Download Part 4

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
4-4-2
© 2008 Pearson Addison-Wesley. All rights reserved
Chapter 1
Section 4-4
Clock Arithmetic and Modular
Systems
4-4-3
© 2008 Pearson Addison-Wesley. All rights reserved
Clock Arithmetic and Modular
Systems
• Finite Systems and Clock Arithmetic
• Modular Systems
4-4-4
© 2008 Pearson Addison-Wesley. All rights reserved
Finite Systems
Because the whole numbers are infinite,
numeration systems based on them are
infinite mathematical systems. Finite
mathematical systems are based on finite
sets.
4-4-5
© 2008 Pearson Addison-Wesley. All rights reserved
12-Hour Clock System
The 12-hour clock system is based on an
ordinary clock face, except that 12 is
replaced by 0 so that the finite set of the
system is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
4-4-6
© 2008 Pearson Addison-Wesley. All rights reserved
Clock Arithmetic
As an operation for this clock system, addition
is defined as follows: add by moving the hour
hand in the clockwise direction.
11 0 1
10
2
9
8
3
4
7 6
5
5+3=8
4-4-7
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Finding Clock Sums by
Hand Rotation
Find the sum: 8 + 7 in 12-hour clock
arithmetic
Solution
Start at 8 and move
the hand clockwise
through 7 more hours.
Answer: 3
11 0 1
10
2
9
8
3
4
7 6
5
4-4-8
© 2008 Pearson Addison-Wesley. All rights reserved
12-Hour Clock Addition Table
4-4-9
© 2008 Pearson Addison-Wesley. All rights reserved
12-Hour Clock Addition Properties
Closure The set is closed under addition.
Commutative For elements a and b, a + b = b + a.
Associative For elements a, b, and c,
a + (b + c) = (a + b) + c.
Identity The number 0 is the identity element.
Inverse Every element has an additive inverse.
4-4-10
© 2008 Pearson Addison-Wesley. All rights reserved
Inverses for 12-Hour Clock Addition
Clock
value a
0
Additive
Inverse -a
0 11 10 9 8 7 6 5 4 3
1
2
3 4 5 6 7 8 9 10 11
2
1
4-4-11
© 2008 Pearson Addison-Wesley. All rights reserved
Subtraction on a Clock
If a and b are elements in clock arithmetic,
then the difference, a – b, is defined as
a – b = a + (–b)
4-4-12
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Finding Clock Differences
Find the difference 5 – 9.
Solution
5–9
= 5 + (–9)
=5+3
=8
Definition of subtraction
Additive inverse of 9
4-4-13
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Finding Clock Products
Find the product 4  5.
Solution
45  5  5  5  5
=8
4-4-14
© 2008 Pearson Addison-Wesley. All rights reserved
Modular Systems
In this area the ideas of clock arithmetic are
expanded to modular systems in general.
4-4-15
© 2008 Pearson Addison-Wesley. All rights reserved
Congruent Modulo m
The integers a and b are congruent modulo
m (where m is a natural number greater than
1 called the modulus) if and only if the
difference a – b is divisible by m.
Symbolically, this congruence is written
a  b (mod m).
4-4-16
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Truth of Modular Equations
Decide whether each statement is true or false.
a) 12  4 (mod 2)
b) 35  4 (mod 7)
c) 11  44 (mod 3)
Solution
a) True. 12 – 4 = 8 is divisible by 2.
b) False. 35 – 4 = 31 is not divisible by 7.
c) True. 11 – 44 = –33 is divisible by 3.
4-4-17
© 2008 Pearson Addison-Wesley. All rights reserved
Criterion for Congruence
a  b (mod m) if and only if the same
remainder is obtained when a and b are
divided by m.
4-4-18
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Solving Modular Equations
Solve the modular equation below for the whole
number solutions.
2  x  5(mod 8)
Solution
Because dividing 5 by 8 has remainder 5, the equation
is true only when 2 + x divided by 8 has remainder 5.
After trying the values 0, 1, 2, 3, 4, 5, 6, 7 we find that
only 3 works. Other solutions can be found by
repeatedly adding 8: {3, 11, 19, 27, …}.
4-4-19
© 2008 Pearson Addison-Wesley. All rights reserved