.

The overflow protection may impose a perceptible performance penalty.

source

Calculates x+y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_subFunction
Base.checked_sub(x, y)

Calculates x-y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_mulFunction
Base.checked_mul(x, y)

Calculates x*y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_divFunction
Base.checked_div(x, y)

Calculates div(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_remFunction
Base.checked_rem(x, y)

Calculates x%y, checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_fldFunction
Base.checked_fld(x, y)

Calculates fld(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_modFunction
Base.checked_mod(x, y)

Calculates mod(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.Checked.checked_cldFunction
Base.checked_cld(x, y)

Calculates cld(x,y), checking for overflow errors where applicable.

The overflow protection may impose a perceptible performance penalty.

source
Base.abs2Function
abs2(x)

Squared absolute value of x.

Examples

julia> abs2(-3)
9
source
Base.copysignFunction
copysign(x, y) -> z

Return z which has the magnitude of x and the same sign as y.

Examples

julia> copysign(1, -2)
-1

julia> copysign(-1, 2)
1
source
Base.signFunction
sign(x)

Return zero if x==0 and $x/|x|$ otherwise (i.e., ±1 for real x).

source
Base.signbitFunction
signbit(x)

Returns true if the value of the sign of x is negative, otherwise false.

Examples

julia> signbit(-4)
true

julia> signbit(5)
false

julia> signbit(5.5)
false

julia> signbit(-4.1)
true
source
Base.flipsignFunction
flipsign(x, y)

Return x with its sign flipped if y is negative. For example abs(x) = flipsign(x,x).

Examples

julia> flipsign(5, 3)
5

julia> flipsign(5, -3)
-5
source
Base.sqrtMethod
sqrt(x)

Return $\sqrt{x}$. Throws DomainError for negative Real arguments. Use complex negative arguments instead. The prefix operator is equivalent to sqrt.

Examples

julia> sqrt(big(81))
9.0

julia> sqrt(big(-81))
ERROR: DomainError with -81.0:
NaN result for non-NaN input.
Stacktrace:
[1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]

julia> sqrt(big(complex(-81)))
0.0 + 9.0im
source
Base.isqrtFunction
isqrt(n::Integer)

Integer square root: the largest integer m such that m*m <= n.

julia> isqrt(5)
2
source
Base.Math.cbrtFunction
cbrt(x::Real)

Return the cube root of x, i.e. $x^{1/3}$. Negative values are accepted (returning the negative real root when $x < 0$).

The prefix operator is equivalent to cbrt.

Examples

julia> cbrt(big(27))
3.0

julia> cbrt(big(-27))
-3.0
source
Base.realMethod
real(z)

Return the real part of the complex number z.

Examples

julia> real(1 + 3im)
1
source
Base.imagFunction
imag(z)

Return the imaginary part of the complex number z.

Examples

julia> imag(1 + 3im)
3
source
Base.reimFunction
reim(z)

Return both the real and imaginary parts of the complex number z.

Examples

julia> reim(1 + 3im)
(1, 3)
source
Base.conjFunction
conj(z)

Compute the complex conjugate of a complex number z.

Examples

julia> conj(1 + 3im)
1 - 3im
source
Base.angleFunction
angle(z)

Compute the phase angle in radians of a complex number z.

Examples

45.0

-45.0

-135.0
source
Base.cisFunction
cis(z)

Return $\exp(iz)$.

Examples

julia> cis(π) ≈ -1
true
source
Base.binomialFunction
binomial(n::Integer, k::Integer)

The binomial coefficient$\binom{n}{k}$, being the coefficient of the $k$th term in the polynomial expansion of $(1+x)^n$.

If $n$ is non-negative, then it is the number of ways to choose k out of n items:

$\binom{n}{k} = \frac{n!}{k! (n-k)!}$

where $n!$ is the factorial function.

