Section 4-1 Historical Numeration Systems. Chapter 4:Numeration Systems Hindus- Arabic 670 AD It is very important moment in development of Mathematics

Section 4-1Section 4-1

Historical Numeration Systems

Chapter 4:Chapter 4: Numeration SystemsNumeration Systems

Hindus- Arabic 670 AD

It is very important moment in development of Mathematics

1-Relatived easy ways to express the numbers using 10 symbols

2-Relatived easy rules for arithmetic operations.

3- It allows several methods and devices to compute arithmetic operations, even useof computer and calculators.

Chapter 4:Chapter 4: Numeration SystemsNumeration Systems Ancient Civilization Ancient Civilization

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Historical Numeration SystemsHistorical Numeration Systems

• Basics of Numeration

• Ancient Egyptian Numeration

• Ancient Roman Numeration

• Classical Chinese Numeration

Numeration SystemsNumeration Systems

The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals.

Two question are, How many symbols we need to represent numbers and what is the optimal way for grouping these symbols.

Example: Counting by TallyingExample: Counting by Tallying

Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing the following:

Counting by GroupingCounting by Grouping

Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system.

Ancient Egyptian Numeration – Simple Ancient Egyptian Numeration – Simple GroupingGrouping

The ancient Egyptian system is an example of a simple grouping system. It uses ten as its base and the various symbols are shown on the next slide.

Ancient Egyptian NumerationAncient Egyptian Numeration

Example: Egyptian NumeralExample: Egyptian Numeral

Write the number below in our system.

Solution2 (100,000) = 200,000 3 (1,000) = 3,000 1 (100) = 100 4 (10) = 40 5 (1) = 5

Answer: 203,145

Ancient Roman NumerationAncient Roman Numeration

The ancient Roman method of counting is a modified grouping system. It uses ten as its base, but also has symbols for 5, 50, and 500.

The Roman system also has a subtractive feature which allows a number to be written using subtraction.

A smaller-valued symbol placed immediately to the left of the larger value indicated subtraction.


The ancient Roman numeration system also has a multiplicative feature to allow for bigger numbers to be written.

A bar over a number means multiply the number by 1000.

A double bar over the number means multiply by 10002 or 1,000,000.


Example: Roman NumeralExample: Roman Numeral


MCMXLVII

Solution M= 1000CM= -100 + 1000XL = -10 + 50 V= 5 I= 1 I= 1

Answer: 1000 + 900 + 40 + 5 + 1 + 1= 1947

Example: Roman NumeralExample: Roman Numeral

Traditional Chinese Numeration – Traditional Chinese Numeration – Multiplicative GroupingMultiplicative Grouping

A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.

Chinese NumerationChinese Numeration

Example: Chinese NumeralExample: Chinese Numeral

Interpret each Chinese numeral.

a) b)


Solution

7000

400

80

2

Answer: 7482

200

0 (tens)

1

Answer: 201


A single symbol rather than a pair denotes as 1 multiplier an when a particular power is missing the omission is denoted with zero symbol.


Section 4-2Section 4-2

• More Historical Numeration Systems

More Historical Numeration SystemsMore Historical Numeration Systems

• Basics of Positional Numeration

• Hindu-Arabic Numeration

• Babylonian Numeration

• Mayan Numeration

• Greek Numeration

Positional NumerationPositional Numeration

A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral.


In a positional numeral, each symbol (called adigit) conveys two things:

1. Face value – the inherent value of the symbol.

2. Place value – the power of the base which is associated with the position that the digit occupies in the numeral.


To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base is not needed.

Hindu-Arabic Numeration – PositionalHindu-Arabic Numeration – Positional

One such system that uses positional form is our system, the Hindu-Arabic system.

The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.

Hindu-Arabic NumerationHindu-Arabic Numeration

Hundr

eds

Thous

ands

Ten th

ousa

nds

Mill

ions

Hundr

ed th

ousa

nds

Tens

Units

Decim

al po

int

7, 5 4 1, 7 2 5 .

Babylonian NumerationBabylonian Numeration

The ancient Babylonians used a modified base 60 numeration system.

The digits in a base 60 system represent the number of 1s, the number of 60s, the number of 3600s, and so on.

The Babylonians used only two symbols to create all the numbers between 1 and 59.

▼ = 1 and ‹ =10

Example: Babylonian NumeralExample: Babylonian Numeral

Interpret each Babylonian numeral.

a) ‹ ‹ ‹ ▼ ▼ ▼ ▼

b) ▼ ▼ ‹ ‹ ‹ ▼ ▼ ▼ ▼ ▼


4 1

Solution

‹ ‹ ‹ ▼ ▼ ▼ ▼

3 10Answer: 34

▼ ▼ ‹ ‹ ‹ ▼ ▼ ▼ ▼ ▼2 1 3 10 5 1

2 60 35 1

Answer: 155






Mayan NumerationMayan Numeration

The ancient Mayans used a base 20 numeration system, but with a twist.

Normally the place values in a base 20 system would be 1s, 20s, 400s, 8000s, etc. Instead, the Mayans used 360s as their third place value.

Mayan numerals are written from top to bottom. Table 1

Mayan NumerationMayan Numeration

Example: Mayan NumeralExample: Mayan Numeral


Solution

Answer: 3619

10 360

0 20

19 1






Write the number below in Mayan Numeral.


Write the number below in Mayan Numeral.

Greek NumerationGreek Numeration

The classical Greeks used a ciphered counting system.

They had 27 individual symbols for numbers, based on the 24 letters of the Greek alphabet, with 3 Phoenician letters added.

The Greek number symbols are shown on the next slide.

Greek NumerationGreek Numeration

Table 2 Table 2(cont.)

Example: Greek NumeralsExample: Greek Numerals

Interpret each Greek numeral.

a)

b)


Solution

Answer: 41

Answer: 689

a)

b)


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Section 4-1 Historical Numeration Systems. Chapter 4:Numeration Systems Hindus- Arabic 670 AD It is very important moment in development of Mathematics