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 `+`

Add (`+`), Plus, Addition, or Sum, is a dyadic scalar function that adds numbers together. As a basic arithmetic operation, Add is subject to the language's number specification. Add shares the glyph `+` with the monadic function Conjugate, and is closely related to Subtract (`-`).

## Examples

```      1 2 3 + 2 3 4
3 5 7
0.1 + 3 4 5
3.1 4.1 5.1```

## Properties

Addition is commutative in almost all number systems. It is associative when performed with no rounding, but is unlikely to be associative when rounded to an inexact precision (see Floating point error).

The identity element for Add is `0`. The inverse of the function `n∘+` or the equivalent function `+∘n` is `-∘n`. Using Commute (`⍨`), we can write `+⍣¯1` ${\displaystyle \Leftrightarrow }$ `-⍨`.

Since adding a number to itself is equivalent to doubling that number, we can express the double function as `+⍨`.

### Reduction

Reduction with Add gives the sum of the whole list.

## Scalar mapping

In mathematics, addition of two identical structures almost always follows the same rules as in APL: it maps over the structures element-wise. This is a fundamental property of a (finite-dimensional) vector space, in which addition of two vectors is equivalent to adding the coefficients of basis vectors one by one. This property likely inspired APL's definition of a scalar function.

Addition of complex and hypercomplex numbers can also be considered an element-wise operation, since each of these types of numbers forms a vector space over the reals. Addition of scalars is always performed within a single domain: mixed-type addition such as adding a real to a complex number treats the real number as complex with imaginary part zero.

Addition in mathematics does not exhibit scalar extension: there is usually no way to add values of different types. The Multiply function on vector spaces does extend scalars, in that a scalar element of a field can be multiplied by a vector over that field.

## Floating point error

When using a limited-precision format such as 8-byte floats, the results of addition are rounded in order to fit into the result type. This rounding can cause the results of adding several numbers to be different depending on the order in which they are added, that is, addition fails to be associative:

```      1e20 + ¯1e20 + 1
0
1e20 + (¯1e20 + 1)      ⍝ Equivalent
0
(1e20 + ¯1e20) + 1      ⍝ Not equivalent
1```

Addition of floating-point numbers may also be subject to overflow, resulting in a DOMAIN ERROR or an infinite result.