Mathematics

# 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.

    +(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

    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
`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.fma` — Function.

    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
`Base.muladd` — Function.

    muladd(x, y, z)

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

    julia> muladd(3, 2, 1)
    7

    julia> 3 * 2 + 1
    7

source
`Base.inv` — Method.

    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.div` — Function.

    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

source
`Base.fld` — Function.

    fld(x, y)

Largest integer less than or equal to `x/y`.

Examples

    julia> fld(7.3,5.5)
    1.0

source
`Base.cld` — Function.

    cld(x, y)

Smallest integer larger than or equal to `x/y`.

Examples

    julia> cld(5.5,2.2)
    3.0

source
`Base.mod` — Function.

    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)`.

See also: `mod1`.

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 `x` ≡ `y` (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.rem` — Function.

    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.rem2pi` — Function.

    rem2pi(x, r::RoundingMode)

Compute the remainder of `x` after integer division by `2π`, 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.mod2pi` — Function.

    mod2pi(x)

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

This function computes a floating point representation of the modulus after division by numerically exact `2π`, 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 `2π`.

Examples

    julia> mod2pi(9*pi/4)
    0.7853981633974481

source
`Base.divrem` — Function.

    divrem(x, y)

The quotient and remainder from Euclidean division. Equivalent to `(div(x,y), rem(x,y))` or `(x÷y, x%y)`.

Examples

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

    julia> divrem(7,3)
    (2, 1)

source
`Base.fldmod` — Function.

    fldmod(x, y)

The floored quotient and modulus after division. Equivalent to `(fld(x,y), mod(x,y))`.

source
`Base.fld1` — Function.

    fld1(x, y)

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

See also: `mod1`, `fldmod1`.

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.mod1` — Function.

    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`.

See also: `fld1`, `fldmod1`.

Examples

    julia> mod1(4, 2)
    2

    julia> mod1(4, 3)
    1

source
`Base.fldmod1` — Function.

    fldmod1(x, y)

Return `(fld1(x,y), mod1(x,y))`.

See also: `fld1`, `mod1`.

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.rationalize` — Function.

    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.numerator` — Function.

    numerator(x)

Numerator of the rational representation of `x`.

Examples

    julia> numerator(2//3)
    2

    julia> numerator(4)
    4

source
`Base.denominator` — Function.

    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 `0`s. 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"

See also `>>`, `>>>`.

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 `0`s if `x >= 0`, `1`s 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"

See also `>>>`, `<<`.

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 `0`s. 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"

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

See also `>>`, `<<`.

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.range` — Function.

    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.OneTo` — Type.

    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.StepRangeLen` — Type.

    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 `ref`erence value, a `step`, and the `len`gth. 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.cmp` — Function.

    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.xor` — Function.

    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.isapprox` — Function.

    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.sin` — Method.

    sin(x)

Compute sine of `x`, where `x` is in radians.

source
`Base.cos` — Method.

    cos(x)

Compute cosine of `x`, where `x` is in radians.

source
`Base.Math.sincos` — Method.

    sincos(x)

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

source
`Base.tan` — Method.

    tan(x)

Compute tangent of `x`, where `x` is in radians.

source
`Base.Math.sind` — Function.

    sind(x)

Compute sine of `x`, where `x` is in degrees.

source
`Base.Math.cosd` — Function.

    cosd(x)

Compute cosine of `x`, where `x` is in degrees.

source
`Base.Math.tand` — Function.

    tand(x)

Compute tangent of `x`, where `x` is in degrees.

source
`Base.Math.sinpi` — Function.

    sinpi(x)

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

source
`Base.Math.cospi` — Function.

    cospi(x)

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

source
`Base.sinh` — Method.

    sinh(x)

Compute hyperbolic sine of `x`.

source
`Base.cosh` — Method.

    cosh(x)

Compute hyperbolic cosine of `x`.

source
`Base.tanh` — Method.

    tanh(x)

Compute hyperbolic tangent of `x`.

source
`Base.asin` — Method.

    asin(x)

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

source
`Base.acos` — Method.

    acos(x)

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

source
`Base.atan` — Method.

