4.1
Historical Numeration Systems
Basics of Numeration
The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols
of a numeration system are called numerals.
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:
Ancient Egyptian Numeration
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.
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 below.
Example 1: Egyptian Numeral
Write the number below in our system.
1
Example 2: Egyptian Numeral
Example 3: Egyptian Numeral
Example 4: Egyptian Numeral
Assignment 32 pg. 144: 1 - 14 ODD, 35-46 ODD, 47
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Ancient Roman Numeration
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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 Numeral
Write the number below in our system.
MCMXLVII
Example 5: Roman Numeral
Example 6: Roman Numeral
Assignment 33 pg. 144: 15-22 ALL
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Ancient Chinese Numerals
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 below.
Example 7: Chinese Numeral
Example 8: Chinese Numeral
Assignment 34 pg. 144: 23-30 ALL, 31, 33, 53-63 ODD
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Section 4-2 – More Historical Number Systems
An additive system is one in which the number represented by a set of numerals is simply the sum of the values
of the numerals.
Examples: Egyptian hieroglyphics and Roman numerals.
Basics of Positional Numeration
A positional system is
In a positional numeral, each symbol (called a digit) conveys two things:
1.
Face value –
2.
Place value –
.
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
One 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.
Digits: In the Hindu-Arabic system, the digits are
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
Positions: In the Hindu-Arabic system, the positional values or place values are
… 105, 104, 103, 102, 10, 1
Example: Fill in the place value for each digit
7,
5
4
1,
7
2
5
5
Babylonian Numeration
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Oldest known numeration system that resembled a place-value system
Developed in about 2500 B.C.
Resembled a place-value system with a base of 60, a sexagesimal system
Not a true place-value system because it lacked a symbol for zero
The lack of a symbol for zero led to a great deal of ambiguity and confusion
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
The positional values in the Babylonian system are
…, (60)3, (60)2, 60, 1
Example 1:
Example 2:
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Example: Interpret each Babylonian numeral.
a) ‹ ‹ ‹ ▼ ▼ ▼ ▼
b) ▼ ▼ ‹ ‹ ‹ ▼ ▼ ▼ ▼ ▼
Example: Write 7223 as a Babylonian numeral.
Assignment 35 page 150: #3,9,11,17,21,23,25,27,29
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Mayan Numeration
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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.
Numerals are written vertically.
Units position is on the bottom. Numeral in bottom row is multiplied by 1.
Numeral in second row is multiplied by 20.
Numeral in third row is multiplied by 18 × 20, or 360.
Numeral in fourth row is multiplied by 18 × 202, or 7200, and so on.
The positional values in the Mayan system are
…, 18 × (20)3, 18 × (20)2, 20, 1
or …, 144,000, 7200,
20, 1
Example 3:
Example 4:
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Example: Write the number below in our system.
Example: Write 4025 as a Mayan numeral.
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.
Example: Interpret each Greek numeral.
a) 
b) 
Assignment 36 page 150 #1,5,7,13,15,19, 31-47 ODD
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4.3 - Arithmetic in the Hindu-Arabic System
Expanded Form
By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made
clear.
Example 1:
Example 2:
Example 3:
Distributive Property: For all real numbers a, b, and c,
b  a    c  a   b  c   a.
For example,
 3 10    2 10    3  2  10
4
4
4
 5  104.
Example 4:
10
Example 5:
Example 6:
Example 7:
Assignment 37 page 157 #1-25 odd
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Historical Calculation Devices
Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem,
meaning ten.
One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar.
Reading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the
value of those below. Beads moved towards the bar are in “active” position.
Example: Which number is shown above?
The Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of
single digits into a diagonalized lattice.
Example: Find the product 38∙ 794 by the lattice method.
Step 1: Set up the grid to the right.
Step 2: Fill in products
Step 3: Add diagonally right to left and carry
as necessary to the next diagonal.
Step 4:
Assignment 38 page 158 #27-38 all, 45-52 all
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4.4 - Conversion Between Number Bases
General Bases Conversions
We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54eight. When a
number appears without a subscript assume it is base ten. Note that 54eight is read “five four base eight.” Do not read it as
“fifty-four.”
Powers of Alternative Bases
Fourth
Power
Third
Power
Second
Power
First
Power
Zero
Power
Base two
16
8
4
2
1
Base five
625
125
25
5
1
Base seven
2401
343
49
7
1
Base eight
4096
512
64
8
1
65,536
4096
256
16
1
Base sixteen
Example: Convert 1342five to decimal form.
Example: Convert 2134five to decimal form.
To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add
the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by
the base.
Example: Use the calculator shortcut to convert 244314five to decimal form.
Example: Use the calculator shortcut to convert 432134five to decimal form.
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Example: Convert 497 from decimal form to base five
Example: Convert 6343seven to decimal for by
expanding in powers, and using the calculator shortcut.
.
Example: Convert 7508 to base seven.
Example: Convert 364seven to base five.
Assignment 39 page 165 # 1-45 odd
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Computer Mathematics
Computers and many other electronic devices use three numeration systems:
Binary – base 2
 Uses only the digits 0 and 1.
 Can be represented with electronic switches that are either off (0) or on (1).
 All computer data can be converted into a series of single binary digits.
 Each binary digit is known as a bit.
Octal – base 8
 Eight bits of data are grouped to form a byte
 American Standard Code for Information Interchange (ASCII) code.
 The byte 01000001 represents A.
 The byte 01100001 represents a.
Hexadecimal – base 16
 Used to create computer languages:
 HTML (Hypertext Markup Language)
 CSS (Cascading Style Sheets).
 Both are used heavily in creating Internet web pages.
Computers easily convert between binary (base 2), octal (base 8), and hexadecimal (base 16) numbers.
Example: Convert 111001two to decimal form.
Example: Convert 9583 to octal form.
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Example: Convert FA5sixteen to decimal form.
Example: Convert 748 from decimal to hexadecimal form.
Example: Convert 473eight to binary form.
Example: Convert 10011110two to octal form.
Example: Convert 8B4Fsixteen to binary form.
Assignment 40 page 166 # 47-95 odd
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