Transcript No Slide Title

Copyright © 2005 Pearson Education, Inc.
SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 4-1
Chapter 4
Systems of Numeration
Copyright © 2005 Pearson Education, Inc.
4.1
Additive, Multiplicative, and
Ciphered Systems of Numeration
Copyright © 2005 Pearson Education, Inc.
Systems of Numeration



A system of numeration consists of a set of
numerals and a scheme or rule for combining
the numerals to represent numbers
A number is a quantity. It answers the question
“how many?”
A numeral is a symbol used to represent the
number (amount).
Copyright © 2005 Pearson Education, Inc.
Slide 4-4
Types Of Numeration Systems

Four types of systems used by different
cultures will be discussed. They are:




Additive (or repetitive)
Multiplicative
Ciphered
Place-value
Copyright © 2005 Pearson Education, Inc.
Slide 4-5
Additive 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.
It is one of the oldest and most primitive types
of systems.
Examples: Egyptian hieroglyphics and Roman
numerals.
Copyright © 2005 Pearson Education, Inc.
Slide 4-6
Multiplicative Systems


Multiplicative systems are more similar to the
Hindu-Arabic system which we use today.
Example: Chinese numerals.
Copyright © 2005 Pearson Education, Inc.
Slide 4-7
Ciphered Systems


In this system, there are numerals for numbers
up to and including the base and for multiples of
the base.
The numbers (amounts) represented by a
specific set of numerals is the sum of the values
of the numerals.
Copyright © 2005 Pearson Education, Inc.
Slide 4-8
Examples of Ciphered Systems:





Ionic Greek system (developed about 3000 B.C.
and used letters of Greek alphabet as
numerals).
Hebrew system
Coptic system
Hindu system
Early Arabic systems
Copyright © 2005 Pearson Education, Inc.
Slide 4-9
4.2
Place-Value or Positional-Value
Numeration Systems
Copyright © 2005 Pearson Education, Inc.
Place-Value System



The value of the symbol depends on its position
in the representation of the number.
It is the most common type of numeration
system in the world today.
The most common place-value system is the
Hindu-Arabic numeration system. This is used
in the United States.
Copyright © 2005 Pearson Education, Inc.
Slide 4-11
Place-Value System


A true positional-value system requires a base
and a set of symbols, including a symbol for
zero and one for each counting number less
than the base.
The most common place-value system is the
base 10 system.

It is called the decimal number system.
Copyright © 2005 Pearson Education, Inc.
Slide 4-12
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, 101, 100.
Copyright © 2005 Pearson Education, Inc.
Slide 4-13
Expanded Form




To evaluate a number in this system, begin with the
rightmost digit and multiply it by 1.
Multiply the second digit from the right by base 10.
Continue by taking the next digit to the left and
multiplying by the next power of 10.
In general, we multiply the digit n places from the right
by 10n-1 in order to show expanded form.
Copyright © 2005 Pearson Education, Inc.
Slide 4-14
Example: Expanded Form
Write the Hindu-Arabic numeral in expanded form.

a) 63
b) 3769
Solution:

63 = (6 x 101 ) + (3 x 1 ) or (6 x 10) + 3

3769 = (3 x 1000) + (7 x 100) + (6 x 10) + 9
or
(3 x 103 ) + (7 x 102 ) + (6 x 101 ) + (9 x 1 )
Copyright © 2005 Pearson Education, Inc.
Slide 4-15
4.3
Other Bases
Copyright © 2005 Pearson Education, Inc.
Bases



Any counting number greater than 1 may be
used as a base for a positional-value
numeration system.
If a positional-value system has a base b,
then its positional values will be
… b 4, b 3, b 2, b 1, b 0.
Hindu-Arabic system uses base 10.
Copyright © 2005 Pearson Education, Inc.
Slide 4-17
Example: Converting from Base 8 to
Base 10
Convert 45368 to base 10.
Solution:

 

45368  4  83  5  82   3  8    6  1
  4  512    5  64    3  8    6  1
 2048  320 
 2398
Copyright © 2005 Pearson Education, Inc.
24  6
Slide 4-18
Example: Converting from Base 5 to
Base 10
Convert 425 to base 10.
Solution:


 20

425  4  51   2  1
2
 22
Copyright © 2005 Pearson Education, Inc.
Slide 4-19
Example: Convert to Base 3
Convert 342 to base 3.
Solution:
The place values in the base 3 system are
…, 36, 35, 34, 33, 32, 3, 1 or …729, 243, 81, 27, 9, 3, 1.
The highest power of the base that is less than or equal
to 342 is 243.
Copyright © 2005 Pearson Education, Inc.
Slide 4-20
Example: Convert to Base 3 continued

Successive division by the powers of the base
gives the following result.
342  243  1 with remainder 99
99  81  1 with remainder 18
18  27  0 with remainder 18
18  9  2 with remainder 0
0  3  0 with remainder 0
Copyright © 2005 Pearson Education, Inc.
Slide 4-21
Example: Convert to Base 3 continued

The remainder, 0, is less than the base, 3, so
further division is necessary.
342  1 243   1 81   0  27    2  9    0  3    0  1
 (1 35 )  1 34    0  33    2  32    0  3    0  1
 1102003
Copyright © 2005 Pearson Education, Inc.
Slide 4-22
Computers

Computers make use of three numeration
systems



Binary
Octal
Hexadecimal
Copyright © 2005 Pearson Education, Inc.
Slide 4-23
Numeration Systems

Binary system




Octal system


Base 2
It is very important because it is the international language
of the computer.
Computers use a two-digit “alphabet” that consists of
numerals 0 and 1.
Base 8
Hexadecimal system

