Numeration Systems

By: Lindsay FredericksOctober 5th, 2010

EDU 290T&TH 8:00am

Three Basic Numeration Systems

I. The Egyptian Numeration System

II. The Babylonian numeration System

III. The Roman Numeration System

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.

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

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

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

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

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)

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) <

II. The Babylonian Numeration System

Examples:

a.) 32<<<▼▼

b.) 5▼▼▼▼▼

c.) 12<▼▼

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”

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

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.”

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

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: ▼ ▼

►►

►► ►

►

II. The Babylonian Numeration System

Like the Egyptian System, this system is additive

Babylonian Numeration System uses placeholders

Therefore this system is positional

►►

II. The Babylonian Numeration System

Let try some!

a. 600

b. 62

c. 120

d. 7321

e. 3601

f. 832

II. The Babylonia Numeration System

Answers:

a. <

b. ▼ ▼▼

c. ▼▼

d. ▼▼ ▼▼ ▼

e. ▼ ▼

f. < ▼▼▼ <<<<<▼▼

►►

►►

►►

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]

III. The Roman Numeration system This system is additive

-to create 6-add V and I : VI

Try one-create 12

Answer: XII

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

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

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)

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