Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 4

Numeration and Mathematical Systems

© 2008 Pearson Addison-Wesley.All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved

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Chapter 4: Numeration and Mathematical Systems

4.1 Historical Numeration Systems

4.2 Arithmetic in the Hindu-Arabic System

4.3 Conversion Between Number Bases

4.4 Clock Arithmetic and Modular Systems

4.5 Properties of Mathematical Systems

4.6 Groups


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Chapter 1

Section 4-2Arithmetic in the Hindu-Arabic System


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Arithmetic in the Hindu-Arabic System

• Expanded Form

• Historical Calculation Devices


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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.


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Example: Expanded Form

Write the number 23,671 in expanded form.

Solution4 3 2 1 02 10 3 10 6 10 7 10 1 10


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Distributive Property

For all real numbers a, b, and c,

For example,

.b a c a b c a

4 4 4

4

3 10 2 10 3 2 10

5 10 .


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Example: Expanded Form

Use expanded notation to add 34 and 45.

1 0

1 0

1 0

34 3 10 4 10

45 4 10 5 10

7 10 9 10 79

Solution


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Decimal System

Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.


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Historical Calculation Devices

One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide.


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Abacus

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.


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Example: Abacus

Which number is shown below?

Solution1000 + (500 + 200) + 0 + (5 + 1) = 1706


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Lattice Method

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.

The method is shown in the next example.


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Example: Lattice Method

Find the product by the lattice method.

38 794

7 9 4

3

8

Solution

Set up the grid to the right.


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Fill in products

2

1

2

7

1

2

5

6

7

2

3

2

7 9 4

3

8


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Add diagonally right to left and carry as necessary to the next diagonal.

2

1

2

7

1

2

5

6

7

2

3

2

1 7 2

0

21

3


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Answer: 30,172

2

1

2

7

1

2

5

6

7

2

3

2

1 7 2

0

21

3


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Napier’s Rods (Napier’s Bones)

John Napier’s invention, based on the lattice method of multiplication, is often acknowledged as an early forerunner to modern computers.

The rods are pictured on the next slide.


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Napier’s Rods

Insert figure 2 on page 174


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Russian Peasant Method

Method of multiplication which works by expanding one of the numbers to be multiplied in base two.


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Nines Complement Method

Step 1 Align the digits as in the standard subtraction algorithm.

Step 2 Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits.

Step 3 Replace each digit in the subtrahend with its nines complement, and then add.

Step 4 Delete the leading (1) and add 1 to the remaining part of the sum.


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Example: Nines Complement Method

Use the nines complement method to subtract 2803 – 647.

Solution

2803 2803 2803 2155

647 0647 +9352 1

12,155 2156

Step 1 Step 2 Step 3 Step 4

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Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved