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Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
Page 1 of 6
Math 365 Lecture Notes
Section 3.1 – Numeration Systems
Numeration Systems
What are some cultural properties that would influence the development of
numeration systems?
Items in their environment (lotus flower symbol), climate (using toes to
count if barefoot), need for larger numbers, calendar (Mayans using 360)
Definitions:
1) Numeration System: a collection of properties and symbols to
represent numbers
2) Numerals: symbols to represent cardinal numbers; the symbols 3,
and 8 are numerals
Types of Numeration Systems:
1) Additive: the value of a number was the sum of the face values of
the numerals
2) Place Value: the value of a number depends on its placement in a
numeral
(the face value is a digit)
Tally Numeration System (additive)
one
two three four five
six seven eight nine ten
Tallies |
||
|||
| | | | | | | | |||| |
|||| ||
|||| ||| |||| |||| |||| ||||
(more primitive tallies did not use cross marks)
Discussion Questions
1) What are some advantages to this numeration system?
Simple system; one-to-one correspondence between tally marks
and objects
2) What are some disadvantages to this numeration system?
Lots of symbols required; as the numbers get larger, it gets
harder to read
Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
Page 2 of 6
Egyptian Numeration System (additive)
Problem 1: Use the Egyptian system to represent 1254.
Solution:
Babylonian Numeration System (place value – base 60)
What numerals are used in the Babylonian system and what are the face
values of the numerals?
Same as Egyptian except:▼ = 1 and < = 10
Problem 2: Find the value of each Babylonian number.
1) ▼▼▼
<<

601
600
2) <▼
3(60) + 20(1) = 200
<<<▼

602

601
3) ▼▼▼
60
600
▼▼

2
<▼▼▼▼
<<▼

1
60
11(60)2 + 34(60) + 14 = 41,654
60
0
3(60)2 + 2(60) + 21 = 10, 941
Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
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Mayan Numeration System (place value – modified base 20)
What numerals are used in the Mayan system and what are the face values of
the numerals? = 1
=5
=0
In a true base 20 system, the third position vertically from the bottom should
be 202 = 400, but this is NOT the case in the Mayan number system. Instead
the Mayan’s used
200, 201, 201 ∙ 18, 202 ∙ 18, 203 ∙ 18, 204 ∙ 18, …
Example of Mayan number:
11(20)
+ 0(1)
220
Problem 3: Convert the following Mayan numbers to Hindu-Arabic (in
expanded form).
a)
9  20(18)
0  20
10  1
3258
b)
12  20(18)
5  20
61
4426
For an explanation of why they chose this number system go to the
following website
http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Mayan_mathematics.html
Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
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Roman Numeration System (additive/subtractive/multiplicative)
What numerals are used in the Roman system and what are the face values
of the numerals?
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
Important properties of Roman numeral system:
1) Additive Property: A property in which the face value of the numerals
are added together using the largest number possible at each stage (i.e. 15 is
written XV not VVV). No numeral is written more than 3 consecutive
times.
2) Subtraction Property
a) Only I, X, and C can be used to subtract.
b) Only one smaller number can be placed to the left.
c) The subtracted value must be no less than a tenth of the value of the
number it is subtracted from. So the only numbers obtained through
the subtraction property are:
IV
IX
XL
5-1=4 10-1=9 50-10=40
XC
CD
CM
100-10=90 500-100=400 1000-100=900
3) Multiplicative Property
CDX 
410(1000) = 410,000
CXI 
CX I 
111(1000)(1000) = 111,000,000
110(1000) + 1 = 110,000
Hindu-Arabic Numeration System (place value – base 10)
What numerals are used in the Hindu-Arabic system?
0 1 2 3 4 5 6 7 8 9
What are the first 5 place values of the Hindu-Arabic system?
Units, tens, hundreds, thousands (right to left)
Write 5,984 in expanded form.
5(103) + 9(102) + 8(101) + 4(100)
Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
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Other Base Systems – Introductory Activity
1. Get a “place-value mat” and thirty-seven pennies.
2. Take the thirty-seven pennies and make as many stacks of ten as
possible out of these pennies. After each stack of ten pennies is made,
place the stack on the “place-value mat” in the spot for (base)1.
3. Place any left over pennies on the “place-value mat” in the spot for
(base)0.
4. Since there are 3 stacks of “tens” and 7 “ones”, we write 37 to
represent the pennies in base 10.
5. The pennies can also be grouped in 3 stacks of eleven and 4 single
pennies. Thus we can write this number as 34eleven.
6. By grouping your pennies as demonstrated in the previous two
examples, write the number in base two through base twelve.
37 = 100101two
37 = 52seven
37 = 1101three
37 = 45eight
37 = 211four
37 = 41nine
37 = 122five
37 = 37ten
37 = 101six
37 = 34eleven
37 = 31twelve
Other Number Base Systems (place value)
Base
2
Digits Used
0, 1
Example of Converting to Base 10
1011two = 1(2)3 + 0(2)2 + 1(2)1 + 1(2)0 = 11
4
0, 1, 2, 3
213four = 2(4)2 + 1(4)0 + 3(4)0 = 39
6
0, 1, 2, 3, 4, 5
250six = 2(6)2 + 5(6)1 + 0(6)0 = 102
12
0, 1, 2,…, 9, T, E
TE39twelve = 10(12)3 + 11(12)2 + 3(12)1 + 9(12)0 = 18909
Math 365 Lecture Notes - © J. Whitfield 6/28/2017
Section 3-1
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Problem 4: Write the first 10 natural numbers in base 2.
1 10 11 100 101 110 111 1000 1001 1010
Problem 5: Change 824 to base five.
824  5 = 164, r. 4 164  5 = 32, r. 4 32  5 = 6, r. 2 6  5 = 1, r. 1
11244
Problem 6: In base twelve, what five numbers come after 19?
18, 19, 1T, 1E, 20