Module-1 Mathematics02-11-2013 Final.doc

8/17/2019 Module-1 Mathematics02-11-2013 Final.doc

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CONTENTS

CHAPTER 1..................................................................................................................1

ARITHMETIC............................................................................................................1

Introduction............................................................................................................1

Arithmetic Terms....................................................................................................1

Directed Numbers.................................................................................................2

Factors...................................................................................................................3

Prime Numbers......................................................................................................4

i!hest Common Factor "c#$...............................................................................4

%o&est Common 'u(ti)(e "%C'$...........................................................................*

Arithmetica( Precedence.......................................................................................*

+odmas E,am)(e..................................................................................................-

Fractions................................................................................................................-

Addition # Fractions............................................................................................./

Subtraction # Fractions........................................................................................0

'u(ti)(ication # Fractions.....................................................................................

Diision # Fractions.............................................................................................

Decima( Fractions................................................................................................1 Addition Subtraction.........................................................................................11

'u(ti)(ication Diision.......................................................................................11

5ei!hts And 'easures.......................................................................................12

Ratio And Pro)ortion...........................................................................................13

Aera!es And Percenta!es.................................................................................14

Aera!es.............................................................................................................14

Percenta!e..........................................................................................................1*

Area And 6o(ume.................................................................................................1-

Area.....................................................................................................................1-

Rectan!u(ar Area.................................................................................................1-

Area o# Trian!(es.................................................................................................1-

Area o# Circu(ar Sha)es......................................................................................1/

Area o# ther Sha)es..........................................................................................10

Ca(cu(ation o# Areas o# Sha)es...........................................................................10

6o(umes...............................................................................................................1

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Po&ers and Roots...............................................................................................21

Po&ers.................................................................................................................21

Roots...................................................................................................................21

CHAPTER 2................................................................................................................23

ALGEBRA..............................................................................................................23

Introduction..........................................................................................................23

)eration.............................................................................................................23

+asic %a&s..........................................................................................................24

E7uations.............................................................................................................2*

So(in! %inear E7uations.....................................................................................2*

E7uations Re7uirin! 'u(ti)(ication or Diision....................................................2-

E7uations Re7uirin! Addition or Subtraction......................................................2-

E7uations Containin! 8n9no&ns on both Sides.................................................2/

E7uations Containin! +rac9ets...........................................................................2/

E7uations Containin! Fractions..........................................................................20

Trans)osition In E7uations..................................................................................20

Construction # E7uations..................................................................................3

Simu(taneous E7uations......................................................................................32

:uadratic E7uations............................................................................................34

N8'+ERS...........................................................................................................3*

Indices and Po&ers.............................................................................................3*

Standard Form.....................................................................................................3/

Numberin! S;stems............................................................................................30

Decima( S;stem # Numeration..........................................................................30

+inar; S;stem # Numeration.............................................................................3

cta( S;stem # Numeration...............................................................................4

Conersion To ther +ases................................................................................41

%o!arithms...........................................................................................................43

CHAPTER 3................................................................................................................45

GEOMETRY............................................................................................................45

An!u(ar 'easurement.........................................................................................4*

An!(es Associated 5ith Para((e( %ines................................................................4-


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Trian!(e................................................................................................................4-

Trian!(e T;)es.....................................................................................................4/

Simi(ar Con!ruent Trian!(es............................................................................4/

Po(;!on................................................................................................................40

:uadri(atera(s......................................................................................................40

Para((e(o!ram......................................................................................................4

Rectan!(e............................................................................................................4

Rhombus.............................................................................................................4

S7uare.................................................................................................................4

Tra)e=ium............................................................................................................*

Circ(es..................................................................................................................*

Radian 'easure..................................................................................................*1


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CHAPTER 1

ARITHMETIC

Introduction

1. 'athematics is the basic (an!ua!e o# science and techno(o!;. It is an e,act(an!ua!e that has a ocabu(ar; and meanin! #or eer; term. Since mathematics#o((o&s de#inite ru(es and behaes in the same &a; eer; time? scientists anden!ineers use it as their basic too(.

