Multiplication of natural numbers

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• Addition of natural numbers
• The recursion theorem
• Multiplication is a well-defined total operation

Used by

• ⌈ m/n ⌉ for naturals m and n ≥ 1: the least q ∈ ℕ with m ≤ n q Definition
• Arithmetic on the integers Definition
• Exponentiation of natural numbers, mⁿ, and its agreement with the integer power in ℝ Definition
• Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ Definition
• The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ Definition
• A floor-division and multiplication adjunction between natural-number preorders Example
• Distributing a finite set over a finite set of boxes, with the ceiling bound computed and attained Example
• FALSE: every list of mn pairwise distinct reals has a strictly increasing sublist of length m+1 or a strictly decreasing sublist of length n+1 False statement
• Cancellation for multiplication by a nonzero factor Lemma
• Distributivity and the successor law for multiplication Lemma
• Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
• Multiplication is associative Lemma
• Multiplication is commutative Lemma
• Order is compatible with multiplication Lemma
• The natural numbers have no zero divisors Lemma
• Zero and one under multiplication Lemma
• A finite set with n elements has exactly C(n, 2) two-element subsets, and 2C(n, 2) = n(n-1) Theorem
• For all m and n there is a list of mn pairwise distinct reals with no strictly increasing sublist of length m+1 and no strictly decreasing sublist of length n+1 Theorem
• ℕ × ℕ ≈ ℕ Theorem
• On ω the ordinal + and · are the Peano operations: ω is closed under ordinal +, · and exponentiation, and for naturals m, n the ordinal m + n and m · n are the natural-number sum and product Theorem
• The product rule: | A × B| = | A| | B|, and |∏_i<m Aᵢ| = ∏_i<m| Aᵢ| Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §2.3 (Multiplication) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)
• B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)
• Mathematics 144: Set Theory (UC Riverside lecture notes) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)