Slide 4-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

Slide 4-1 Copyright © 2005 Pearson Education, Inc.

SEVENTH EDITION and EXPANDED SEVENTH EDITION

Copyright © 2005 Pearson Education, Inc.

Chapter 4

Systems of Numeration


4.1

Additive, Multiplicative, and Ciphered Systems of Numeration


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).


Types Of Numeration Systems

Four types of systems used by different cultures will be discussed. They are:

Additive (or repetitive) Multiplicative Ciphered Place-value


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.


Multiplicative Systems

Multiplicative systems are more similar to the Hindu-Arabic system which we use today.

Example: Chinese numerals.


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.


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


4.2

Place-Value or Positional-Value Numeration Systems


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.


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.


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.


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.


Example: Expanded Form

Write the Hindu-Arabic numeral in expanded form. a) 63 b) 3769Solution: 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 )


4.3

Other Bases


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

… b4, b3, b2, b1, b0. Hindu-Arabic system uses base 10.


Example: Converting from Base 8 to Base 10

Convert 45368 to base 10.

Solution:

3 284536 4 8 5 8 3 8 6 1

4 512 5 64 3 8 6 1

2048 320 24 6

2398


Example: Converting from Base 5 to Base 10


Solution:

1542 4 5 2 1

20 2

22


Example: Convert 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.


Example: Convert to Base 3 continued

Successive division by the powers of the base gives the following result.

342 243 1 with remainder 99






Example: Convert to Base 3 continued

The remainder, 0, is less than the base, 3, so further division is necessary.

5 4 3 2

3

342 1 243 1 81 0 27 2 9 0 3 0 1

(1 3 ) 1 3 0 3 2 3 0 3 0 1

110200


Computers

Computers make use of three numeration systems

Binary Octal Hexadecimal


Numeration Systems

Binary 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. Octal system

Base 8 Hexadecimal system

Base 16


4.4

Computation in Other Bases


Addition

An addition table can be made for any base and it can be used to add in that base.

Base 5 Addition Table

1312111044

121110433

11104322

1043211

432100

43210+


Add 445

235

Solution:

From the table 45 + 35 = 125 Record the 2 and carry the 1.

1445

235

25

Example: Using the Base 5 Addition Table


Example: Using the Base 5 Addition Table continued Add the numbers in the second column, (15 +

45) + 25 = 105 + 25 = 125. Record the 12.

1445

235

1225

The sum is 1225.


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.


Multiplication

Multiplication table for the given base is extremely helpful.

Base 5 Multiplication Table

312213404

221411303

13114202

443211

000000

43210x


Example: Using the Base 5 Multiplication TableMultiply 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.


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.


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


4.5

Early Computation Methods


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.


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.


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

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


Duplation and Mediation continued

Now, add the remaining numbers in the right-hand column, obtaining 22 + 352 = 374.

To check 17 22 = 374


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.


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 2-

digit number on the right of the boxes. Place a diagonal in each box. Complete by multiplying.


Example: The Galley Method continued

Add the numbers along the diagonals. The number is read down the left-handed column and along the bottom, as shown by the arrow.

The answer is 27,690.

2

4 2 6

5

62

4

0

3

0

10

2

1

6

32

7

096


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.


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.

Thus, 6 284 = 1704.

1

2 48

246 1

7

2

4048

Slide 4-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

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