    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.asind` — Function.

    asind(x)

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

source
`Base.Math.acosd` — Function.

    acosd(x)

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

source
`Base.Math.atand` — Function.

    atand(y)
    atand(y,x)

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

source
`Base.Math.sec` — Method.

    sec(x)

Compute the secant of `x`, where `x` is in radians.

source
`Base.Math.csc` — Method.

    csc(x)

Compute the cosecant of `x`, where `x` is in radians.

source
`Base.Math.cot` — Method.

    cot(x)

Compute the cotangent of `x`, where `x` is in radians.

source
`Base.Math.secd` — Function.

    secd(x)

Compute the secant of `x`, where `x` is in degrees.

source
`Base.Math.cscd` — Function.

    cscd(x)

Compute the cosecant of `x`, where `x` is in degrees.

source
`Base.Math.cotd` — Function.

    cotd(x)

Compute the cotangent of `x`, where `x` is in degrees.

source
`Base.Math.asec` — Method.

    asec(x)

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

source
`Base.Math.acsc` — Method.

    acsc(x)

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

source
`Base.Math.acot` — Method.

    acot(x)

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

source
`Base.Math.asecd` — Function.

    asecd(x)

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

source
`Base.Math.acscd` — Function.

    acscd(x)

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

source
`Base.Math.acotd` — Function.

    acotd(x)

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

source
`Base.Math.sech` — Method.

    sech(x)

Compute the hyperbolic secant of `x`.

source
`Base.Math.csch` — Method.

    csch(x)

Compute the hyperbolic cosecant of `x`.

source
`Base.Math.coth` — Method.

    coth(x)

Compute the hyperbolic cotangent of `x`.

source
`Base.asinh` — Method.

    asinh(x)

Compute the inverse hyperbolic sine of `x`.

source
`Base.acosh` — Method.

    acosh(x)

Compute the inverse hyperbolic cosine of `x`.

source
`Base.atanh` — Method.

    atanh(x)

Compute the inverse hyperbolic tangent of `x`.

source
`Base.Math.asech` — Method.

    asech(x)

Compute the inverse hyperbolic secant of `x`.

source
`Base.Math.acsch` — Method.

    acsch(x)

Compute the inverse hyperbolic cosecant of `x`.

source
`Base.Math.acoth` — Method.

    acoth(x)

Compute the inverse hyperbolic cotangent of `x`.

source
`Base.Math.sinc` — Function.

    sinc(x)

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

source
`Base.Math.cosc` — Function.

    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
`Base.Math.deg2rad` — Function.

    deg2rad(x)

Convert `x` from degrees to radians.

Examples

    julia> deg2rad(90)
    1.5707963267948966

source
`Base.Math.rad2deg` — Function.

    rad2deg(x)

Convert `x` from radians to degrees.

Examples

    julia> rad2deg(pi)
    180.0

source
`Base.Math.hypot` — Function.

    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.log` — Method.

    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.log` — Method.

    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.log2` — Function.

    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.log10` — Function.

    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.log1p` — Function.

    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.frexp` — Function.

    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.exp` — Method.

    exp(x)

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

Examples

    julia> exp(1.0)
    2.718281828459045

source
`Base.exp2` — Function.

    exp2(x)

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

Examples

    julia> exp2(5)
    32.0

source
`Base.exp10` — Function.

    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.ldexp` — Function.

    ldexp(x, n)

Compute $x \times 2^n$.

Examples

    julia> ldexp(5., 2)
    20.0

source
`Base.Math.modf` — Function.

    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.expm1` — Function.

    expm1(x)

Accurately compute $e^x-1$.

source
`Base.round` — Method.

    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 `RoundingMode` `r` 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.RoundingMode` — Type.

    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:

- `RoundNearest` (default)
- `RoundNearestTiesAway`
- `RoundNearestTiesUp`
- `RoundToZero`
- `RoundFromZero` (`BigFloat` only)
- `RoundUp`
- `RoundDown`

source
`Base.Rounding.RoundNearest` — Constant.

    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.RoundNearestTiesAway` — Constant.

    RoundNearestTiesAway

Rounds to nearest integer, with ties rounded away from zero (C/C++ `round` behaviour).

source
`Base.Rounding.RoundNearestTiesUp` — Constant.

    RoundNearestTiesUp

Rounds to nearest integer, with ties rounded toward positive infinity (Java/JavaScript `round` behaviour).

source
`Base.Rounding.RoundToZero` — Constant.

    RoundToZero

`round` using this rounding mode is an alias for `trunc`.

source
`Base.Rounding.RoundFromZero` — Constant.

    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.Rounding.RoundUp` — Constant.

    RoundUp

`round` using this rounding mode is an alias for `ceil`.

source
`Base.Rounding.RoundDown` — Constant.

    RoundDown

`round` using this rounding mode is an alias for `floor`.

source
`Base.round` — Method.