2. %on! be#ore an; meta( is cut #or a ne& aircra#t desi!n? there are (itera((;mi((ions o# mathematica( com)utations made. Aiation maintenance technicians)er#orm their duties &ith the aid o# man; di##erent too(s. %i9e the &rench or scre&drier? mathematics is an essentia( too( in the maintenance? re)air and#abrication o# re)(acement )arts. 5ith this in mind? ;ou can see &h; ;ou must becom)etent in mathematics to an acce)tab(e (ee(. These notes coer the com)(etemathematics s;((abus re7uired to com)(; &ith the @AR>-- +1 and +2 (icence (ee(.

3. Arithmetic is the basic (an!ua!e o# a(( mathematics and uses rea(? non>ne!atie numbers. These are sometimes 9no&n as countin! numbers. n(; #our o)erations are used? addition? subtraction? mu(ti)(ication and diision. 5hi(st theseo)erations are &e(( 9no&n to ;ou? a reie& o# the terms and o)erations used &i((ma9e (earnin! the more di##icu(t mathematica( conce)ts easier.

Arit!"tic T"r!#

4. The most common s;stem o# numbers in use is the d"ci!$% s;stem? &hichuses the ten di!its ? 1? 2? 3? 4? *? -? /? 0? .

*. These ten &ho(e numbers #rom =ero to are ca((ed int"&"r#. Aboe thenumber nine? the di!its are re>used in arious combinations to re)resent (ar!er numbers. This is accom)(ished b; arran!in! the numbers in co(umns based on amu(ti)(e o# ten. 5ith the addition o# a minus ">$ si!n? numbers sma((er than =ero areindicated.

-. To describe 7uantities that #a(( bet&een &ho(e numbers? #ractions are used.

Co!!on 'r$ction# are used &hen the s)ace bet&een t&o inte!ers is diided intoe7ua( se!ments? such as 7uarters. 5hen the s)ace bet&een inte!ers is diided intoten se!ments? d"ci!$% 'r$ction# are t;)ica((; used.

/. Students &i(( be #ami(iar &ith this s;stem and the basic o)erations? &hich ma;ino(e Addition? Subtraction? 'u(ti)(ication and Diision.

5hen numbers are added? the; #orm a #u!.

5hen numbers are subtracted? the; create a di''"r"nc".

5hen numbers are mu(ti)(ied? the; #orm a (roduct.

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5hen one number "the di)id"nd$ is diided b; another "the di)i#or $? the resu(t is a*uoti"nt.

0. It is use#u( i# a student is )ro#icient at sim)(e menta( arithmetic? and this is on(;)ossib(e i# one has a #ee(B #or numbers? and the si=e o# numbers. A 9no&(ed!e o#

sim)(e times tab(esB is a(so use#u(.

TIME TABLE

1 2 3 4 5 + , - 1/

1 1 2 3 4 * - / 0 1

2 2 4 - 0 1 12 14 1* 10 2

3 3 - 12 1* 10 21 24 2/ 3

4 4 0 12 1- 2 24 20 32 3- 4

* * 1 1* 2 2* 3 3* 4 4* *

- - 12 10 24 3 3- 42 40 *4 -

/ / 14 21 20 3* 42 4 *- -3 /

0 0 1- 24 32 4 40 *- -4 /2 0

10 2/ 3- 4* *4 -3 /2 01

1 1 2 3 4 * - / 0 1

. The #o((o&in! sim)(e tests #or diisibi(it; ma; be use#u(. A number is diisib(eb;

"a$ 2 i# it is an een number.

"b$ 3 i# the sum o# the di!its that #orm the number is diisib(e b; 3.

"c$ 4 i# the (ast t&o di!its are diisib(e b; 4.

"d$ * I# the (ast di!it is or *.

"e$ 1 i# the (ast di!it is

0ir"ct"d Nu!"r#

1. Directed numbers are numbers &hich hae a or si!n attached to them.Directed numbers can be added? subtracted? etc. etc? but care shou(d be ta9en toensure a correct so(ution. The #o((o&in! ru(es shou(d assist.

11. To $dd seera( numbers o# the #$!" si!n? add them to!ether and ensure si!no# the sum is the same as the si!n o# the numbers.