Base 16
Copyright © 2005 Pearson Education, Inc.
Slide 4-24
4.4
Computation in Other Bases
Copyright © 2005 Pearson Education, Inc.
Addition


An addition table can be made for any base and it can
be used to add in that base.
Base 5 Addition Table
+
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
10
2
2
3
4
10
11
3
3
4
10
11
12
4
4
10
11
12
13
Copyright © 2005 Pearson Education, Inc.
Slide 4-26
Example: Using the Base 5 Addition
Table
Add 445
235
Solution:
From the table 45 + 35 = 125 Record the 2 and
carry the 1.
144
5
235
25
Copyright © 2005 Pearson Education, Inc.
Slide 4-27
Example: Using the Base 5 Addition
Table continued


Add the numbers in the second column, (15 +
45) + 25 = 105 + 25 = 125.
Record the 12.
144
5
235
1225
The sum is 1225.
Copyright © 2005 Pearson Education, Inc.
Slide 4-28
Subtraction



Subtraction can also be performed in other
bases.
When you “borrow” you borrow the amount of
the base given in the subtraction problem.
Example: If you are subtracting in base 5,
when you borrow, you borrow 5.
Copyright © 2005 Pearson Education, Inc.
Slide 4-29
Multiplication


Multiplication table for the given base is extremely
helpful.
Base 5 Multiplication Table
x
0
1
2
3
4
0
0
0
0
0
0
1
1
2
3
4
4
2
0
2
4
11
13
3
0
3
11
14
22
4
0
4
13
22
31
Copyright © 2005 Pearson Education, Inc.
Slide 4-30
Example: Using the Base 5
Multiplication Table
Multiply
125
x 35
Solution:
Use the base 5 multiplication table to find the products.
When the product consists of two digits, record the right
digit and carry the left digit.
Copyright © 2005 Pearson Education, Inc.
Slide 4-31
Example: Using the Base 5
Multiplication Table continued
Record the 1 carry the 1.
125
x 35
1
(35 x 15) + 15 = 45 Record the 4.
125
x 35
415
The product is 415.

Copyright © 2005 Pearson Education, Inc.
Slide 4-32
Division


Division is carried out much the same way as
long division in base 10.
A division problem can be checked by
multiplication.

(quotient
 divisor) + remainder = dividend
Copyright © 2005 Pearson Education, Inc.
Slide 4-33
4.5
Early Computation Methods
Copyright © 2005 Pearson Education, Inc.
Early Civilizations


Early civilizations used a variety of methods for
multiplication and division.
Multiplication was performed by duplation and
mediation, by the galley method, and by Napier
rods.
Copyright © 2005 Pearson Education, Inc.
Slide 4-35
Duplation and Mediation
Duplation and mediation uses a pairing method for
multiplication.
Example: Multiply 13  22 using duplation and mediation.
Solution: Write 13 and 22 with a dash to separate. Divide
the number on the left in half, drop the remainder and
place the quotient under the 13. Double the number on
the right, and place it under the 22.

Copyright © 2005 Pearson Education, Inc.
Slide 4-36
Duplation and Mediation continued

17 – 22
8 – 44
Continue this process until a 1
appears in the left hand
column.
17 – 22
8 – 44
4 – 88
2 – 176
1 – 352
Copyright © 2005 Pearson Education, Inc.

Cross out all the even
numbers in the left-hand
column and the corresponding
numbers in the right-hand
column.
17 – 22
8 – 44
4 – 88
2 – 176
1 – 352
Slide 4-37
Duplation and Mediation continued


Now, add the remaining numbers in the righthand column, obtaining 22 + 352 = 374.
To check 17  22 = 374
Copyright © 2005 Pearson Education, Inc.
Slide 4-38
The Galley Method


The Galley method is also referred to as the
Gelosia method.
This method uses a rectangle split into columns
and rows with each newly-formed square split in
half by a diagonal.
Copyright © 2005 Pearson Education, Inc.
Slide 4-39
Example: The Galley Method

Multiply 426 65.
Solution:

Construct a rectangle consisting of 3 columns and 2
rows.

Place the 3-digit number above the boxes and the 2digit number on the right of the boxes.

Place a diagonal in each box.

Complete by multiplying.
Copyright © 2005 Pearson Education, Inc.
Slide 4-40
Example: The Galley Method
continued
4
2
2
6
1
2
2
0
6
6

The answer is 27,690.
6
3
1
7
Add the numbers along
the diagonals. The
number is read down
the left-handed column
and along the bottom,
as shown by the arrow.
3
4
2

0
9
Copyright © 2005 Pearson Education, Inc.
0
0
5
Slide 4-41
Napier Rods



John Napier developed in the 17th century.
Napier rods, proved to be one of the forerunners of the
modern-day computer.
Napier developed a system of separate rods numbered
0 through 9 and an additional strip for an index,
numbered vertically 1 through 9.

Each rod is divided into 10 blocks. Each block below
contains a multiple of a the number in the first block, with a
diagonal separating its digits. The units are placed to the
right of the diagonals and the tens digits to the left.
Copyright © 2005 Pearson Education, Inc.
Slide 4-42
Example: Napier Rods

Multiply 6 284, using Napier rods.
Solution: Line up the rods 2, 8, 4, using 6 as the index. To
obtain the answer, add along the diagonals as in the
galley method.
2
6
Thus, 6
 284 = 1704.
Copyright © 2005 Pearson Education, Inc.
1 1 2
7
8
4
4
8
0
2
4
4
Slide 4-43