# [Conversion and Promotion](@id conversion-and-promotion)

Julia has a system for promoting arguments of mathematical operators to a common type, which has
been mentioned in various other sections, including [Integers and Floating-Point Numbers](@ref),
[Mathematical Operations and Elementary Functions](@ref), [Types](@ref man-types), and [Methods](@ref).
In this section, we explain how this promotion system works, as well as how to extend it to new
types and apply it to functions besides built-in mathematical operators. Traditionally, programming
languages fall into two camps with respect to promotion of arithmetic arguments:

  * **Automatic promotion for built-in arithmetic types and operators.** In most languages, built-in
    numeric types, when used as operands to arithmetic operators with infix syntax, such as `+`,
    `-`, `*`, and `/`, are automatically promoted to a common type to produce the expected results.
    C, Java, Perl, and Python, to name a few, all correctly compute the sum `1 + 1.5` as the floating-point
    value `2.5`, even though one of the operands to `+` is an integer. These systems are convenient
    and designed carefully enough that they are generally all-but-invisible to the programmer: hardly
    anyone consciously thinks of this promotion taking place when writing such an expression, but
    compilers and interpreters must perform conversion before addition since integers and floating-point
    values cannot be added as-is. Complex rules for such automatic conversions are thus inevitably
    part of specifications and implementations for such languages.
  * **No automatic promotion.** This camp includes Ada and ML -- very "strict" statically typed languages.
    In these languages, every conversion must be explicitly specified by the programmer. Thus, the
    example expression `1 + 1.5` would be a compilation error in both Ada and ML. Instead one must
    write `real(1) + 1.5`, explicitly converting the integer `1` to a floating-point value before
    performing addition. Explicit conversion everywhere is so inconvenient, however, that even Ada
    has some degree of automatic conversion: integer literals are promoted to the expected integer
    type automatically, and floating-point literals are similarly promoted to appropriate floating-point
    types.

In a sense, Julia falls into the "no automatic promotion" category: mathematical operators are
just functions with special syntax, and the arguments of functions are never automatically converted.
However, one may observe that applying mathematical operations to a wide variety of mixed argument
types is just an extreme case of polymorphic multiple dispatch -- something which Julia's dispatch
and type systems are particularly well-suited to handle. "Automatic" promotion of mathematical
operands simply emerges as a special application: Julia comes with pre-defined catch-all dispatch
rules for mathematical operators, invoked when no specific implementation exists for some combination
of operand types. These catch-all rules first promote all operands to a common type using user-definable
promotion rules, and then invoke a specialized implementation of the operator in question for
the resulting values, now of the same type. User-defined types can easily participate in this
promotion system by defining methods for conversion to and from other types, and providing a handful
of promotion rules defining what types they should promote to when mixed with other types.

## Conversion

The standard way to obtain a value of a certain type `T` is to call the type's constructor, `T(x)`.
However, there are cases where it's convenient to convert a value from one type to another
without the programmer asking for it explicitly.
One example is assigning a value into an array: if `A` is a `Vector{Float64}`, the expression
`A[1] = 2` should work by automatically converting the `2` from `Int` to `Float64`, and
storing the result in the array.
This is done via the [`convert`](@ref) function.

The `convert` function generally takes two arguments: the first is a type object and the second is
a value to convert to that type. The returned value is the value converted to an instance of given type.
The simplest way to understand this function is to see it in action:

```jldoctest
julia> x = 12
12

julia> typeof(x)
Int64

julia> convert(UInt8, x)
0x0c

julia> typeof(ans)
UInt8

julia> convert(AbstractFloat, x)
12.0

julia> typeof(ans)
Float64

julia> a = Any[1 2 3; 4 5 6]
2×3 Array{Any,2}:
 1  2  3
 4  5  6

julia> convert(Array{Float64}, a)
2×3 Array{Float64,2}:
 1.0  2.0  3.0
 4.0  5.0  6.0
```

Conversion isn't always possible, in which case a no method error is thrown indicating that `convert`
doesn't know how to perform the requested conversion:

```jldoctest
julia> convert(AbstractFloat, "foo")
ERROR: MethodError: Cannot `convert` an object of type String to an object of type AbstractFloat
[...]
```

Some languages consider parsing strings as numbers or formatting numbers as strings to be conversions
(many dynamic languages will even perform conversion for you automatically), however Julia does
not: even though some strings can be parsed as numbers, most strings are not valid representations
of numbers, and only a very limited subset of them are. Therefore in Julia the dedicated [`parse`](@ref)
function must be used to perform this operation, making it more explicit.

### When is `convert` called?

The following language constructs call `convert`:

  * Assigning to an array converts to the array's element type.
  * Assigning to a field of an object converts to the declared type of the field.
  * Constructing an object with [`new`](@ref) converts to the object's declared field types.
  * Assigning to a variable with a declared type (e.g. `local x::T`) converts to that type.
  * A function with a declared return type converts its return value to that type.
  * Passing a value to [`ccall`](@ref) converts it to the corresponding argument type.