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12. To $dd 2 numbers &ith di''"r"nt si!ns? #utr$ct the sma((er #rom the (ar!er.The si!n o# the resu(tant "the di##erence$ is the same as the si!n o# the (ar!e number.

e!. >12 - "12 > -$ - G >-

13. I# there are more than 2 numbers? carr; out the o)eration 2 numbers at a time?or )roduce t&o numbers b; addin! u) a(( the numbers &ith (i9e si!ns. And then a))(;the ru(es aboe.

e!. >1* > 0 13 > 1 - ">1* > 0$ >23 13 >1 > 1 >2 - 23

or >1* ">0$ ">1$ >42 and 13 - 1

>42 1 23

14. To #utr$ct directed numbers? chan!e the si!n o# the number to be

subtracted and add the resu(tin! numbers.

e!. >1 > ">-$ > 1 - > 4

/ > "10$ / > 0 >11

1*. A minus in #ront o# brac9ets shou(d be ta9en to mean 1. 8sin! the aboee,am)(e ">-$ shou(d be read as 1">-$ i.e. minus 1 times minus si,. Simi(ar(;? a)ositie si!n in #ront o# brac9ets shou(d be read as 1? so ">-$ shou(d be read as1">-$ i.e. )(us 1 times minus -.

1-. The (roduct o# t&o numbers &ith (i9e si!ns is )ositie "e$? the )roduct o# numbers &ith un(i9e si!ns is ne!atie ">e$.

1/. 5hen di)idin& numbers &ith (i9e si!ns? the 7uotient o# the resu(t is e.5hen. diidin! numbers &ith un(i9e si!ns? the 7uotient is e.

10. This can be summarised as #o((o&s

6 7 6 7 6 7 6 7

8$ctor#

1. 5e 9no& that 2 , - 12. 2 and - are '$ctor# o#12. 5e cou(d a(so state that?as 3 , 4 12? 3 and 4 are a(so #actors o# 12. Simi(ar(; 12 and 1.

2. This ma; seem obious? but it is sometimes use#u( to H#actoriseH a number? i.e.determine the #actors that ma9e u) the number. 'ore common(; it is necessar; to#ind the #actors o# an a(!ebraic e,)ression.

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Find the )ossib(e #actors o# -.

"in other &ords? #ind the inte!ers that diide into -$.

The #actors &i(( be

1? 2? 3? 4? *? -? 1? 12? 1*? 2? 3 and -

Chec9 them ;ourse(#.

Pri!" Nu!"r#

21. A )rime number is a number &hose on(; #actors are 1 and itse(#.

22. The )rime numbers bet&een 1 and 3 are

1? 2? 3? *? /? 11? 13? 1/? 1? 23 and 2.

Chec9 them ;ourse(#.

23. It is sometimes use#u( to e,)ress the #actors o# a !ien number in terms o# )rimenumbers.

24. For e,am)(e? (et us (oo9 at the #actors o# - a!ain? ta9in! 4 and 1* as 2#actors."4 , 1* -$? but 4 has #actors o# 2 and 2? and 1* has #actors o# * and 3. ence thenumber - can be e,)ressed as 2 , 2 , 3 , *? &hich are a(( #actors o# -.

Note &e hae no& #actorised the number - in terms o# )rime numbers.

Hi&"#t Co!!on 8$ctor Hc'

2*. The hi!hest common #actor is the bi!!est #actor "number$ that &i(( diide intothe numbers bein! e,amined. Su))ose that &e ta9e 3 numbers? 1/-4? 21 and24. The hi!hest common #actor o# these numbers is 04. In some instances ;ou &i((be ab(e to identi#; this a(ue sim)(; b; (oo9in! at the numbers? in others ;ou &i((need to ca(cu(ate it. To ca(cu(ate the CF? &e must identi#; the #actors o# eachnumber in terms o# )rime numbers

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2-. 5e then se(ect the common )rime #actors and mu(ti)(; them to!ether to)roduce the i!h Common Factor? in this case

2 × 2 × 3 × / = 04

Lo9"#t Co!!on Mu%ti(%" LCM

2/. The (o&est common mu(ti)(e o# a set o# numbers is the sma((est number into&hich each o# the !ien numbers &i(( diide e,act(;. The %C' can be #ound b;mu(ti)(;in! to!ether $%% o# the #actors common to each o# the indiidua( numbers.

