multiplication
Let an agent explain multiplication and its properties across number systems, describe generalisations such as scalar, matrix and Hadamard products, relay algorithms and teaching approaches, and help compute or check products.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain multiplication and its properties across number systems, describe generalisations such as scalar, matrix and Hadamard products, relay algorithms and teaching approaches, and help compute or check products.
A binary arithmetic operation that combines two numbers to give their product, defined for natural numbers as repeated addition and extended to integers, rationals, reals and complex numbers, and generalised to scalar multiplication of vectors, matrix multiplication, the element-wise Hadamard product and multiplication in rings and groups; multiplication is commutative and associative for numbers, distributes over addition, and is a foundation of arithmetic and algebra.
What it is for: Combining quantities by scaling.
It can be explain the operation and properties; describe generalisations; relay algorithms and teaching; compute or check products.
Distinguishing features
Commutative and associative for numbers
Distributive over addition
Identity one and inverses for nonzero numbers
Non-commutative generalisations
What it looks like
Not a visible object; expressions with times signs, dots or juxtaposition.
How it is recognised
Binary operation giving a product
Numbers, vectors, matrices and abstract structures
Addition combines by summing; exponentiation is repeated multiplication
Related models
is a kind of - in registry terms
is inverse to - for nonzero divisors
generalises to - in linear algebra
distributes over - the distributive law
In practice
Families and kinds
multiplication of natural numbers, integers, rationals, reals and complex numbers
scalar multiplication of vectors
matrix multiplication
Hadamard element-wise product
multiplication in rings, fields and groups
doubling as a special case
Identifiers
symbols times sign, middle dot, asterisk notation
Standards and regulation
ISO 80000-2 notation for multiplication
Curriculum standards for arithmetic
Failure modes and hazards
Assuming commutativity in matrix multiplication
Overflow in computer arithmetic
Confusing element-wise and matrix products
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Define What multiplication is.
Definition.
Definition
Definition and properties.
Definition
Definition.
- What is multiplication, and what properties does it have across number systems? definition
- Is the question about number multiplication or a generalisation? boundary
Generalisations
Generalisations.
Generalisations
Generalisations.
- How do scalar, matrix, Hadamard and abstract multiplications differ? definition
- Which entry fits the specific generalisation? action
Compute Computation.
Computation.
Algorithms
Algorithms.
Algorithms
Algorithms.
- How are products computed by hand and by algorithms such as long multiplication and Karatsuba? action
- What is the result for the values in question? measurement
Computers
Computer arithmetic.
Computers
Computers.
- How do computers multiply integers and floating-point numbers, and where do errors arise? provenance
- Which entry fits computer arithmetic? action
Theory Algebraic structure.
Theory.
Structure
Rings and groups.
Structure
Structure.
- How is multiplication axiomatised in rings, fields and groups? provenance
- Which references are standard? provenance
Matrices
Matrix multiplication.
Matrices
Matrices.
- Why is matrix multiplication non-commutative, and how does it represent composition? provenance
- Which entry fits linear algebra? action
Context History and teaching.
Context.
History
History.
History
History.
- How did multiplication methods develop, from Egyptian doubling to modern algorithms? provenance
- Which entry fits the history of arithmetic? action
Teach
Teaching.
Teach
Teaching.
- How is multiplication taught, and which misconceptions arise? provenance
- Which sources are cited? provenance
What the second pass must settle
- Should matrix multiplication be a separate primary entry?
- How should notation standards be linked?
- The registry entry has merged aliases naming generalisations; should they be split off?