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Numeration Systems
By: Lindsay FredericksOctober 5th, 2010
EDU 290T&TH 8:00am
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Three Basic Numeration Systems
I. The Egyptian Numeration System
II. The Babylonian numeration System
III. The Roman Numeration System
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I. The Egyptian Numeration System
• Symbols used were called Hieroglyphics
• Carved symbols into their monuments (like temples, obelisks, and tombs)
• Hieroglyphic writing arose roughly five-thousand years ago
Egyptians used ancient symbols
to represen
t quantitie
s.
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I. The Egyptian Numeration System
The Egyptian numeration system used the following symbols to represent basic quantities:
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html
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I. They Egyptian Numeration
1 = Stroke10 = Arch100 = Coiled Rope (Coil)1,000 = Lotus Flower10,000 = Pointed Finger100,000 = Tadpole1,000,000 = Man with Arms Raised
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I. The Egyptian Numeration System
This system is a strictly additional system. There are multiple ways to represent quantities. • 221 could be written : 1+10+10+100+100, or
1+100+10+100+10 using the symbols. • This was probably confusing.
Multiple representations were used up until the twenty-seventh century BCE when it became more typical to write basic symbols in descending order.• Because strictly addition, simple matter of style
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I. The Egyptian Numeration System
Examples: 276
Two coilsSeven archesSix stokes
• 4622
Four lotus flowersSix coilsTwo archesTwo stokesPictures from:
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html
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I. The Egyptian SystemRecap
• Egyptians used symbol: Hieroglyphics
• The numerations system is strictly additive, descending order.
• Carved symbols into monuments.
• There are seven symbols used in the system (stoke, arch, coil, lotus flower, pointed finger, tadpole, and man with raised arms)
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II. The Babylonian Numeration System
Mathematicians and astronomers of Babylon developed a numeration system based on much older system inherited from the Sumerians.
There were two basic symbols used1. Upright wedge – representing one (1)
▼2. Sideways wedge – representing ten
(10) <
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II. The Babylonian Numeration System
Examples:
a.) 32<<<▼▼
b.) 5▼▼▼▼▼
c.) 12<▼▼
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II. The Babylonian Numeration System
Once the quantity being represented reached sixty this became a group.
In a new place, ▼ represents not one, but one group of sixty (hence place values).
In a new place, < represents not ten, but ten groups of sixty.
“sixties place”
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II. The Babylonian Numeration System
Upright wedge represents one times the “sixties” place.
Sideways wedge represents ten times the “ones” place.
Upright wedge represents one times the “ones” place.
Upright wedge represents one times the “thousandths” place.
Upright wedge represents one times the “sixties” place.
Upright wedge represents one times the “ones” place.
a. 723600 60 1 . ▼ <▼▼
b. 36613600 60 1
. ▼ ▼ ▼
x60
x60
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II. The Babylonian Numeration System
“Think of time.
we write nine fifty-nine as 9:59.
What happens when another minute passes?
we DON’T write 9:60.
Instead , we think of those sixty minutes as one hour (one group of sixty minutes) the nine increases by
one.
So, we write 10:00.
Meaning ten hours and no leftover minutes.”
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II. The Babylonian Numeration System
Place Values:
3600 60 1
Example: Example:a.) 72 b.) 3661
▼ < ▼ ▼ ▼ ▼ ▼
1x60 10x1 1x1 1x1 3600x1 60x1 1x1
60 + 10 + 1 + 1 = 72 3600 + 60 + 1 = 3661
X 60 X 60
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II. The Babylonian Numeration System There is some confusion when leaving spaces
Deciding whether it’s in the “sixties” place or the “thousandths” place.
New Symbol for this “empty space”
Example: 36013600 60 1 . = 3601 ▼ ▼ ▼ ▼
Instead of: ▼ ▼
►►
►► ►
►
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II. The Babylonian Numeration System
Like the Egyptian System, this system is additive
Babylonian Numeration System uses placeholders
Therefore this system is positional
►►
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II. The Babylonian Numeration System
Let try some!
a. 600
b. 62
c. 120
d. 7321
e. 3601
f. 832
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II. The Babylonia Numeration System
Answers:
a. <
b. ▼ ▼▼
c. ▼▼
d. ▼▼ ▼▼ ▼
e. ▼ ▼
f. < ▼▼▼ <<<<<▼▼
►►
►►
►►
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III. The Roman Numeration System
Includes the following symbols:
I = [1] M = [1000]V = [5] V = [5000]X = [10] X = [10000]L = [50]C = [100]D = [500]
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III. The Roman Numeration system This system is additive
-to create 6-add V and I : VI
Try one-create 12
Answer: XII
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III. The Roman Numeration System This system is also subtractive
- to create 9-place I in front of X: IX
* Smaller number in front of the larger number to subtract
Try one-create 499
Answer: ID
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III. The Roman Numeration System
To Review. This system is both additive, and
subtractive. Because of the placement of the symbols
matters the system is positional
A few more examples:a. CLVI c. MMDCCCLVI
b. CDLXI d. DXVII
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III. The Roman Numeration SystemAnswers:a. CLVI = 156
c = 100 + L =50 + V =5 + I =1
b. CDLXI = 461C= 100 - D= 500 + L= 50 + X= 10 + I = 1
c. MMDCCCLVI = 2,856(M=1000) + (M=1000) + (D=500) +(C=100) + (C=100) + (C=100)+ (L=50) + (V=5)+ (I=1)
d. DXVII = 517(D = 500) + (X = 10) + (V = 5) + (I = 1) + (I = 1)
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The END
References: Dr. Christine PhelpsFall 2010MTH 151Central Michigan University Course # 22129396
Pictures on slide seven fromhttp://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html
Pictures on slide four from http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html