20. Consider the )reious three numbers? 1/-4? 21 and 24 and their #actors.

2. The %o&est Common 'u(ti)(e o# these three numbers &i(( be

2 , 2 "in a(($ , 3 , 3 "in 1/-4$ , * , * "in 21$ , / , / "in 1/-4 and 24$

2 × 2 × 3 × 3 × * × * × / × / = 44?1

So 2 , 2 , 3 , 3 , * , * , / , / 44?1 is the %.C.'

1/-4 , 2* 44?1

21 , 21 44?1

24 , 1* 44?1

Arit!"tic$% Pr"c"d"nc"

3. The term Arithmetic Precedence means the order in &hich &e carr; outarithmetic #unctions. Sometimes it doesnt matter &hat order &e carr; them out.

31. Consider the e,)ression 2 3 *. It ma9es no di##erence i# &e &rite 32*. A!ain? consider 3 , 4 12? there is no di##erence i# &e &rite 4 , 3 12.

o&eer? i# I &rite 2 3 , 4? &hat is the ans&erJ

I# &e #irst add 2 3? &e &i(( !et * and then * , 4 2.

A(ternatie(;? mu(ti)(;in! 3 , 4 12? addin! 2 &e !et 14.

32. I# &e are !oin! to a!ree on the ans&er &e must #irst a!ree on the ru(es &e

use. This introduces the to)ic 9no&n as arithmetica( )recedence? and is most easi(;

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remember b; the term BO0MAS. +D'AS indicates the )recedence? or the order in &hich &e )er#orm our ca(cu(ations

B stands #or +rac9ets

O stands #or H#H

0 stands #or Diision

M stands #or 'u(ti)(ication

A stands #or Addition

S stands #or Subtraction

Bod!$# E6$!(%"

Find the a(ue o# -4 K ">1-$ ">/ >12$ > ">2 3-$">2 $

This e,)ression becomes

-4 K ">1-$ ">1$ > "/$"/$ +

">4$ ">1$ > "/$"/$ D

">4$ ">1$ > 4 '

> 23 > 4 A

> /2 S

8r$ction#

is an e,am)(e o# a Pro("r 8r$ction? !enera((; abbreiated to #raction.

It has the same meanin! as 11 K 1-? that is? 11 diided b; 1-.

It has the same meanin! as 11 K 1-? that is? 11 diided b; 1-.

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Addition o' 8r$ction#

33. The im)ortant thin! to remember here is that on(; #ractions &ith the same"a common$ denominator can be added or subtracted.

34. I# the denominators are not the same? then it is necessar; to #ind the %o9"#tCo!!on 0"no!in$tor LC0 and to )ut each #raction in terms o# this a(ue.

Findin! the %o&est Common Denominator is essentia((; the same as #indin! the%o&est Common 'u(ti)(e? &hich &as coered in a )reious to)ic.

3*. In this e,am)(e? the %CD o# 1-? 12 and 0 is 40. In some cases it ma; be7uic9er to #ind a common denominator b; sim)(; mu(ti)(;in! the denominatorsto!ether i.e. 1- , 12 , 0 1*3-. Note? this is not the %CD.

3-. ain! #ound the %CD? each #raction no& needs to be e,)ressed in terms o# the %CD. This is achieed b; diidin! the %CD b; the denominator and mu(ti)(;in!the resu(t b; the numerator.

A(ternatie(;? diide the %CD b; the denominator 40 K 1- 3

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And mu(ti)(; to) and bottom o# the #raction b; the resu(t

Sutr$ction o' 8r$ction#

3/. The basic )rocedure is er; simi(ar to that used #or additionL #ind the %C'?conert the indiidua( #ractions? but subtract the numerators instead o# addin!. Therema; be one di##erence &hich is im)ortant.

