2.0 THE FIRST TEN NUMBERS OF EACH

NUMERATION SYSTEM

2.1 EGYPTIAN NUMERATION SYSTEM

One of the earliest examples of a numeral system is the Egyptian

numeral system, based on the following hieroglyphs:

If we look at the diagram above, we will notice that the first nine

numerals are pictographic in character, but the remaining ones are

logographic in character. Notice also that this is a decimal system.

However, it does not tell us how the Egyptians wrote compound

numerals. As it turns out, the Egyptians used a simple additive system,

as illustrated in the following diagrams.

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Note carefully that although the Egyptian numeral system does

not especially require a symbol for zero, the Egyptians nevertheless

had a symbol for zero

which they used for a variety of engineering and accounting purposes,

including some rather astonishing projects, such as the Pyramids

which were constructed during 2550 BC.

HINDU-ARABIC

NUMERATION

SYSTEM

EGYPTIAN NUMERATION SYSTEM

1

2

3

4

3

5

6

7

8

9

10

2.2 BABYLONIAN NUMERATION SYSTEM

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The Babylonians lived in Mesopotamia, which is between the Tigris

and Euphrates rivers. They began a numbering system about 5,000

years ago. It is one of the oldest numbering systems.

The first mathematics can be traced to the ancient country of

Babylon, during the third millennium B.C. Tables were the Babylonians

most outstanding accomplishment which helped them in calculating

problems. The Babylonian numeration system was developed between

3000 and 2000 BCE.

It uses only two numerals or symbols, a one and a ten to represent

numbers and they looked these:

To represent numbers from 2 to 59, the system was simply additives.

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.

The Babylonians had a very advanced number system even for

today's standards. It was a base 60 system (sexagesimals) rather than

a base ten (decimal). Base ten is what we use today.

The Babylonians divided the day into twenty-four hours, each

hour into sixty minutes, and each minute to sixty seconds. This form of

counting has survived for four thousand years.

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Example#1:

5 is written as shown:

12 are written as shown:

Notice how the ones, in this case two ones are shown on the right just

like the Hindu-Arabic numeration system

45 is written as shown:

For number bigger than 59, the Babylonian used a place value system

with a base of 60

62 is written as shown:

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Notice this time the use of a big space to separate the space value

Without the big space, things look like this:

However, what is that number without this big space? Could it be 2 ×

60 + 1 or 1× 602 + 1 × 60 + 1 or??? 

The Babylonians introduced the big space after they became aware of

this ambiguity.

The number 4871 could be represented as follow: 3600 + 1260 + 11 =

4871 

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Even after the big space was introduced to separate place value, the

Babylonians still faced a more serious problem?

Since there was no zero to put in an empty position, the number 60

would thus have the same representation as the number 1

How did they make the difference? All we can say is that the context

must have helped them to establish such difference yet the Babylonian

numeration system was without a doubt a very ambiguous numeral

system.

If this had become a major problem, no doubt the Babylonians were

smart enough to come up with a working system.

HINDU-ARABIC

NUMERATION

SYSTEM

BABYLONIAN NUMERATION SYSTEM

8

1

2

3

4

5

6

7

9

8

9

10

2.3 ROMAN NUMERATION SYSTEM

Before Rome, the most developed civilization on the Italic Peninsula

was the Etruscan civilization, who copied their numerals from the early

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Greek (Attic) system. These numerals were adopted and adapted by

the Romans, who formulated the Roman numeral system, still in wide

use today for a variety of purposes. There are other Roman numerals

that most of us never learn, but can be found in Latin dictionaries – for

example:

5000 I>>

10000 ==I>>

50000 I>>>

100000 ===I>>>

500000 I>>>>

1000000 ====I>>>>

As every grade school child knows, the Roman numeral system

is based on the following seven atomic numerals:

I V X L C D M

1 5 10 50 100 500 1000

The Roman numeral system is not a simple additive system, but

is rather an additive-subtractive system. In fact, the subtractive aspect

is frequently a source of worry when reading large numerals – for

example:

MCMXCIX

By saying that the Roman system is (partly) subtractive, we

mean that some combinations of symbols require us to apply

subtraction in order to interpret them. For example, IV stands for “one

before five”, which is four [i.e., 5 minus 1]. Similarly, the numeral XC

stands for “ten before one-hundred”, which is ninety [i.e., 100 minus

10]. On the other hand, the string IC is officially ill-formed, although it

could be understood to mean “one before one-hundred”, which would

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then be ninety-nine. So how do we interpret a Roman numeral such as

‘MCMXCIX’?