### Conversion vs. Construction

Note that the behavior of `convert(T, x)` appears to be nearly identical to `T(x)`.
Indeed, it usually is.
However, there is a key semantic difference: since `convert` can be called implicitly,
its methods are restricted to cases that are considered "safe" or "unsurprising".
`convert` will only convert between types that represent the same basic kind of thing
(e.g. different representations of numbers, or different string encodings).
It is also usually lossless; converting a value to a different type and back again
should result in the exact same value.

There are four general kinds of cases where constructors differ from `convert`:

#### Constructors for types unrelated to their arguments

Some constructors don't implement the concept of "conversion".
For example, `Timer(2)` creates a 2-second timer, which is not really a
"conversion" from an integer to a timer.

#### Mutable collections

`convert(T, x)` is expected to return the original `x` if `x` is already of type `T`.
In contrast, if `T` is a mutable collection type then `T(x)` should always make a new
collection (copying elements from `x`).

#### Wrapper types

For some types which "wrap" other values, the constructor may wrap its argument inside
a new object even if it is already of the requested type.
For example `Some(x)` wraps `x` to indicate that a value is present (in a context
where the result might be a `Some` or `nothing`).
However, `x` itself might be the object `Some(y)`, in which case the result is
`Some(Some(y))`, with two levels of wrapping.
`convert(Some, x)`, on the other hand, would just return `x` since it is already
a `Some`.

#### Constructors that don't return instances of their own type

In *very rare* cases it might make sense for the constructor `T(x)` to return
an object not of type `T`.
This could happen if a wrapper type is its own inverse (e.g. `Flip(Flip(x)) === x`),
or to support an old calling syntax for backwards compatibility when a library is
restructured.
But `convert(T, x)` should always return a value of type `T`.

### Defining New Conversions

When defining a new type, initially all ways of creating it should be defined as
constructors.
If it becomes clear that implicit conversion would be useful, and that some
constructors meet the above "safety" criteria, then `convert` methods can be added.
These methods are typically quite simple, as they only need to call the appropriate
constructor.
Such a definition might look like this:

```julia
convert(::Type{MyType}, x) = MyType(x)
```

The type of the first argument of this method is a [singleton type](@ref man-singleton-types),
`Type{MyType}`, the only instance of which is `MyType`. Thus, this method is only invoked
when the first argument is the type value `MyType`. Notice the syntax used for the first
argument: the argument name is omitted prior to the `::` symbol, and only the type is given.
This is the syntax in Julia for a function argument whose type is specified but whose value
does not need to be referenced by name. In this example, since the type is a singleton, we
already know its value without referring to an argument name.

All instances of some abstract types are by default considered "sufficiently similar"
that a universal `convert` definition is provided in Julia Base.
For example, this definition states that it's valid to `convert` any `Number` type to
any other by calling a 1-argument constructor:

```julia
convert(::Type{T}, x::Number) where {T<:Number} = T(x)
```

This means that new `Number` types only need to define constructors, since this
definition will handle `convert` for them.
An identity conversion is also provided to handle the case where the argument is
already of the requested type:

```julia
convert(::Type{T}, x::T) where {T<:Number} = x
```

Similar definitions exist for `AbstractString`, [`AbstractArray`](@ref), and [`AbstractDict`](@ref).

## Promotion

Promotion refers to converting values of mixed types to a single common type. Although it is not
strictly necessary, it is generally implied that the common type to which the values are converted
can faithfully represent all of the original values. In this sense, the term "promotion" is appropriate
since the values are converted to a "greater" type -- i.e. one which can represent all of the
input values in a single common type. It is important, however, not to confuse this with object-oriented
(structural) super-typing, or Julia's notion of abstract super-types: promotion has nothing to
do with the type hierarchy, and everything to do with converting between alternate representations.
For instance, although every [`Int32`](@ref) value can also be represented as a [`Float64`](@ref) value,
`Int32` is not a subtype of `Float64`.

Promotion to a common "greater" type is performed in Julia by the [`promote`](@ref) function, which takes
any number of arguments, and returns a tuple of the same number of values, converted to a common
type, or throws an exception if promotion is not possible. The most common use case for promotion
is to convert numeric arguments to a common type:

```jldoctest
julia> promote(1, 2.5)
(1.0, 2.5)

julia> promote(1, 2.5, 3)
(1.0, 2.5, 3.0)

julia> promote(2, 3//4)
(2//1, 3//4)

julia> promote(1, 2.5, 3, 3//4)
(1.0, 2.5, 3.0, 0.75)

julia> promote(1.5, im)
(1.5 + 0.0im, 0.0 + 1.0im)

julia> promote(1 + 2im, 3//4)
(1//1 + 2//1*im, 3//4 + 0//1*im)
```

Floating-point values are promoted to the largest of the floating-point argument types. Integer
values are promoted to the larger of either the native machine word size or the largest integer
argument type. Mixtures of integers and floating-point values are promoted to a floating-point
type big enough to hold all the values. Integers mixed with rationals are promoted to rationals.
Rationals mixed with floats are promoted to floats. Complex values mixed with real values are
promoted to the appropriate kind of complex value.

