What You Will Learn
Additive, multiplicative, and ciphered
systems of numeration
4.1-1
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Systems of Numeration
A number is a quantity. It answers the
question “How many?”
A numeral is a symbol such as  ,10 or
 used to represent the number.
4.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Systems of Numeration
A system of numeration consists of
1.
a set of numerals and
2.
a scheme or rule for combining the
numerals to represent numbers.
4.1-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Additive Systems
Examples:
•
Egyptian hieroglyphics
•
Roman numerals.
4.1-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Egyptian Hieroglyphics
4.1-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: From Egyptian to
Hindu-Arabic Numerals
Write the following numeral as a HinduArabic numeral.
Solution
10,000 + 10,000 + 10,000 + 100 +
100 + 100 + 10 + 1 + 1 + 1 + 1
= 30,314
4.1-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Roman Numerals
Roman Numerals
I
V
X
L
C
D
M
4.1-7
Hindu-Arabic Numerals
1
5
10
50
100
500
1000
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Roman Numerals
Two advantages over Egyptian system:
1. Uses the subtraction principle as well as
addition principle
DC = 500 + 100 = 600
CD = 500 – 100 = 400
2.
Uses the multiplication principle for
numerals greater than 1000
V  5  1000  5000
CD  400  1000  400,000
4.1-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: From Roman to
Hindu-Arabic Numerals
Write CMLXIV as a Hindu-Arabic
numeral.
Solution
It’s an additive system so,
=
CM
+ L + X+
IV
= (1000 – 100) + 50 + 10 + (5 – 1)
= 900 + 50 + 10 + 4
= 964
4.1-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplicative Systems
Read if interested
Chinese numerals
4.1-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Ciphered Systems
•
4.1-11
Read if interested.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Place-Value or Position-Value
Numeration Systems
4.2-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Place-Value System
(or Positional-Value System)
•
The most common type of numeration
system in the world today.
•
The most common place-value
system is the Hindu-Arabic
numeration system.
4.2-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Place-Value System
The most common place-value system
is the base 10 system.
It is called the decimal number
system.
4.2-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Hindu-Arabic System
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
4.2-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Expanded Form
In expanded form, 1234 is written
1234 = (1 × 103) + (2 × 102)
+ (3 × 10) + (4 × 1)
or
= (1 × 1000) + (2 × 100)
+ (3 × 10) + 4
4.2-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Babylonian Numerals
Read if interested.
4.2-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Babylonian Numerals
The positional values in the Babylonian
system are
…, (60)3, (60)2, 60, 1
4.2-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Mayan Numerals
Read if interested.
4.2-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Mayan Numerals
The positional values in the Mayan
system are
…, 18 × (20)3, 18 × (20)2, 20, 1
or …, 144,000,
7200,
20, 1
4.2-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Converting base 10 numerals to
numerals in other bases
Converting numerals in other bases to
base 10 numerals
4.3-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Positional Values
•
The positional values in the Hindu-Arabic numeration
system are
… 105, 104, 103, 102, 10, 1
10 is called the base of the Hindu-Arabic numeration
system
•
Any counting number greater than 1 may be used as a
base.
If a positional-value system has base b, then its
positional values will be
…, b4, b3, b2, b, 1
4.3-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Bases Less Than 10
A place-value system with base b has b distinct
objects, one for zero and one for each numeral less
than the base.
Example. Base 8 system has 8 distinct objects:
0, 1, 2, 3, 4, 5, 6, 7.
All numerals in base 8 are constructed from these
8 symbols.
4.3-23
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Bases Less Than 10
A numeral in a base other than base 10 will
be indicated by a subscript to the right of
the numeral.
1235 : a base 5 numeral.
1236 : a base 6 numeral.
Base 10 numerals can be written without a
subscript: 123 means 12310.
4.3-24
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Converting from other Base to Base 10
To change a numeral from one base to base 10,
multiply each digit by its respective positional value,
then find the sum of the products.
4.3-25
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Converting from
Base 5 to Base 10
Convert 2435 to base 10.
