Number Theory: primer in numbers

Terms

• number

• mathematical object

• mathematical primitive

• axiom

• abstract object

• mathematical abstraction

• mathematical notion

• mathematical concept

• mathematical representation

• numeral

• numeral system

• positional numeral system

• non-positional numeral system

• place value

• number base, radix

• radix

• zero as placeholder

• decimal number representation

• binary number representation

• octal number representation

• hexadecimal number representation

• base32 number representation

• base64 number representation

• roman numerals

• greek numerals

• digit

• bit

• binary digit

• amount

• quantity

• enumeration

• counting

• labelling

• fundamental sets of numbers

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Number

Number is...

• a concept, an idea

• an abstract object without physical manifestation

• mathematical concept

• abstract mathematical object

• expression of quantity

• abstract representation of quantity

• arithmetical value

  expressed in some way (word, symbol)

  representing a particular quantity

• defininition of numbers is in terms of sets (as is everything else assuming set theory as the FOM)

• Peano's axioms define ℕ - the set of the natural numbers, stating that:

  • the number zero exists, ∃n.n=0

    • error: variable n ranges over as-of-yet undefined set ℕ

    • error: = symbol not found

  • the number zero is a natural number, 0 ∈ ℕ

  • there is

  • if the number n is a natural number, n ∈ ℕ

    then so is the successor of n

Types of number

miriad divisions:

• by type: nat, int, rat, real, complex

• naturals:

  • prime vs composite

  • odd vs even

• ints:

  • positive vs zero vs negative; nonnegative, nonpositive

• numerous groups: fiendly, abundant, happy, ...

ordinal cardinal

Number representation

• Representing numbers

  • in writing

  • in speach

  • in other forms of human communication (gesture, flags, lights, etc.)

  • in other interactions (forms, mediums, etc.); human-computer, computer

• Symbolic (written) representations:

  • Positional numeral system

    • Hindu-Arabic symbols

    • binary digits: {T,⊥} or {0,1} or {+5v,-5v}

  • Non-Positional numeral systems

    • Roman numerals: I,V,X,L,C,D,M

    • Formal definitions:

    • Lambda Calculus:

      • 0 := λsz.z

      • S := λnfx.f(nfx)

      • 1 := λsz.sz

      • 2 := λsz.s(sz)

      • 3 := λsz.s(s(sz))

    • Sets:

      • 0 := ∅

      • S(n) := n U {n}

      • 1 := { ∅ }

      • 2 := { ∅, {∅} }

      • 3 := { ∅, {∅}, {∅,{∅}} }

• There is a sharp distinction betwen the properties of a number and the properties of its corresponding numerals.

• In linguistics, a numeral, or number word, in the broadest sense is a word or phrase that describes a numerical quantity.

• Representing numbers

• A numeral is a written, agreed-upon symbol used to represent a number

• numerals come in many shapes; they are organized into a particular set

• a particular set of decimal numerals consists of the familiar numerals

• decimal numbers use the Hindu-Arabic numeral symbols

• dec numerals form a set of 10 elements, kind of numeric alphabet, {0..9}

numeral set

• a depends on the numerals system used

Natural numbers

• this subtitle is stylistic, "the" is missing

• there are 2 "the" occurances:

  • the set of... (it is a unique set; "the" unique things, they're definite

  • the natural numbers (specific kind of numbers)

    ℕ the set of the natural numbers