INSTITUT PERGURUAN PERSEKUTUAN PULAU PINANG.

MATHEMATICS EVALUATION PROJECT

NUMERATION SYSTEM

NAME : NOOR FATEEN BINTI TUGIMIN(890214105914)

NURUL SAKINAH BINTI MOHD ADNAN(900618105378)

COURSE : PPISMP

CLASS : 1 BMMT1

LECTURER : MISTER TEO CHUEN TICK

SUBMISSION DATE : 15th of SEPTEMBER 2008

CONTENTS

TOPIC

Numeration System

Egyptian Numeration System

Babylonian Numeration System

Mayan Numeration System

Number Chart

Advantages and Disadvantages of Numeration System

Comparison of Numeration System with Hindu-Arabic System

The Choosen Numeration System

Broteen Numeration System

NUMERATION SYSTEM

What is Numeration System?

Numeration systems are methods for representing quantities. In order to perform any

simple mathematical operations, you would have to begin with some kind of numeration

system.

Why numeration systems exist?

First, it is often necessary to tell the number of items contained in a collection or set of

those items. To do that, you have to have some method for counting the items. The total

number of items is represented by a number known as a cardinal number.

Numbers can also be used to express the rank or sequence or order of items. Numbers

used in this way are known as ordinal numbers.

Finally, numbers can be used for purposes of identification. Some method must be

devised to keep checking and savings accounts, credit card accounts, drivers' licenses,

and other kinds of records for different people separated from each other.

EGYPTIAN NUMERATION SYSTEM

Egyptian numeration system was a numeral system used in ancient Egypt. It was a

decimal system, often rounded off to the higher power, written in hieroglyphs. The

hieratic form of numerals stressed an exact finite series notation, being ciphered one: one

onto the Egyptian alphabet. The Egyptian system used bases of ten.

The Egyptian method for recording quantitities is based on 10 with a symbol for 1, ten,

and each successive power of ten. A distinct hieroglypic was used for each power of 10.

There was no symbol for zero, therefore a particular symbol was omitted in a numeral

when that multiple of ten was not part of the number

Multiples of these values were expressed by repeating the symbol as many times as

needed. For instance, a stone carving from Karnak shows the number 4622 as

Egyptian hieroglyphs could be written in both directions (and even vertically). This

example is written left-to-right and top-down; on the original stone carving, it is right-to-

left, and the signs are thus reversed.

Fractions

Rational numbers could also be expressed, but only as sums of unit fractions, i.e., sums of

reciprocals of positive integers, except for 2/3 and 3/4. The hieroglyph indicating a

fraction looked like a mouth, which meant "part":

Fractions were written with this fractional solidus, i.e., the numerator 1, and the positive

denominator below. Thus, 1/3 was written as:

There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently)

and 3/4 (used less frequently):

   

If the denominator became too large, the "mouth" was just placed over the beginning of

the "denominator":

Addition and subtraction

For plus and minus signs, the hieroglyphs

and

were used: if the feet pointed into the direction of writing, it signified addition, otherwise

subtraction.

BABYLONIAN NUMERATION SYSTEM

Origin

The Babylonian civilisation in Mesopotamia replaced the Sumerian civilisation

and the Akkadian civilisation. We give a little historical background to these events in

our article Babylonian mathematics. Certainly in terms of their number system the

Babylonians inherited ideas from the Sumerians and from the Akkadians. From the

number systems of these earlier peoples came the base of 60, that is the sexagesimal

system. Yet neither the Sumerian nor the Akkadian system was a positional system and

this advance by the Babylonians was undoubtedly their greatest achievement in terms of

developing the number system.

The Babylonian number system began with tally marks just as most of the ancient

math systems did. The Babylonians developed a form of writing based on cuneiform.

Cuneiform means "wedge shape" in Latin. They wrote these symbols on wet clay tablets

which were baked in the hot sun. Many thousands of these tablets are still around today.

The Babylonians used a stylist to imprint the symbols on the clay since curved lines

could not be drawn.

Symbols

Only two symbols (one similar to a "Y" to count units, and another similar to a

"<" to count tens) were used to notate the 59 non-zero digits. These symbols and their

values were combined to form a digit in a sign-value notation way similar to that of

Roman numerals; for example, the combination "<<YYY" represented the digit for 23. A

space was left to indicate a place without value, similar to the modern-day zero.

Babylonians later devised a sign to represent this empty place. They lacked a symbol to

serve the function of radix point, so the place of the units had to be inferred from context.

