# 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&#x20;
  * 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