If $n$ is negative, then it is defined in terms of the identity

$\binom{n}{k} = (-1)^k \binom{k-n-1}{k}$

Examples

julia> binomial(5, 3)
10

julia> factorial(5) ÷ (factorial(5-3) * factorial(3))
10

julia> binomial(-5, 3)
-35

source
Base.factorialFunction
factorial(n::Integer)

Factorial of n. If n is an Integer, the factorial is computed as an integer (promoted to at least 64 bits). Note that this may overflow if n is not small, but you can use factorial(big(n)) to compute the result exactly in arbitrary precision.

Examples

julia> factorial(6)
720

julia> factorial(21)
ERROR: OverflowError: 21 is too large to look up in the table; consider using factorial(big(21)) instead
Stacktrace:
[...]

julia> factorial(big(21))
51090942171709440000

source
Base.gcdFunction
gcd(x,y)

Greatest common (positive) divisor (or zero if x and y are both zero). The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

julia> gcd(6,9)
3

julia> gcd(6,-9)
3
source
Base.lcmFunction
lcm(x,y)

Least common (non-negative) multiple. The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

julia> lcm(2,3)
6

julia> lcm(-2,3)
6
source
Base.gcdxFunction
gcdx(x,y)

Computes the greatest common (positive) divisor of x and y and their Bézout coefficients, i.e. the integer coefficients u and v that satisfy $ux+vy = d = gcd(x,y)$. $gcdx(x,y)$ returns $(d,u,v)$.

The arguments may be integer and rational numbers.

Julia 1.4

Rational arguments require Julia 1.4 or later.

Examples

julia> gcdx(12, 42)
(6, -3, 1)

julia> gcdx(240, 46)
(2, -9, 47)
Note

Bézout coefficients are not uniquely defined. gcdx returns the minimal Bézout coefficients that are computed by the extended Euclidean algorithm. (Ref: D. Knuth, TAoCP, 2/e, p. 325, Algorithm X.) For signed integers, these coefficients u and v are minimal in the sense that $|u| < |y/d|$ and $|v| < |x/d|$. Furthermore, the signs of u and v are chosen so that d is positive. For unsigned integers, the coefficients u and v might be near their typemax, and the identity then holds only via the unsigned integers' modulo arithmetic.

source
Base.ispow2Function
ispow2(n::Integer) -> Bool

Test whether n is a power of two.

Examples

julia> ispow2(4)
true

julia> ispow2(5)
false
source
Base.nextpowFunction
nextpow(a, x)

The smallest a^n not less than x, where n is a non-negative integer. a must be greater than 1, and x must be greater than 0.

Examples

julia> nextpow(2, 7)
8

julia> nextpow(2, 9)
16

julia> nextpow(5, 20)
25

julia> nextpow(4, 16)
16

source
Base.prevpowFunction
prevpow(a, x)

The largest a^n not greater than x, where n is a non-negative integer. a must be greater than 1, and x must not be less than 1.

Examples

julia> prevpow(2, 7)
4

julia> prevpow(2, 9)
8

julia> prevpow(5, 20)
5

julia> prevpow(4, 16)
16

source
Base.nextprodFunction
nextprod([k_1, k_2,...], n)

Next integer greater than or equal to n that can be written as $\prod k_i^{p_i}$ for integers $p_1$, $p_2$, etc.

Examples

julia> nextprod([2, 3], 105)
108

julia> 2^2 * 3^3
108
source
Base.invmodFunction
invmod(x,m)

Take the inverse of x modulo m: y such that $x y = 1 \pmod m$, with $div(x,y) = 0$. This is undefined for $m = 0$, or if $gcd(x,m) \neq 1$.

Examples

julia> invmod(2,5)
3

julia> invmod(2,3)
2

julia> invmod(5,6)
5
source
Base.powermodFunction
powermod(x::Integer, p::Integer, m)

Compute $x^p \pmod m$.

Examples

julia> powermod(2, 6, 5)
4

julia> mod(2^6, 5)
4

julia> powermod(5, 2, 20)
5

julia> powermod(5, 2, 19)
6

julia> powermod(5, 3, 19)
11
source
Base.ndigitsFunction

Compute the number of digits in integer n written in base base (base must not be in [-1, 0, 1]), optionally padded with zeros to a specified size (the result will never be less than pad).

