History of Numeration Systems Mte3101

8/11/2019 History of Numeration Systems Mte3101

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September 28, 2014 1

Numeration Systems

MTE 3101

PISMP


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The Tally Numeration System

Virtually all numeration starts as tallies,using single strokes to represent each

additional unit: / for one, // for two, etc. Evidence of tallies has been found on bone

fragmentsfrom as early as 15,000 BC.

A tally system can exist before a languagedevelops words for numbers. Reference: Bunch and Hellemans, The Timetables of Technology,

Simon & Schuster, 1993


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Tally Stick


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Tokens

Early societies developed tokensto represent quantities.

By 4000 BC, tokens existed for ten sheep ( say:) and

for one sheep(say:)

Given the following tokens:How many sheep are represented?

There were different tokens for different commodities!

Three horseswould be represented as and not, which is three sheep!


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Concept ofNumber

Around 4000 BC, traders in Uruk (Babylon orthe present Iraq) were discovering that thesame numbercould be used to mean tensheep,

tenbags of grain, or tentalents of copper.

About 3000 BC, Egyptian tallies show items

grouped atten;these tallies were regrouped at a hundred,

and regrouped again at one thousand.


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Hieroglyphic numbers

Source: 195.8.72.23/numbers.htm Mark Millmore

used with permission


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Two examples




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12,425 Birds




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The Egyptians Numeration System

Hieroglyphics(pictographic symbols) 1 = Stick /

10 = Arch

100 = Coiled Rope 1000 = Lotus Flower

10,000 = Finger (pointing to sky) 100,000 = Tadpole (from the Nile)

1,000,000 = Man (arms reaching to heaven)


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Hieroglyphics (addition)

/ / / / / /

/ / / / / / / /

4561

38


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Hieroglyphics (addition)

/ / / / / /

/ / / / / / / /

-----------------------------------

/ / / / / / / / / // / / /----------------------------------

/ / / /

456138

---------------------144

---------------------144


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Multiplication by Doubling (23 X 13)

Number multiplier / / / 1

// / / / / 2

4

//

8

/ / / /

Number multiplier23 1

46 2

92 4

184 8


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Multiplication by Doubling (23 X 13)

Number multiplier / / / 1

// / / / / 2

4

//

8

/ / / /

Number multiplier23 1

46 2

92 4

184 8

=================

299 13


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Check and verify!!!

23

times 13

6 9

2 3_

2 9 9


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Greeks and Romans

The Greeks adapted their alphabet for

numerals; others followed their example.

Roman numerals are also alphabetical, but

they did not originate as such. Early

artifacts show that the Xfor ten, originated

from the way in which scribes drew a

slanted line through the number for four:

///// + / became X; one half of Xwas V,

and the habit of putting a circle around the

tenth Xto indicate one hundred became C


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The Roman Numeration System

Roman numerals use a basic set of seven symbols:I 1 (one) (unus)

V 5 (five) (quinque)X 10 (ten) (decem)

L 50 (fifty) (quinquaginta)

C 100 (one hundred) (centum)

D 500 (five hundred) (quingenti)

M 1000 (one thousand) (mille)


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Roman Numerals

I XI XXX XXXX L

II XII

III XIII

IV XIV

V XV

VI XVI

VII XVII

VIII XVIII

IX XIX

X XX


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Roman Numerals

L = 50

C = 100

D = 500

M = 1000

V (bar)= 5000 vee bar

XV (bar) = 15,000

L (bar) = 50,000

C (bar) = 100,000

M (bar)= 1,000,000


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Addition using Roman Numerals

2318 MM CCC X V III

+821 DCCC XX I______3139 MM DCCCXXX V IIII

CCC--------------------------------MM IX

D

collecting DDC XXXterms --------------------------------

MMM C XXX IX


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Subtraction using RomanNumerals

2486 MM CCCC L XXX V I

-1343 M CCC XXXX III

1143 expand: MM CCCCXXXXX I I I I I I| XXX

| minus: M CCC XXXX I I I

||-------------------

M C XXXX III


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Multiplication using Roman NumeralsI ____V___ _X___L____ C____D___ M

V | V XXV L CCL D MMD V-bar

X | X L C D M V-bar VV-bar

2 8 XXVIII

times 1 25 6 XXVIII times 1 = XXVIII

2 8__ XXVIII times 1 = XXVIII

3 3 6 XXVIII times 10 = CCLXXX

CC L XXXXXXX VV I I I I I I

Collect terms: CCC XXX VI


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The Mayan Numeration System

The numerals are made up

of three symbols; zero

(shell shape), one(a dot)

and five (a bar).

For example, nineteen

(19) is written as four dots

in a horizontal row above

three horizontal lines

stacked upon each other
http://en.wikipedia.org/wiki/Image:Maya.svg


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The Mayan Numeration System

400s

20s

1s

33 429 5125

Numbers after 19 were

written vertically down in

powers of twenty. For

example, thirty-three wouldbe written as one dot above

three dots, which are in turn

atop two lines. The first dot

represents "one twenty" or

"120", which is added tothree dots and two bars, or

thirteen. Therefore, (120) +

13 = 33.
http://en.wikipedia.org/wiki/Image:Maya_5.pnghttp://en.wikipedia.org/wiki/Image:Maya_9.pnghttp://en.wikipedia.org/wiki/Image:Maya_13.pnghttp://en.wikipedia.org/wiki/Image:Maya_16.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.pnghttp://en.wikipedia.org/wiki/Image:Maya_12.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.pnghttp://en.wikipedia.org/wiki/Image:Maya_5.pnghttp://en.wikipedia.org/wiki/Image:Maya_9.pnghttp://en.wikipedia.org/wiki/Image:Maya_16.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.pnghttp://en.wikipedia.org/wiki/Image:Maya_12.pnghttp://en.wikipedia.org/wiki/Image:Maya_1.png


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Addition and Subtraction
http://en.wikipedia.org/wiki/Image:Maya_add.png


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Hindu-Arabic notation The Indians used horizontal tallies (/) for one,

two and three, and special symbols for four

through nine.

Around 600 CE, the Indians started usingplace

values, i.e., instead of writing the equivalent of100 + 80 + 7, they wrote 187

Only nine digits were used along with a symbol

for zero, probably derived from astronomersmarking empty places.

A famous inscription dated 870 CE contains the first

zero that has survived.


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Evolution of symbols
http://en.wikipedia.org/wiki/Image:Arabic_numerals-en.svg


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Hindu-Arabic numerals

Ancient Hindus:

zeroplace values and decimal system (base 10)

Positional Notation: 4, 4 2 8

Arab traders brought the system to Europewhere it became known as Arabic numerals

base: X5 X4 X3 X2 X1 X0

10 100000 10000 1000 100 10 1

2 32 16 8 4 2 1


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Multiplication by doubling

297times 22

5 9 4

5 9 4__

6 5 3 4

297 1

594 2

1188 4

2376 8

4752 16

6534 22


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Assignment 1

You are required to write out the

Reflective Log individually andsubmit to lecturer for comments

Happy Writing

Documents

History of Numeration Systems Mte3101