Solution
2435 = (2 × 52) + (4 × 5) + (3 × 1)
= (2 × 25) + (4 × 5) + (3 × 1)
= 50 + 20 + 3
= 73
4.3-26
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Converting from
Base 2 to Base 10
Convert 1100102 to base 10.
1100102 = 50
4.3-27
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Converting Base 10 to other Base
•
Divide the base 10 numeral by the highest power of the
new base that is less than or equal to the given base 10
numeral and record this quotient.
•
Then divide the remainder by the next smaller power of
the new base and record this quotient.
•
Repeat this procedure until the remainder is less than
the new base.
•
The answer is the set of quotients listed from left to
right, with the remainder on the far right.
4.3-28
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: Converting from
Base 10 to Base 3
Convert 273 to base 3.
4.3-29
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
273 = 1010103
Bases Greater Than 10
Now consider base > 10. We will need single digit
symbols to represent the numbers ten, eleven, twelve, ...
up to one less than the base.
In this textbook,
•use
A to represent 10
•use
B to represent 11
•And
so on
4.3-30
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Bases Greater Than 10
For base 16, we use the symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.
4.3-31
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 7: Converting to and from
Base 16
Convert 7DE16 to base 10.
4.3-32
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
7DE16 = 2014
Example 7: Converting to and from
Base 16
Convert 6713 to base 16.
4.3-33
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
6713 = 1A3916
What You Will Learn
Performing addition, subtraction,
multiplication and division in other
bases
4.4-34
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Addition
An addition table can be made for any
base and be used to add in that base.
Base 5 Addition Table
+
0
1
2
3
4
4.4-35
0
0
1
2
3
4
1
1
2
3
4
10
2
2
3
4
10
11
3
3
4
10
11
12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
4
4
10
11
12
13
Example 2: Using the Base 5
Addition Table
Like adding base 10 numbers. Need to
carry.
Add
4.4-36
345
+ 235
345 + 235 = 1125
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Subtraction
Like adding base 10 numbers. Need to
borrow.
4.4-37
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 6: Subtracting in Base 12
Subtract
4.4-38
91B12
– 2A212
91B12 + 2A212 = 63912
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Units Digits in Different Bases
Notice that 35 has the same value as
310, since both are equal to 3 units.
That is, 35 = 310.
If n is a digit less than the base b, and
the base b is less than or equal to 10,
then nb = n10.
4.4-39
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplication
Multiplication can be performed in bases other than 10.
•
In base 10, 4 × 3 means four groups of three units.
•
In base 5, 45 × 35 means four groups of three units.
(1+1+1)+(1+1+1)+(1+1+1)+(1+1+1)
Regroup into groups of five:
(1+1+1+1+1)+(1+1+1+1+1)+(1+1)
45 × 35 = 225
4.4-40
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplication
Multiplication table for the given base
is extremely helpful.
Base 5 Multiplication Table
×
0
1
2
3
4
4.4-41
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
11
13
3
0
3
11
14
22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
4
0
4
13
22
31
Multiplication
Another way to multiply: 45 × 35 = 225
It is easier to
First multiply the values in the base 10 system
4 × 3 = 12
Then change the product to base 5,
12 = 225
4.4-42
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 8: Multiplying in Base 7
Multiply 437
× 257
4.4-43
437  257 = 15017
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Division
Division is performed in much the
same way as long division in base 10.
A division problem can be checked by
multiplication.
(quotient × divisor) + remainder = dividend
4.4-44
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 10: Dividing in Base 8
Divide 68 40718
Solution
The multiples
68 × 18 = 68
68 × 38 = 228
68 × 58 = 368
68 × 78 = 528
5368
68 40718
of 6 in base 8:
68 × 28 = 148
68 × 48 = 308
68 × 68 = 448
Quotient is 5368, remainder 58
4.4-45
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
36
27
22
51
44
5
Example 10: Dividing in Base 8
Check
Does (5368 × 68) + 58= 40718?
True
4.4-46
Copyright 2013, 2010, 2007, Pearson, Education, Inc.