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== Scalar mapping == | == 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) [ | 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 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 of [[Complex number|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. | ||