Historical Numeration Systems

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Section 4-1

• Historical Numeration Systems

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Chapter 4 Numeration 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 use of computer and calculators.
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Chapter 4 Numeration Systems
Ancient Civilization
European
Hindu- Arabic 670 AD
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Chapter 4 Numeration Systems
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Hindu- Arabic 670 AD
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Chapter 4 Numeration Systems
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Chapter 4 Numeration Systems
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Chapter 4 Numeration Systems
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Chapter 4 Numeration Systems
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Hindu- Arabic 670 AD
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Chapter 4 Numeration Systems
Mayan 2000 BC-1546 AD
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Chapter 4 Numeration Systems
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Historical Numeration Systems

• Basics of Numeration
• Ancient Egyptian Numeration
• Ancient Roman Numeration
• Classical Chinese Numeration

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Numeration 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.
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Example 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
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Counting 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.
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Ancient Egyptian Numeration Simple Grouping
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.
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Ancient Egyptian Numeration
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Example Egyptian Numeral
Write the number below in our system.
Solution
2 (100,000) 200,000 3 (1,000) 3,000
1 (100) 100 4 (10)
40 5 (1) 5
Answer 203,145
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Ancient 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.

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Ancient Roman Numeration

• 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.

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Ancient Roman Numeration
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Example Roman Numeral
Write the number below in our system. MCMXLVII
Solution
M 1000 CM -100 1000 XL -10 50 V
5 I 1 I 1
Answer 1000 900 40 5 1 1 1947
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Example Roman Numeral
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Traditional Chinese Numeration Multiplicative
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.
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Chinese Numeration
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Example Chinese Numeral
Interpret each Chinese numeral. a) b)
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Example Chinese Numeral
Solution
7000
200
400
0 (tens)
1
80
Answer 201
2
Answer 7482
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Example Chinese Numeral
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.
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Example Chinese Numeral
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Section 4-2

• More Historical Numeration Systems

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More Historical Numeration Systems

• Basics of Positional Numeration
• Hindu-Arabic Numeration
• Babylonian Numeration
• Mayan Numeration
• Greek Numeration

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Positional 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.
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Positional Numeration
In a positional numeral, each symbol (called
a digit) 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.
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Positional Numeration
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.
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Hindu-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.
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Hindu-Arabic Numeration
Hundred thousands
Thousands
Ten thousands
Millions
Decimal point
Hundreds
Tens
Units
7, 5 4 1, 7 2
5 .
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Babylonian 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

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Example Babylonian Numeral

• Interpret each Babylonian numeral.
• a) ? ? ? ?
• b) ? ? ? ? ? ? ?

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Example Babylonian Numeral
Solution
? ? ? ?
Answer 34
? ? ? ? ? ? ?
Answer 155
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Example Babylonian Numeral
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Example Babylonian Numeral
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Example Babylonian Numeral
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Example Babylonian Numeral
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Example Babylonian Numeral
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Mayan 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
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Mayan Numeration
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Example Mayan Numeral
Write the number below in our system.
Solution
Answer 3619
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Example Mayan Numeral
Write the number below in our system.
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Example Mayan Numeral
Write the number below in our system.
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Example Mayan Numeral
Write the number below in Mayan Numeral.
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Example Mayan Numeral
Write the number below in Mayan Numeral.
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Greek 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.

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Greek Numeration
Table 2
Table 2 (cont.)
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Example Greek Numerals

• Interpret each Greek numeral.
• a) ma
• b) cpq

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Example Greek Numerals
Solution
a) ma b) cpq
Answer 41
Answer 689
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Example Greek Numerals