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Question: 1416 only...

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MODERN ALGEBRA NOTES SPRING 2010 OHN C.MAYER 1. CLOCK ARİTİMrtC In this section, we exanine the operatices of addition and mnltiplication in a bew atting clock arithanetic In the process, we will review the terns coemutative, tributive、identity e lennett, additive imre, multiplientiw, mverse. with refrmre to tIsr operations of addition +) and multiplication ). We may abo consider subtraction (-) and division +) when those sre well-defined on this set of munbers. (We will revisit thse clock nmbers more forrmally in Section & MODCLAR ARITHMTc.) To begin, we hove to define the zaumbens of clock aritlhnsctie and the operatins on t Consider the sis hour clock in Figure 1, wlich we call o. 5 4 2 3 FIGURE i. The 6-hour clock C, Definition 1.1 (Addition and Multiplication on a Clock). We define the operation of sddition on the umbers of Cli as follows: a+bgo from a on the clock b units clockwise to the rosult. We ddine the operation of mult-pliation on cas trpeated adatíma t- go from 0 on the clock b units clockwise, u times, one after the other, to the resalt Debe: Decesber 3, 2018 Gencral Page 6

J. G. MAYE For eamiple, 3+5-2 and 35+5+3 in 6-hour clock arithmetic Problem 1.2. Construet addition und maltiplicadion tables for 6-hoor clock arithmctic These ill i by taides) Problem 1.3. Find dditne and uultiplicutive identity eiements, if any, for 6-hour clock arithmetic. Are suck elements unique? C Problem 1 atiom, and distribulivily of multiplication oser addition, hold for 6-bhour clock arithmetic I.4. Determéne sbether or not comauuaiy and associatinity for each oper Problem 1.5. Determine which elements, if omp, heve additihe inverses, and swhat thes are, for 6 bour plock arithmetic Ars shucb elements ssique? Problem 1.6. Detersuine swhich clements, if any, have multiplicutce iwverses, and sha thep ure for 6-hou clock arithmete. Are such elements asique? O Problem 1.7. Iuvestigute fo sohst ertent gou can find solatione to linour equations of the Jorn az + er d uere ar is the unkmom (in C6) and a, b, c,d are knosm Ci nembers Note that +จาะ 0, az-0, end ar-l are important special cases. Are solutions, if any, unique? D Problem 1.8. Ertend the above studp to C5, the S-hour clock Problem 1.9. Estend the adove study to C12, the 18-hor clock Problem 1.10. For udat clocks do you think unique solutiona criat for every lincur oquation You have completed this section if you have a justification (proof, explained solution) for all items 1.2-1.1n. E-mail address, John C. Maper: sagerdeath.uas.eds A AL 35294-1170

1.4-1.6 only

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