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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) [[wikipedia:vector space|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. | 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) [[wikipedia:vector space|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 number|complex]] and [[hypercomplex | Addition of [[Complex number|complex]] and [[hypercomplex number]]s 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. | 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. | ||