Results 1 to 2 of 2

Math Help - Origin of Numbers and Order

  1. #1
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    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 1s 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)

    -----------------------------------
    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.
    Last edited by Hartlw; January 15th 2013 at 05:12 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,545
    Thanks
    1394

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 9th 2011, 08:09 PM
  2. Sequential Order of Numbers
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: July 3rd 2010, 09:39 AM
  3. Order of Numbers Proofs
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 28th 2009, 08:36 PM
  4. Order Among the Numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 26th 2009, 09:53 PM
  5. order numbers
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 25th 2008, 03:56 PM

Search Tags


/mathhelpforum @mathhelpforum