Their system clearly used internal decimal to represent digits, but it was not really a

mixed-radix system of bases 10 and 6, since the ten sub-base was used merely to

facilitate the representation of the large set of digits needed, while the place-values in a

digit string were consistently 60-based and the arithmetic needed to work with these digit

strings was correspondingly sexagesimal.

A common theory is that 60, a superior highly composite number, was chosen due

to its prime factorization: 2×2×3×5, which makes it divisible by 1, 2, 3, 4, 5, 6, 10, 12,

15, 20, and 30. In fact, it is the smallest integer divisible by all integers from 1 to 6.

Integers and fractions were represented identically — a radix point was not written but

rather made clear by context.

Now given a positional system one needs a convention concerning which end of the

number represents the units. For example the decimal 12345 represents

1 104 + 2 103 + 3 102 + 4 10 + 5.

Now if the empty space caused a problem with integers then there was an even

bigger problem with Babylonian sexagesimal fractions. The Babylonians used a system

of sexagesimal fractions similar to our decimal fractions. For example if we write 0.125

then this is 1/10 + 2/100 + 5/1000 = 1/8. Of course a fraction of the form a/b, in its lowest form,

can be represented as a finite decimal fraction if and only if b has no prime divisors other

than 2 or 5. So 1/3 has no finite decimal fraction. Similarly the Babylonian sexagesimal

fraction 0;7,30 represented 7/60 + 30/3600 which again written in our notation is 1/8.

Since 60 is divisible by the primes 2, 3 and 5 then a number of the form a/b, in its

lowest form, can be represented as a finite decimal fraction if and only if b has no prime

divisors other than 2, 3 or 5. More fractions can therefore be represented as finite

sexagesimal fractions than can as finite decimal fractions. Some historians think that this

observation has a direct bearing on why the Babylonians developed the sexagesimal

system, rather than the decimal system, but this seems a little unlikely. If this were the

case why not have 30 as a base? We discuss this problem in some detail below.

Now we have already suggested the notation that we will use to denote a

sexagesimal number with fractional part. To illustrate 10,12,5;1,52,30 represents the

number

10 602 + 12 60 + 5 + 1/60 + 52/602 + 30/60

3

which in our notation is 36725 1/32. This is fine but we have introduced the notation of the

semicolon to show where the integer part ends and the fractional part begins. It is the

"sexagesimal point" and plays an analogous role to a decimal point. However, the

Babylonians has no notation to indicate where the integer part ended and the fractional

part began. Hence there was a great deal of ambiguity introduced and "the context makes

it clear" philosophy now seems pretty stretched. If I write 10,12,5,1,52,30 without having

a notation for the "sexagesimal point" then it could mean any of:

0;10,12, 5, 1,52,30

10;12, 5, 1,52,30

10,12; 5, 1,52,30

10,12, 5; 1,52,30

10,12, 5, 1;52,30

10,12, 5, 1,52;30

10,12, 5, 1,52,30

in addition, of course, to 10, 12, 5, 1, 52, 30, 0 or 0 ; 0, 10, 12, 5, 1, 52, 30 etc.

MAYAN NUMERATION SYSTEM

The Mayan Indians lived on the Yucatan Peninsula in central America from about

200 B.C. to 1540 A.D. The Mayans used a vigesimal system, which had a base 20. This

system is believed to have been used because the Mayans lived in such a warm climate

and there was rarely a need to wear shoes, thus 20 was the total number of visible fingers

and toes making the system workable. Therefore two important markers in this system

are 20, which relates to the fingers and toes, and five, which relates to the number of

digits on one hand or foot. The pre-Columbian Mayans developed a fairly sophisticated

system of numeration, primarily for the purpose of making calenders and keeping

track of time. (A concern for quantifying the passage of time, and minding the calender,

seems to have been a characteristic of many primitive peoples, and prompted much of the

early record-keeping.

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.

Addition and Subtraction

Adding and subtracting numbers using Maya numerals is very simple.

Addition is performed by combining the numeric symbols at each level:

If five or more dots result from the combination, five dots are removed and replaced by a

bar. If four or more bars result, four bars are removed and a dot is added to the next

higher column.

Similarly with subtraction, remove the elements of the subtrahend symbol from the

minuend symbol:

If there are not enough dots in a minuend position, a bar is replaced by five dots. If there

are not enough bars, a dot is removed from the next higher minuend symbol in the

column and four bars are added to the minuend symbol being worked on.