    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 `RoundingMode`s. 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.ceil` — Function.

    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.floor` — Function.

    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.trunc` — Function.

    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_trunc` — Function.

    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.min` — Function.

    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.max` — Function.

    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.minmax` — Function.

    minmax(x, y)

Return `(min(x,y), max(x,y))`. See also: `extrema` that returns `(minimum(x), maximum(x))`.

Examples

    julia> minmax('c','b')
    ('b', 'c')

source
`Base.Math.clamp` — Function.

    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.abs` — Function.

    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_abs` — Function.

    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_neg` — Function.

    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
`Base.Checked.checked_add` — Function.

    Base.checked_add(x, y)

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

The overflow protection may impose a perceptible performance penalty.

source
`Base.Checked.checked_sub` — Function.

    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_mul` — Function.

    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_div` — Function.

    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_rem` — Function.

    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_fld` — Function.

    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_mod` — Function.

    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_cld` — Function.

    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.Checked.add_with_overflow` — Function.

    Base.add_with_overflow(x, y) -> (r, f)

Calculates `r = x+y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.Checked.sub_with_overflow` — Function.

    Base.sub_with_overflow(x, y) -> (r, f)

Calculates `r = x-y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.Checked.mul_with_overflow` — Function.

    Base.mul_with_overflow(x, y) -> (r, f)

Calculates `r = x*y`, with the flag `f` indicating whether overflow has occurred.

source
`Base.abs2` — Function.

    abs2(x)

Squared absolute value of `x`.

Examples

    julia> abs2(-3)
    9

source
`Base.copysign` — Function.

    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.sign` — Function.

    sign(x)

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

source
`Base.signbit` — Function.

    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.flipsign` — Function.

    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.sqrt` — Method.

    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.isqrt` — Function.

    isqrt(n::Integer)

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

    julia> isqrt(5)
    2

source
`Base.Math.cbrt` — Function.

    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.real` — Method.

    real(z)

Return the real part of the complex number `z`.

Examples

    julia> real(1 + 3im)
    1

source
`Base.imag` — Function.

    imag(z)

Return the imaginary part of the complex number `z`.

Examples

    julia> imag(1 + 3im)
    3

source
`Base.reim` — Function.

    reim(z)

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

Examples

    julia> reim(1 + 3im)
    (1, 3)

source
`Base.conj` — Function.

    conj(z)

Compute the complex conjugate of a complex number `z`.

Examples

    julia> conj(1 + 3im)
    1 - 3im

source
`Base.angle` — Function.

    angle(z)

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

Examples

    julia> rad2deg(angle(1 + im))
    45.0

    julia> rad2deg(angle(1 - im))
    -45.0

    julia> rad2deg(angle(-1 - im))
    -135.0

source
`Base.cis` — Function.

    cis(z)

Return $\exp(iz)$.

Examples

    julia> cis(π) ≈ -1
    true

source
`Base.binomial` — Function.

    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

See also

- `factorial`

External links

- Binomial coeffient on Wikipedia.

source
`Base.factorial` — Function.

    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

See also

- `binomial`

External links

- Factorial on Wikipedia.

source
`Base.gcd` — Function.

    gcd(x,y)

Greatest common (positive) divisor (or zero if `x` and `y` are both zero).

Examples

    julia> gcd(6,9)
    3

    julia> gcd(6,-9)
    3

source
`Base.lcm` — Function.

    lcm(x,y)

Least common (non-negative) multiple.

Examples

    julia> lcm(2,3)
    6

    julia> lcm(-2,3)
    6

source
`Base.gcdx` — Function.

    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 $(d,u,v)$.

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.ispow2` — Function.

    ispow2(n::Integer) -> Bool

Test whether `n` is a power of two.

Examples

    julia> ispow2(4)
    true

    julia> ispow2(5)
    false

source
`Base.nextpow` — Function.

    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

See also `prevpow`.

source
`Base.prevpow` — Function.

    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

See also `nextpow`.

source
`Base.nextprod` — Function.

    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

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`Base.invmod` — Function.

    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.powermod` — Function.

    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.ndigits` — Function.

    ndigits(n::Integer; base::Integer=10, pad::Integer=1)

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"

    julia> ndigits(123, pad=5)
    5

source
`Base.widemul` — Function.

    widemul(x, y)

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

Examples

    julia> widemul(Float32(3.), 4.)
    12.0

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`Base.Math.@evalpoly` — Macro.

    @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.@fastmath` — Macro.

    @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

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