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E6$!(%" 4

Mu%ti(%ic$tion o' 8r$ction#

30. These ca(cu(ations are !enera((; easier to )er#orm than addition andsubtraction.

E6$!(%" 1

0i)i#ion o' 8r$ction#

3. To diide t&o #ractions &e inert the di)i#or "the number &e are diidin! b;$and mu(ti)(;.

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First(;? conert into im)ro)er #ractions. Then inert the second #raction and mu(ti)(;.

Not". Eer; o))ortunit; shou(d be ta9en to sim)(i#; b; Hcance((in!H numbers aboeand be(o& the (ine &hereer )ossib(e.

"a / aboe and be(o& the (ine cance(s? as does an 0$.

0"ci!$% 8r$ction#

4. Decima( #ractions are #ractions &here the Denominator is e7ua( to some)o&er o# 1? i.e. 1? 1? 1 etc.

For e,am)(e? 12 * is a decima( #raction.

Decima( #ractions are usua((; re>&ritten as decima(s. This is er; easi(; done b;

usin! a 0"ci!$% Point. Ta9e the e,am)(e

41. P(ace a decima( )oint to the ri!ht o# the numerator "to) number$. Then moethe decima( )oint to the (e#t? b; a number o# )(aces e7ua( to the number o# Hnou!htsHin the denominator "bottom number$. Remoe one nou!ht #rom the denominator #or each moe.

An: 'r$ction c$n " 'or!"d into $ d"ci!$%; : di)idin& t" nu!"r$tor : t"d"no!in$tor

For e,am)(e becomes .0/*. Found b; a )rocess o# (on! diision .

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Addition < Sutr$ction

42. The main thin! to remember &hen addin! or subtractin! decima( numbers isto ensure the; are correct(; (ined u) usin! the decima( )oint as a re#erence.

E6$!(%" 1

Mu%ti(%ic$tion < 0i)i#ion

43. 'u(ti)(ication o# Decima(s is the same as ordinar; H(on!H mu(ti)(ication? but thenumber o# decima( )(aces in the ans&er must e7ua( the sum o# decima( )(aces in thenumbers bein! mu(ti)(ied.

E6$!(%" 1

-.24 , 3.121

44. There are t&o di!its a#ter the decima( )(ace in the #irst number and 3 in the

second. There#ore? there must be * di!its a#ter the decima( )(ace in the ans&er? sothe ans&er becomes 1M4/*4.

4*. "Common sense he()s here. A number s(i!ht(; !reater then - is mu(ti)(ied b;another number s(i!ht(; !reater then 3. %o!ica((; the ans&er shou(d bea))ro,imate(; 10$.

4-. Diision is a(so the same as ordinar; (on! diision? but a!ain a sim)(e ru(ehe()s to sim)(i#; the )rocess Do not tr; to diide b; a #raction. 'u(ti)(; both thediisor and diidend b; a )o&er o# ten "moe the decima( )(ace to the ri!ht$ so thatthe diisor becomes a &ho(e number.

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E6$!(%"#

3-* K 4*.*- > 'u(ti)(; both numbers b; 1 "12$ to !ie 3-* K 4**-

/- K ⋅3-4 > 'u(ti)(; both numbers b; 1 "13$ to !ie /- K 3-4

="i&t# $nd M"$#ur"#

4/. A &ide number o# di##erent &ei!hts and measures are used durin! themaintenance o# aircra#t. The ones that come to mind #irst are )robab(; #ue(ca)acities? t;re )ressures? tem)eratures and s)eeds. There are ho&eer er; man;others? &hich ;ou &i(( meet as ;ou )ro!ress throu!h ;our course.

40. First(;? the most common(; used s;stem in aiation toda; is the S:#t"!"Int"rn$tion$%" "SI$. This s;stem is based on mu(ti)(es o# 1 and has been acce)ted&ide(;? &ith one or t&o e,ce)tions. It consists o# a standard set o# units #or %"n&t

!"tr"? !$## >i%o&r$!? ti!" #"cond? t"!("r$tur" ?"%)in? curr"nt $!("r"and %i&t c$nd"%$. There are seera( other units &hich? &hi(st not bein! )art o# thebasic S.I. ones aboe? are in common use and sti(( use the metric s;stem #or ca(cu(ations.

