Transcript Mathematical Ideas

Chapter 4
Numeration
and
Mathematical
Systems
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 4: Numeration and
Mathematical Systems
4.1
4.2
4.3
4.4
4.5
4.6
Historical Numeration Systems
Arithmetic in the Hindu-Arabic System
Conversion Between Number Bases
Clock Arithmetic and Modular Systems
Properties of Mathematical Systems
Groups
4-2-2
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Chapter 1
Section 4-2
Arithmetic in the Hindu-Arabic System
4-2-3
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Arithmetic in the Hindu-Arabic
System
• Expanded Form
• Historical Calculation Devices
4-2-4
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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.
4-2-5
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Example: Expanded Form
Write the number 23,671 in expanded form.
Solution
2 10  3 10  6 10  7 10  110
4
3
2
1
0
4-2-6
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Distributive Property
For all real numbers a, b, and c,
b  a  c  a  b  c   a.
For example,

 

3 104  2 104   3  2  104
 5 10 .
4
4-2-7
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Example: Expanded Form
Use expanded notation to add 34 and 45.
Solution
  

 45   4 10    5 10 
 7 10   9 10   79
34  3 10  4 10
1
0
1
0
1
0
4-2-8
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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.
4-2-9
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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.
4-2-10
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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.
4-2-11
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Example: Abacus
Which number is shown below?
Solution
1000 + (500 + 200) + 0 + (5 + 1) = 1706
4-2-12
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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.
4-2-13
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Example: Lattice Method
Find the product 38  794 by the lattice
method.
Solution
7
9
Set up the grid
to the right.
4
3
8
4-2-14
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Example: Lattice Method
Fill in products
7
9
2
4
2
1
5
1
7
7
6
2
3
2
2
3
8
4-2-15
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Example: Lattice Method
Add diagonally right to left and carry as
necessary to the next diagonal.
1
2
2
3
2
1
5
7
7
6
0
1
1
2
3
2
7
2
2
4-2-16
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Example: Lattice Method
1
2
3
2
2
1
5
7
7
6
0
1
1
2
3
2
7
2
2
Answer: 30,172
4-2-17
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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.
4-2-18
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Napier’s Rods
Insert figure 2 on page 174
4-2-19
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Russian Peasant Method
Method of multiplication which works by
expanding one of the numbers to be multiplied
in base two.
4-2-20
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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.
4-2-21
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Example: Nines Complement Method
Use the nines complement method to subtract
2803 – 647.
Solution
Step 1
Step 2 Step 3 Step 4
2803
2803 2803 2155
 647  0647 +9352
1
12,155 2156
4-2-22
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