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.