M is not before a larger numeral, so it reads: + 1000 1000

C is before a larger numeral, so it reads: - 100

M is after a negative prefix, so it reads: + 1000 900

X is before a larger numeral, so it reads: - 10

C is after a negative prefix, so it reads: + 100 90

I is before a larger numeral, so it reads: - 1

X is after a negative prefix, so it reads: + 10 9

Thus, ‘MCMXCIX’ represents the number 1999.

HINDU-ARABIC

NUMERATION

SYSTEM

ROMAN NUMERATION SYSTEM

1 I

2 II

3 III

4 IV

5 V

6 VI

7 VII

8 VIII

9 IX

10 X

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2.4 MAYAN NUMERATION SYSTEM

The Mayan number system dates back to the fourth century and was

approximately 1,000 years more advanced than the Europeans of that time.

This system is unique to our current decimal system, which has a base 10, in

that the Mayan's used a base 20.

This system is believed to have been used because, since the Mayan's

lived in such a warm climate and there was rarely a need to wear shoes, 20

was the total number of fingers and toes, thus 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 Mayan numeration system evolved around A.D. 300. It uses 3

basic numerals to represent any possible number: a dot for one, a horizontal

bar for 5, and a conch shell for zero.

The Mayan's were also the first to symbolize the concept of nothing (or

zero). The most common symbol was that of a shell ( ) but there were several

other symbols (e.g. a head). It is interesting to learn that with all of the great

mathematicians and scientists that were around in ancient Greece and Rome,

it was the Mayan Indians who independently came up with this symbol which

usually meant completion as opposed to zero or nothing.

They used the 3 symbols above to represent the numbers from 0

through 19 as shown below:

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For number bigger than 19, a number is written in a vertical position so

that it becomes a vertical place value system. Initially, the base used in the

Mayan numeration system was base 20 and their place values were 1, 20,

202, 203,

Then, they changed their place values to 1, 20, 20 × 18, 202 × 18, 203× 18, ...

Using the base 20, 1, 20, 202,203, ..., we can write 20 as follow:

In the ones place we have 0 and in the twenties place we have 1, so

the number is

0 × 1 + 1 × 20 = 0 + 20 = 20

Still using a base of 20, we can write 100 as follow:

0 × 1 + 5 × 20 = 0 + 20 = 100

Below is how to represent 2007

5 × 202 + 0 × 20 + 7 = 5 × 400 + 0 + 7 = 2000 + 7 = 2007

It is started from the bottom, a place value must have a number from the list

above.

14

(1-19)

Look carefully and see how it was separated into the place values.

Again, it was separated according to numbers that are the list above from 1

through 19

The number is: 14 + 7 × 20 + 1 × 202 + 3 × 20 3 + 0 × 20 4 + 15 × 20 5 + 5 ×

20 6

The number is 14 + 140 + 1 × 400 + 3 × 8,000 + 0 + 15 × 3,200,000 + 5×

64,000,000

The number is = 14 + 140 + 400 + 24,000 + 0 + 48,000,000 + 320,000,000 =

368024554

With the base 1, 20, 20 × 18, 202 × 18, 203× 18, ... computation is done the

exact same way!

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Group as shown below:

The number is 11 × 1 + 1 × 20 + 10 × 20 × 18 = 11 + 20 + 3600 = 3631

No doubt; the Mayan numeration system was sophisticated. 

HINDU-ARABIC

NUMERATION

SYSTEM

MAYAN NUMERATION SYSTEM

1

2

3

4

5

6

7

8

9

10

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2.5 WORD PROBLEM AND ITS SOLUTION

Question:

One of the natives on the island named Karu. One day, he went out to

find food. He collected 41 carrots, 26 clams, 13 fishes and a dozen

bananas in a big rattan basket. He felt hungry so he decided to eat 3

carrots and 5 fishes. As he was about to go back home, he fell onto

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the ground and lost 20 clams and 24 carrots. On seeing Karu was

injured, a monkey quickly stole 3 bananas. How many carrots, clams,

fishes and bananas left in the rattan basket at last?

Answer:

Carrots:

41- 3 – 24 = 14

Clams:

26 – 20 = 6

Fishes:

13 – 5 = 8

Bananas:

12 – 3 = 9

TRANSLATION INTO BABYLONIAN WRITING

:

18

.

.

,

,

19

.

.

20

,

.

,

21

.

,

,

?

:

22

:

- - =

:

- =

:

- =

:

- =

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