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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
Source: 195.8.72.23/numbers.htm Mark Millmore
used with permission
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12,425 Birds
Source: 195.8.72.23/numbers.htm Mark Millmore
used with permission
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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.svg8/11/2019 History of Numeration Systems Mte3101
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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.png8/11/2019 History of Numeration Systems Mte3101
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Addition and Subtraction
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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.svg8/11/2019 History of Numeration Systems Mte3101
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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