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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