Math 2.0

Mathematical logic and foundations Definition

Multiplication of natural numbers

Builds on Natural numbersSuccessor functionAddition of natural numbersDefinition by recursion

Definition

Let ℕ\mathbb{N} be the set of natural numbers, let SS be the successor function and let ++ be the addition of ℕ\mathbb{N}. Multiplication is the function ⋅ ⁣:ℕ×ℕ→ℕ\cdot \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}, (a,b)↦a⋅b(a, b) \mapsto a \cdot b, with these two properties for all aa and bb in ℕ\mathbb{N}: a⋅0=0,a⋅S(b)=a⋅b+a.a \cdot 0 = 0, \qquad a \cdot S(b) = a \cdot b + a.

Remarks

For each aa, these equations are a definition by recursion on bb. Thus exactly one such function exists. From the definition, a⋅1=a⋅0+a=aa \cdot 1 = a \cdot 0 + a = a. We often write abab for a⋅ba \cdot b.

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Definition by recursion, concept, no evidence levelDefinition by recursi…conceptNatural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionAddition of natural numbers, definition, no evidence levelAddition of natural n…definitionMultiplication of natural numbers, definition, no evidence levelMultiplication of nat…definition
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  1. Definition by recursion concept
  2. Natural numbers object
  3. Peano axioms concept
  4. Successor function definition
  5. Addition of natural numbers definition
  6. Multiplication of natural numbers definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Multiplication, Product of two natural numbers
  2. Web https://en.wikipedia.org/wiki/Peano_axioms, Multiplication
About this page
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Kinddefinition
WritingW0 stub
MSC03F30
DefinesMultiplication of natural numbers
UsesNatural numbers, Successor function, Addition of natural numbers, Definition by recursion

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