4. An o(der s;stem that is sti(( used in some countries toda;? is the I!("ri$%S:#t"!; &hich uses a mi,ture o# o(d units such as #eet and inches #or (en!th?)ounds #or &ei!ht? !a((ons #or ca)acit; and Fahrenheit #or tem)erature.

*. Oou &i(( occasiona((; meet a mi,ture o# s;stems? &hich &i(( re7uire conersion#rom one to another. A !ood e,am)(e is the amount o# #ue( )ut into an aircra#ts tan9s.Oou &i(( #ind this bein! measured in im)eria( !a((ons? American !a((ons? im)eria()ounds? SI 9i(o!rams or metric (itres.

*1. Chan!in! a 7uantit; in one unit to a 7uantit; in another unit re7uires acon)"r#ion '$ctor . 5hen the 7uantit; in the #irst unit is mu(ti)(ied b; the conersion#actor? the resu(t is the 7uantit; in the second units. For e,am)(e? to conert im)eria(!a((ons to (itres? the; must be mu(ti)(ied b; 4.*4-

E6$!(%" 1

Conert 2* !a((ons into (itres.

2* , 4.*4- 113.-* %itres.

E6$!(%" 2

Conert 1* mi(es into 9i(ometres usin! the conersion #actor 1⋅-4

1* , 1⋅-4 2413⋅ Qi(ometres.

Note Oou &i(( norma((; be !ien the conersion #actor? ho&eer? ;ou ma; hae to

trans)ose a #ormu(a in order to use it.

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R$tio $nd Pro(ortion

*2. This to)ic is an e,tension o# seera( )reious to)ics. Ratio and )ro)ortion areessentia((; statements that (in9 t&o or more H7uantitiesH to!ether. For e,am)(e? a 3to1 mi, o# sand and cement? &hich ma; be &ritten as a 31 mi, o# sand and cement?

means mi, 3 )arts o# sand to 1 )art o# cementH. This is a common(; used statement&hich ;ou &i(( notice has no #orma( units? a(thou!h o(ume is in#erred. Parts cou(d bere)resented b; shoe(s #u((? buc9ets #u((? &hee(barro&s #u(( etc.

*3. A ratio there#ore sim)(; )roides a means o# com)arin! one a(ue &ith

another. For e,am)(e? i# an en!ine turns at 4r)m and the )ro)e((er turns at24r)m? the ratio o# the t&o s)eeds is 4 to 24? or * to 3 &hen reduced to its(o&est terms. This re(ationshi) can a(so be e,)ressed as *3 or *3.

*4. The use o# ratios is common in aiation? such as &hen considerin! thecom)ression ratio in an en!ine. This is the ratio o# c;(inder dis)(acement? &hen the)iston is at the bottom o# its stro9e com)ared &ith the dis)(acement &hen it is at theto). For e,am)(e? i# the o(ume o# the c;(inder at the bottom o# its stro9e is 24 cm2and at the to) becomes 3 cm2 the ratio is 243 or? reduced to its (o&est terms?01.

**. Another t;)ica( ratio is that o# di##erent !ear si=es. For e,am)(e? the ratio o# adrie !ear &ith 1* teeth to a drien !ear &ith 4* teeth is 1*4* or 13 &hen reduced.This means that #or eer; one tooth o# the drie !ear there are three teeth on thedrien !ear. o&eer? &hen &or9in! &ith !ears? the ratio o# teeth is o))osite theratio o# reo(utions. In other &ords? since the drie !ear has one third as man; teethas the drien !ear? the drie !ear must com)(ete three reo(utions to turn the drien!ear once. This resu(ts in a r")o%ution r$tio o# 31? &hich is the o))osite o# the ratioo# teeth.

