The bound can be derived axiomatically as a function of the shape from two rules:
- The bound of a vector is its length, and
- If two shapes are catenated (for instance by an outer product), the resulting bound is the product of their individual bounds.
A scalar has rank 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating
⍬ with some other shape leaves that shape unchanged, so multiplying any bound by a scalar's bound cannot change it either. Therefore a scalar's bound must be the multiplicative identity, 1.
|APL features |
|Built-ins||Primitive function ∙ Primitive operator ∙ Quad name|
|Array model||Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index (Indexing) ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype|
|Data types||Number (Boolean, Complex number) ∙ Character (String) ∙ Box ∙ Namespace|
|Concepts and paradigms||Leading axis theory ∙ Scalar extension ∙ Conformability ∙ Scalar function ∙ Pervasion ∙ Glyph ∙ Identity element ∙ Complex floor ∙ Total array ordering|
|Errors||LIMIT ERROR ∙ RANK ERROR ∙ SYNTAX ERROR ∙ DOMAIN ERROR ∙ LENGTH ERROR ∙ INDEX ERROR ∙ VALUE ERROR|