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🔗 Source: math.mit.edu
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📄 Pages: 10 pages
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Summary

Number theory is the study of numbers, but what is a number. The set of natural numbers N can be defined in several ways, including finite ordinals, finite cardinals, and strings over a unary alphabet. N is a commutative semiring with addition and multiplication satisfying usual properties. The set of integers Z is a commutative ring containing N, and the set of rational numbers Q is a field containing Z. The real numbers R are the completion of Q, making it a complete ordered field. Each of these algebraic structures is canonical, meaning every non-trivial structure of the same type contains a copy of N, Z, Q, or R. The construction of R as equivalence classes of Cauchy sequences relies on properties of the absolute value function, which determines which sequences are Cauchy and which are equivalent. Absolute values can be defined on any field, and examples include the trivial absolute value, the archimedean absolute value, and the p-adic absolute value.

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Number theory is the study of numbers, but what is a number.

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  • File Format: PDF
  • File Size: 247 KB
  • Pages: 10
  • Language: EN
  • Total Downloads: 303
  • Last Updated: 9 hours ago

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