That is really all there is to using promotions. The rest is just a matter of clever application,
the most typical "clever" application being the definition of catch-all methods for numeric operations
like the arithmetic operators `+`, `-`, `*` and `/`. Here are some of the catch-all method definitions
given in [`promotion.jl`](https://github.com/JuliaLang/julia/blob/master/base/promotion.jl):

```julia
+(x::Number, y::Number) = +(promote(x,y)...)
-(x::Number, y::Number) = -(promote(x,y)...)
*(x::Number, y::Number) = *(promote(x,y)...)
/(x::Number, y::Number) = /(promote(x,y)...)
```

These method definitions say that in the absence of more specific rules for adding, subtracting,
multiplying and dividing pairs of numeric values, promote the values to a common type and then
try again. That's all there is to it: nowhere else does one ever need to worry about promotion
to a common numeric type for arithmetic operations -- it just happens automatically. There are
definitions of catch-all promotion methods for a number of other arithmetic and mathematical functions
in [`promotion.jl`](https://github.com/JuliaLang/julia/blob/master/base/promotion.jl), but beyond
that, there are hardly any calls to `promote` required in Julia Base. The most
common usages of `promote` occur in outer constructors methods, provided for convenience, to allow
constructor calls with mixed types to delegate to an inner type with fields promoted to an appropriate
common type. For example, recall that [`rational.jl`](https://github.com/JuliaLang/julia/blob/master/base/rational.jl)
provides the following outer constructor method:

```julia
Rational(n::Integer, d::Integer) = Rational(promote(n,d)...)
```

This allows calls like the following to work:

```jldoctest
julia> Rational(Int8(15),Int32(-5))
-3//1

julia> typeof(ans)
Rational{Int32}
```

For most user-defined types, it is better practice to require programmers to supply the expected
types to constructor functions explicitly, but sometimes, especially for numeric problems, it
can be convenient to do promotion automatically.

### Defining Promotion Rules

Although one could, in principle, define methods for the `promote` function directly, this would
require many redundant definitions for all possible permutations of argument types. Instead, the
behavior of `promote` is defined in terms of an auxiliary function called [`promote_rule`](@ref), which
one can provide methods for. The `promote_rule` function takes a pair of type objects and returns
another type object, such that instances of the argument types will be promoted to the returned
type. Thus, by defining the rule:

```julia
promote_rule(::Type{Float64}, ::Type{Float32}) = Float64
```

one declares that when 64-bit and 32-bit floating-point values are promoted together, they should
be promoted to 64-bit floating-point. The promotion type does not need to be one of the argument
types, however; the following promotion rules both occur in Julia Base:

```julia
promote_rule(::Type{BigInt}, ::Type{Float64}) = BigFloat
promote_rule(::Type{BigInt}, ::Type{Int8}) = BigInt
```

In the latter case, the result type is [`BigInt`](@ref) since `BigInt` is the only type
large enough to hold integers for arbitrary-precision integer arithmetic. Also note that
one does not need to define both `promote_rule(::Type{A}, ::Type{B})` and
`promote_rule(::Type{B}, ::Type{A})` -- the symmetry is implied by the way `promote_rule`
is used in the promotion process.

The `promote_rule` function is used as a building block to define a second function called [`promote_type`](@ref),
which, given any number of type objects, returns the common type to which those values, as arguments
to `promote` should be promoted. Thus, if one wants to know, in absence of actual values, what
type a collection of values of certain types would promote to, one can use `promote_type`:

```jldoctest
julia> promote_type(Int8, Int64)
Int64
```

Internally, `promote_type` is used inside of `promote` to determine what type argument values
should be converted to for promotion. It can, however, be useful in its own right. The curious
reader can read the code in [`promotion.jl`](https://github.com/JuliaLang/julia/blob/master/base/promotion.jl),
which defines the complete promotion mechanism in about 35 lines.

### Case Study: Rational Promotions

Finally, we finish off our ongoing case study of Julia's rational number type, which makes relatively
sophisticated use of the promotion mechanism with the following promotion rules:

```julia
promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{Rational{S}}) where {T<:Integer,S<:Integer} = Rational{promote_type(T,S)}
promote_rule(::Type{Rational{T}}, ::Type{S}) where {T<:Integer,S<:AbstractFloat} = promote_type(T,S)
```

The first rule says that promoting a rational number with any other integer type promotes to a
rational type whose numerator/denominator type is the result of promotion of its numerator/denominator
type with the other integer type. The second rule applies the same logic to two different types
of rational numbers, resulting in a rational of the promotion of their respective numerator/denominator
types. The third and final rule dictates that promoting a rational with a float results in the
same type as promoting the numerator/denominator type with the float.

This small handful of promotion rules, together with the type's constructors and the default
`convert` method for numbers, are sufficient to make rational numbers interoperate completely
naturally with all of Julia's other numeric types -- integers, floating-point numbers, and complex
numbers. By providing appropriate conversion methods and promotion rules in the same manner, any
user-defined numeric type can interoperate just as naturally with Julia's predefined numerics.
