MMW_UNIT-10 (2).pdf mathematics modern world

MATHEMATICS

1. MMW
SECTION VIII. MATHEMATICAL SYSTEMS
PREPARED BY:
ENGR. MELOWIN M. PEÑA, ECT, MSIT

2. MODULAR ARITHMETIC
❑Overview:Modulararithmetic,oftenreferredtoas"clockarithmetic,"isa
systemofarithmeticforintegers,wherenumbers"wraparound"upon
reachingacertainvalue,knownasthemodulus.Inthissystem,thenumbers
areconsideredequivalentiftheydifferbyamultipleofthemodulus.

3. MODULAR ARITHMETIC
•Example:Considera12-hourclock.After12hours,theclockresetsto1.If
it's10o'clocknow,in5hours,itwillbe3o'clock.Mathematically,thiscan
beexpressedas:10+5≡3 (mod 12)
•KeyConcepts:
•CongruenceRelation:Twointegersaandbarecongruentmodulonif
theirdifferencea−bisdivisiblebyn,denotedasa≡b (mod n)
•ModularAdditionandMultiplication:Operationscanbeperformed
underamodulus.Forexample,(7+8)mod  10=5and(7×8)mod  10=6

4. MODULAR ARITHMETIC
•Applications:Modulararithmeticisfoundationalinvariousareasof
mathematicsanditsapplications,including:
•Cryptography:AlgorithmslikeRSAencryptionrelyheavilyonmodular
arithmeticforsecuringdigitalcommunication.
•ComputerScience:Hashfunctions,cyclicredundancychecks(CRCs),
andmanyalgorithmsusemodulararithmetictomanageoverflowand
optimizeoperations.
•NumberTheory:Solvingcongruencesandunderstandingpropertiesof
integersthroughmodularsystemsisakeyareaofresearch.

5. MODULAR ARITHMETIC
•ApplicationsofModularArithmetic
•Cryptography:Oneofthemostcriticalapplicationsofmodulararithmeticis
incryptography,particularlyinpublic-keysystemslikeRSA.Thesecurityof
RSAdependsonthedifficultyoffactoringlargenumbers,whicharederived
frommodulararithmetic.Theencryptionanddecryptionprocessuses
modularexponentiationtoencodeanddecodemessages.
•ComputerAlgorithms:Incomputerscience,modulararithmeticisusedto
managetheoverflowofdatatypeswithlimitedsize,suchasintegersin
programminglanguages.Forexample,modulararithmetichelpsin
implementingefficienthashfunctionswherelargeinputsarereducedtoa
manageablesize(e.g.,withinafixednumberofbuckets).

6. MODULAR ARITHMETIC
•ApplicationsofModularArithmetic
•CalendarCalculations:Determiningthedayoftheweekforagivendate
canbedoneusingmodulararithmetic.Forinstance,Zeller'sCongruenceisa
famousformulathatcalculatesthedayoftheweekforanygivendateusing
modulo7.
•ErrorDetection:Indatatransmission,errorscanbedetectedusing
checksumsthatarebasedonmodulararithmetic.Forexample,CRC(Cyclic
RedundancyCheck)isatechniqueusedtodetectaccidentalchangestoraw
dataindigitalnetworksandstoragedevices.

7. GROUP THEORY
•Overview:GroupTheoryisabranchofmathematicsthatstudiesalgebraic
structuresknownasgroups.Agroupisasetofelementsequippedwithan
operationthatcombinesanytwoelementstoformathirdelement,andthis
operationsatisfiesfourconditions:closure,associativity,thepresenceofan
identityelement,andthepresenceofinverseelements.
•Example:Thesetofintegersunderaddition(Z,+)formsagroup.Here,the
operationisaddition,theidentityelementis0,andeachintegerhasan
inverse(e.g.,theinverseofais−a).

8. GROUP THEORY
•KeyConcepts:
•Closure:Foranytwoelementsaandbinthegroup,theresultofthe
operationa∗bmustalsobeinthegroup.
•Associativity:Foranythreeelementsa,b,andc,(a∗b)∗c=a∗(b∗c).
•IdentityElement:Thereexistsanelementeeeinthegroupsuchthat
a∗e=e∗a=aforallainthegroup.
•InverseElement:Foreachelementa,thereexistsanelementbsuchthat
a∗b=b∗a=e.

9. GROUP THEORY
•Applications:Grouptheoryhasprofoundimplicationsinvariousfields:
•Physics:Inquantummechanicsandrelativity,symmetrygroupsdescribethe
symmetriesofphysicalsystems,leadingtoconservationlawsandfundamental
insightsintothenatureofparticlesandforces.
•Chemistry:Thesymmetriesofmoleculesaredescribedbygrouptheory,
whichhelpsinunderstandingmolecularvibrationsandthepropertiesof
crystals.
•Cryptography:Manycryptographicsystemsarebuiltuponthealgebraic
structuresdescribedbygrouptheory,particularlyintheconstructionofelliptic
curvecryptography.
•Mathematics:Grouptheoryprovidesaframeworkforstudyingpolynomial
equations,geometrictransformations,andevennumbertheory.

10. GROUP THEORY
•DiscussionPoints:
1.ConnectionsBetweenModularArithmeticandGroupTheory:Howdoes
modulararithmeticformagroupunderadditionormultiplicationmodulon?
Discusstheconditionsunderwhichthisformsagroupandtheimplicationsfor
numbertheoryandcryptography.
2.Real-WorldApplications:Explorehowmodulararithmeticandgrouptheory
playrolesineverydaytechnology,suchassecurecommunication,dataintegrity,
andeveninunderstandingnaturalphenomenalikecrystalstructures.
3.AdvancedTopics:Delveintohowgrouptheoryextendstootheralgebraic
structureslikeringsandfields,andtherelevanceofthesestructuresinmore
advancedmathematicalandphysicaltheories.

11. •Conclusion:Modulararithmeticandgrouptheoryarepowerfulmathematical
toolswithextensiveapplicationsacrossscience,technology,andeverydaylife.
Understandingtheseconceptsopensdoorstodeeperinsightsintoboththeoretical
andappliedmathematics,withfar-reachingconsequencesinfieldsrangingfrom
cryptographytoquantumphysics.

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