Examples

julia> ndigits(12345)
5

julia> ndigits(1022, base=16)
3

julia> string(1022, base=16)
"3fe"

5
source
Base.widemulFunction
widemul(x, y)

Multiply x and y, giving the result as a larger type.

Examples

julia> widemul(Float32(3.), 4.)
12.0
source
Base.Math.evalpolyFunction
evalpoly(x, p)

Evaluate the polynomial $\sum_k p[k] x^{k-1}$ for the coefficients p[1], p[2], ...; that is, the coefficients are given in ascending order by power of x. Loops are unrolled at compile time if the number of coefficients is statically known, i.e. when p is a Tuple. This function generates efficient code using Horner's method if x is real, or using a Goertzel-like [DK62] algorithm if x is complex.

Julia 1.4

This function requires Julia 1.4 or later.

Example

julia> evalpoly(2, (1, 2, 3))
17
source
Base.Math.@evalpolyMacro
@evalpoly(z, c...)

Evaluate the polynomial $\sum_k c[k] z^{k-1}$ for the coefficients c[1], c[2], ...; that is, the coefficients are given in ascending order by power of z. This macro expands to efficient inline code that uses either Horner's method or, for complex z, a more efficient Goertzel-like algorithm.

Examples

julia> @evalpoly(3, 1, 0, 1)
10

julia> @evalpoly(2, 1, 0, 1)
5

julia> @evalpoly(2, 1, 1, 1)
7
source
Base.FastMath.@fastmathMacro
@fastmath expr

Execute a transformed version of the expression, which calls functions that may violate strict IEEE semantics. This allows the fastest possible operation, but results are undefined – be careful when doing this, as it may change numerical results.

This sets the LLVM Fast-Math flags, and corresponds to the -ffast-math option in clang. See the notes on performance annotations for more details.

Examples

julia> @fastmath 1+2
3

julia> @fastmath(sin(3))
0.1411200080598672
source

## Customizable binary operators

Some unicode characters can be used to define new binary operators that support infix notation. For example ⊗(x,y) = kron(x,y) defines the (otimes) function to be the Kronecker product, and one can call it as binary operator using infix syntax: C = A ⊗ B as well as with the usual prefix syntax C = ⊗(A,B).

Other characters that support such extensions include \odot and \oplus

The complete list is in the parser code: https://github.com/JuliaLang/julia/blob/master/src/julia-parser.scm

Those that are parsed like * (in terms of precedence) include * / ÷ % & ⋅ ∘ × |\\| ∩ ∧ ⊗ ⊘ ⊙ ⊚ ⊛ ⊠ ⊡ ⊓ ∗ ∙ ∤ ⅋ ≀ ⊼ ⋄ ⋆ ⋇ ⋉ ⋊ ⋋ ⋌ ⋏ ⋒ ⟑ ⦸ ⦼ ⦾ ⦿ ⧶ ⧷ ⨇ ⨰ ⨱ ⨲ ⨳ ⨴ ⨵ ⨶ ⨷ ⨸ ⨻ ⨼ ⨽ ⩀ ⩃ ⩄ ⩋ ⩍ ⩎ ⩑ ⩓ ⩕ ⩘ ⩚ ⩜ ⩞ ⩟ ⩠ ⫛ ⊍ ▷ ⨝ ⟕ ⟖ ⟗ and those that are parsed like + include + - |\|| ⊕ ⊖ ⊞ ⊟ |++| ∪ ∨ ⊔ ± ∓ ∔ ∸ ≏ ⊎ ⊻ ⊽ ⋎ ⋓ ⧺ ⧻ ⨈ ⨢ ⨣ ⨤ ⨥ ⨦ ⨧ ⨨ ⨩ ⨪ ⨫ ⨬ ⨭ ⨮ ⨹ ⨺ ⩁ ⩂ ⩅ ⩊ ⩌ ⩏ ⩐ ⩒ ⩔ ⩖ ⩗ ⩛ ⩝ ⩡ ⩢ ⩣ There are many others that are related to arrows, comparisons, and powers.

• DK62Donald Knuth, Art of Computer Programming, Volume 2: Seminumerical Algorithms, Sec. 4.6.4.