082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.2 Numbers and intervals 55
in order to achieve a simple, easy-to-use model. A judgement is made as to when the
right blend of accuracy and conciseness is achieved. For example, the most common
mathematicalmodelforaresistorusesOhm’slawwhichstatesthatthevoltageacrossa
resistor equals the current through the resistor multiplied by the resistance value of the
resistor,thatisV =IR.However,thismodelisbasedonanumberofsimplifications.It
ignores any variation in current density across the cross-section of the resistor and assumes
a singlecurrent value isacceptable. Italsoignores the factthatifalargeenough
voltageisplacedacrosstheresistorthentheresistorwillbreakdown.Inmostcasesitis
worthaccepting these simplifications inorder toobtain a concise model.
Havingobtainedamathematicalmodel,itisthenusedtopredicttheeffectofchanging
elements or conditions within the actual system. Using the model to examine these
effects isoften cheaper, safer and more convenient than usingthe actual system.
2.2 NUMBERSANDINTERVALS
Numberscanbegroupedintovariousclasses,orsets.Theintegersarethesetofnumbers
{...,−3,−2,−1,0,1,2,3,...}
denotedby Z.Thenaturalnumbersare {0,1,2,3,...}andthissetisdenotedby N.The
positive integers, denoted by N + , are given by {1,2,3,...}. Note that some numbers
occur inmore than one set,thatisthe setsoverlap.
A rational number has the form p/q, where p andqare integers withq ≠ 0. For
example,5/2,7/118, −1/9and3/1areallrationalnumbers.Thesetofrationalnumbers
isdenotedby Q.Whenrationalnumbersareexpressedasadecimalfractiontheyeither
terminate orrecur infinitely.
}
5
can be expressed as2.5 These decimal fractions terminate,
2
1
can be expressed as0.125 thatisthey are of finitelength.
8
}
1
can be expressed as0.111111... These areinfinitely
9
1
can be expressed as 0.090909... recurring decimal fractions.
11
A number which cannot be expressed in the form p/q is called irrational. When
writtenasadecimalfraction,anirrationalnumberisinfiniteinlengthandnon-recurring.
The numbers π and √ 2 areboth irrational.
It is useful to introduce the factorial notation. We write 3! to represent the product
3 ×2×1. The expression 3! is read as ‘factorial 3’. Similarly 4! is a shorthand way of
writing4 ×3×2×1.Ingeneral, forany positive integer,n, we can write
n! =n(n −1)(n −2)(n −3)...(3)(2)(1)
Itisusefultorepresentnumbersbypointsontherealline.Figure2.1illustratessome
rational and irrational numbers marked on the real line. Numbers which can be represented
by points on the real line are known as real numbers. The set of real numbers
is denoted by R. This set comprises all the rational and all the irrational numbers. In
Chapter9weshallmeetcomplexnumberswhichcannotberepresentedaspointsonthe
–2 –– 3
5
2
–1 0 1 √ 2 2 2
– 3 p 4
Figure2.1
Both rationalandirrational numbers are representedonthe real line.