Zero

The Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder

within its vigesimal positional numeral system. A shell glyph -- -- was used as a zero

symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo,

Chiapas) has a date of 36 BCE.

However, since the eight earliest Long Count dates appear outside the Maya homeland, it

is assumed that the use of zero predated the Maya, and was possibly the invention of the

Olmec. Indeed, many of the earliest Long Count dates were found within the Olmec

heartland. However, the Olmec civilization had come to an end by the 4th century BCE,

several centuries before the earliest known Long Count dates--which suggests that zero

was not an Olmec discovery.

EGYPTIAN NUMERATION SYSTEM

ADVANTAGES DISADVANTAGES

The system the Egyptians used was

simpler, and far less tedious

Could put symbols in any order,

because the value of a symbol did

not depend on its position.

The Egyptian Numeration System

compressed at ten. That is to say,

ten was the count they used to

bundle or group lower level

symbols

Uncommon for most numbers other

than one and two, and also the signs

were used a lot in their time.

There was no symbol for zero;

therefore a particular symbol was

omitted in a numeral when that

multiple of ten was not part of the

number.

Egyptian arithmetic has long been

devalued because it lacks a sign for

zero and has no place-system.

BABYLONIAN NUMERATION SYSTEM

ADVANTAGES DISADVANTAGES

The Babylonians were quite able to

divide by numbers other than

regular ones, and approximated the

results to three or four sexagesimal

places.

The Babylonians had more regular

numbers in the above sense than we

have—any number with prime

factors other than 2 and 5 lands us

with an unending decimal fraction:

e.g. ⅓=0.333… .)

Only two symbols (one similar to a

"Y" to count units, and another

similar to a "<" to count tens) were

used to notate the 59 non-zero

digits

Babylonian civilisations did invent

a symbol to indicate an empty place

so the lack of a zero could not have

been totally satisfactory.

The Babylonians did not technically

have a digit for, nor a concept, of

the number zero.

There was no zero to put into an

empty position.

When we read the first digit we do

not know its value until we have

read the complete number to find

out how many powers of 10 are

associated with this first place.

MAYAN NUMERATION SYSTEM

ADVANTAGES DISADVANTAGES

The numerals are made up of three

symbols; zero (shell shape), one (a

dot) and five (a bar).

Maya numerals can be illustrated by

face type glyphs.they used a "place"

system (another impressive

invention), with

the lowest digit signifying 1's, and

the higher places signifying more

powers of the base which was

nominally always 20.

Fairly sophisticated system of

numeration,

One consequence of this anomaly is

that the possible representations of

a given number are not necessarily

unique.

The system had one anomaly in that

the denomination increased by a

factor of 18 instead of 20 when

rising from the second to the third

digit.

EGYPTIAN NUMERATION SYSTEM

BABYLONIAN NUMERATION SYSTEM

MAYAN NUMERATION SYSTEM

THE CHOOSEN NUMERATION SYSTEM

After doing the analysis of these four numeration system, we had choose Mayan

Numeration System rather than Hindu-Arabic System. Why we choose this Numeration

System? This is because:

It is easier for us to understand this Mayan Numeration System rather than the

other three systems which are Egyptian Numeration System, Babylonian

Numeration System and also Hindu-Arabic Numeration System. Although

recently we use Hindu-Arabic System in our education, for us, Mayan

Numeration System is easier for us.

The numerals are made up of three symbols; zero (shell shape), one (a dot) and

five (a bar). It is easy for us to write the number.

Maya numerals can be illustrated by face type glyphs.they used a "place" system

(another impressive invention), with

The lowest digit signifying 1's, and the higher places signifying more powers of

the base which was nominally always 20.

BROTEEN NUMERATION SYSTEM

After doing the researches of all these four numeration systems and making our

discussion, finally we had come out with our own numeration system which the name of

our numeration system is Broteen Numeration System. This name is taken from our name

and had been created based from Hindu-Arabic Numeration System and the dimension

shape that we always use nowadays.

The rules of the system

Addition

= 4 + 5 = 9

Substraction

= 8 – 4 = 4

Multiple

= 10 x 2 = 20

Division

= 10 ÷ 2 = 5

The advantages of this number system are

The shape of the number is simple but interesting.

As the shape is simple, so it is easy for us to write.

This numeration system has the easiest and the simplest rules.

It does not have complicated calculation. All calculation problems can be solve by

the easiest way than the other number system.

The advantages of this number system is

The symbol of the operation is quite complicated.