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Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 1
Chapter 4
Systems of Numeration
Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 2
WHAT YOU WILL LEARN
• Additive, multiplicative, and ciphered
systems of numeration
• Place-value systems of numeration
• Egyptian, Hindu-Arabic, Roman,
Chinese, Ionic Greek, Babylonian,
and Mayan numerals
• Converting base 10 numerals to
numerals in other bases
Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 3
WHAT YOU WILL LEARN
• Converting numerals in other bases to
base 10 numerals
• Performing addition, subtraction,
multiplication and division in other
bases
• Other computational methods such as
duplation and mediation, the lattice
method and Napier’s rods
Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 4
Section 4
Computation in Other Bases
Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 5
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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 6
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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 7
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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 8
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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 9
Multiplication


Multiplication table for the given base is extremely
helpful.
Base 5 Multiplication Table

0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
11
13
3
0
3
11
14
22
4
0
4
13
22
31
Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 10
Example: Using the Base 5
Multiplication Table
Multiply
125
 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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 11
Example: Using the Base 5
Multiplication Table (continued)

Multiply 35 times the number in the first column:
35 ´ 25 = 115
Record the 1 carry the 1.
112
5
 35
1
 Multiply 35 times the number in the second column and
add the carry:
(35  15) + 15 = 45 Record the 4.
125
 35
415
The product is 415.

Copyright © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 12
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 © 2009 Pearson Education, Inc.
Chapter 4 Section 4 - Slide 13