Origin of Numbers and Order

It starts with scratches on a rock (permanent) or in the dirt (temporary).

Integers: 1, 11, 111, 1111, …..

Equality: 11 = 11

Order: 111 > 11, (1111, 1, 11) → (1, 11, 1111) Well Ordering Principle

Pos Integers: 01, 011, 0111, …

Neg Integers: 10, 110, 1110, …..

Addition:

011 + 0111 = 011111,

110 + 011 = 0,

1110 + 011 = 10

0111 +110 = 01

110 + 1110 = 111110 (a+b)=(b+a)

Left Hand Multiplication by a pos Integer:

(011)(0111) = (0111) + (0111) = 0111111

(0111)(011) = (011) + (011) + (011) = 0111111 (This does not prove ab=ba)

(011)(110) = (110) + (110) = 11110

Definition: -01 = 10

A symbol a, b, can be associated with each successive row of 1’s using digits and placement:

0 ---- 0

01---- 1

011 ---- 2

0111 --- 3

Ex: 10344

Order: a > b iff a + (-b) positive.

There is no greatest number: I can always add a scratch. → Archimedes Postulate

(111) + (111) + (111) +1 → Euclids Algorithm (10/3)

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From here on definitions and proofs are not completely visually obvious.

ab = ba. Assume ab=ba for some b. Then a(b+1) = ab + a = ba + a = (b + 1)a (visual proof and ab = ba). True for all b by induction.

a(b+c) = ab + ac, etc

The next step is to define rational and real numbers using the definitions of integer and order above. For ex a/b > c/d means the integer ab is greater than the integer cd.

An early abstraction was to let an integer be a characteristic of a class: 0111 for three apples, or three pears, or three oranges, which introduces counting.

There is a message here. The basis of the real number system is visually obvious and accessible to everyone (first-graders). The real number system as started with abstractions is accessible to very few, and with understanding, by even fewer.

Re: Origin of Numbers and Order

Except that what you are talking about "numeral", not numbers. The way we happen to represent numbers does not, itself, tell us about the numbers. It is often true of numbers that the way we represent them is designed to **reflect** properties of numbers, but it is the numbers, not the numerals, that come first. We can continue making scratches to represent nubers **because** numbers are unbounded, not the other way around.