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  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    The First Computations The History of Mathematics, Part 2

    Chuck Garner, Ph.D.

    Department of Mathematics Rockdale Magnet School for Science and Technology

    January 13, 2014

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Outline

    Ancient Numerals Egyptian Babylonian Chinese Greek

    Arithmetic Babylonian Egyptian Egyptian Fractions

    The Method of False Position

    Homework

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Outline

    Ancient Numerals Egyptian Babylonian Chinese Greek

    Arithmetic Babylonian Egyptian Egyptian Fractions

    The Method of False Position

    Homework

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Ancient Geography

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Ancient Numerals

    I Nothing like ours I Mixture of additive and positional number systems I Symbols for numerals were symbols from culture

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Egyptian Numerals

    I Two kinds: I hieratic, for writing on parchment and daily use I hieroglyphic, for carvings

    I Additive notation for both I Symbols for powers of 10

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Hieratic

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Hieroglyphic

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Hieroglyphic

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Babylonian Numerals

    I Used a base-60 positional system I Cuneiform writing I Only two numerals

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Babylonian Numerals

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Chinese Numerals

    I Two kinds: I oracular, additive notation I rods, positional base-10 notation

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Oracular Chinese Numerals

    Earliest known use from 1045 bc

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Rod Chinese Numerals

    Earliest known use from 4th century bc

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Greek Numerals

    I Two kinds: acrophonic and alphabetic, both additive I Alphabetic numerals are le�ers I Distinguish numbers from words through context

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Greek Acrophonic Numerals

    Used as far back as 1000 bc

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Greek Alphabetic Numerals

    Used from 4th century bc

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Outline

    Ancient Numerals Egyptian Babylonian Chinese Greek

    Arithmetic Babylonian Egyptian Egyptian Fractions

    The Method of False Position

    Homework

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Babylonian Arithmetic

    I Info found on clay tablets c.2100-1600 bc I Addition and subtraction similar to present

    procedure I Used tables to multiply and divide I Extensive tables of reciprocals since division by n

    was multiplication by 1n I This may explain why the base was 60

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Egyptian Arithmetic

    I Info from Moscow papyrus (c.1850 bc) and Rhind papyrus (c.1650 bc)

    I Hieroglyphic addition and subtraction similar to present

    I Hieratic arithmetic may have relied on tables I Multiplication and division achieved by doubling

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    The Rhind Papyrus

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Doubling

    Multiply 12 by 25.

    1 12 2 24 4 48 8 96

    16 192 25 300

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Doubling

    Multiply 12 by 25.

    1 12

    2 24 4 48 8 96

    16 192 25 300

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Doubling

    Multiply 12 by 25.

    1 12 2 24 4 48 8 96

    16 192

    25 300

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Doubling

    Multiply 12 by 25.

    1′ 12 2 24 4 48 8′ 96

    16′ 192 25 300

  • First Computations

    Garner

    Ancient Numerals Egyptian

    Babylonian

    Chinese

    Greek

    Arithmetic Babylonian

    Egyptian

    Egyptian Fractions

    The Method of False Position

    Homework

    Doubling

    Divide 858 by 26.

    1

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