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

© 2008 Pearson Addison-Wesley. All rights reserved 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 4.6Groups

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 4-4 Clock Arithmetic and Modular Systems

© 2008 Pearson Addison-Wesley. All rights reserved Clock Arithmetic and Modular Systems Finite Systems and Clock Arithmetic Modular Systems

© 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.

© 2008 Pearson Addison-Wesley. All rights reserved 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}.

© 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 = 8

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Clock Sums by Hand Rotation Find the sum: in 12-hour clock arithmetic Solution Start at 8 and move the hand clockwise through 7 more hours. Answer: 3 Plus 7 hours

© 2008 Pearson Addison-Wesley. All rights reserved Hour Clock Addition Table

© 2008 Pearson Addison-Wesley. All rights reserved 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.

© 2008 Pearson Addison-Wesley. All rights reserved Inverses for 12-Hour Clock Addition Clock value a Additive Inverse -a

© 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)

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Clock Differences Find the difference 5 – 9. Solution 5 – 9 = 5 + (–9)Definition of subtraction = Additive inverse of 9 = 8

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Clock Products Find the product Solution = 8

© 2008 Pearson Addison-Wesley. All rights reserved Modular Systems In this area the ideas of clock arithmetic are expanded to modular systems in general.

© 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

© 2008 Pearson Addison-Wesley. All rights reserved Example: Truth of Modular Equations Decide whether each statement is true or false. 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.

© 2008 Pearson Addison-Wesley. All rights reserved Criterion for Congruence if and only if the same remainder is obtained when a and b are divided by m.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Solving Modular Equations Solve the modular equation below for the whole number solutions. 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, …}.