*-. A )ro)ortion is a statement o# e7ua(it; bet&een t&o or more ratios andre)resents a conenient &a; to so(e )rob(ems ino(in! ratios. For e,am)(e? i# an

en!ine has a reduction !ear ratio bet&een the cran9sha#t and the )ro)e((er o# 32?and the en!ine is turnin! at 2/r)m? &hat is the rotationa( s)eed o# the )ro)e((erJIn this )rob(em (et @( re)resent the un9no&n a(ue? &hich in this case is the s)eedo# the )ro)e((er. Ne,t? set u) a )ro)ortiona( statement usin! the #ractiona( #orm? 32 2/6). To so(e this e7uation? cross mu(ti)(; to arrie at the e7uation 36) 2 ,2/? or *4r)m. To so(e #or 6) diide *4 b; 3. Thus? the )ro)e((er s)eed is10r)m.

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E6$!(%" Diide 24 bet&een 4 men in the ratio o# 11131*.

The norma( )rocedure #or this t;)e o# )rob(em is to

"a$ Add a(( o# the indiidua( )ro)ortions to #ind the tota( number o# )arts.

"b$ Diide the tota( amount b; the number o# )arts to #ind the a(ue o# each)art.

"c$ 'u(ti)(; each ratio b; the a(ue o# each )art.

So 11 13 1* 40

24 diided b; 40 *. There#ore each )art is &orth *.

, * 4*

11 , * **

13 , * -*

1* , * /*

The )ro)ortions are there#ore 4*? **? -* and /*

A use#u( chec9 is to add the indiidua( )arts to!ether? to ensure the tota( is theamount ;ou started &ith.

A)"r$&"# $nd P"rc"nt$&"#

A)"r$&"#

*/. 5hen &or9in! &ith numerica( in#ormation? it is sometimes use#u( to #ind theaera!e a(ue. 5hen estimatin! the time a )articu(ar ourne; &ou(d be no )oint inbasin! the time on the s(o&est s)eed or the hi!hest s)eed? a(&a;s use an aera!es)eed.

*0. 5e &ou(d a(so use aera!e #ue( consum)tion to estimate ho& much #ue( anaircra#t &ou(d use #or a )articu(ar #(i!ht.

*. In both o# these t;)es o# ca(cu(ation? &e can on(; &or9 out the aera!e b;diidin! the tota( distance or #ue( used b; the time.

E6$!(%" 1

An aircra#t trae(s a tota( distance o# /* 9m in a time o# 3 hours 4* minutes. 5hat isthe aera!e s)eed in 9mhrJ

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E6$!(%" 2

An aircra#t uses 3 !a((ons o# #ue( #or a #(i!ht o# duration 4 hours. 5hat is theaera!e #ue( consum)tionJ

5e o#ten need to ca(cu(ate aera!es based on man; data items.

E6$!(%" 3

The &ei!ht o# si, items are as #o((o&s

.*? 1.3? 0.? .4? 11.2? 1.1 5hat is the aera!e &ei!htJ

To ca(cu(ate this &e sim)(; add the tota( &ei!hts and diide b; the number o# items.

P"rc"nt$&"

-. Percenta!es are s)ecia( #ractions &hose denominator is 1. The decima(

#raction .33 is the same as 331 and is e7uia(ent to 33 )ercent or 33U. Oou canconert common #ractions to )ercenta!es b; #irst conertin! them to decima(#ractions and then mu(ti)(;in! b; 1. For e,am)(e? *0 e,)ressed as a decima( is.-2*? and is conerted to a )ercenta!e b; moin! the decima( )oint t&o )(aces tothe ri!ht? the same as mu(ti)(;in! b; 1. This becomes -2.*U.

-1. To #ind the )ercenta!e o# a number? mu(ti)(; the number b; the decima(e7uia(ent o# the )ercenta!e. For e,am)(e? to #ind 1U o# 2? be!in b; conertin!1U to its decima( e7uia(ent? &hich is .1. This is achieed b; diidin! the)ercenta!e #i!ure b; 1. No& mu(ti)(; 2 b; .1 to arrie at the a(ue o# 2.

-2. I# ;ou &ant to #ind the )ercenta!e one number is o# another? ;ou must diidethe #irst number b; the second and mu(ti)(; the 7uotient b; 1. For instance? anen!ine )roduces 0*h) #rom a )ossib(e 12*h). 5hat )ercenta!e o# the tota(horse)o&er aai(ab(e is bein! dee(o)edJ To so(e this? diide the 0* b; 12* andmu(ti)(; the 7uotient b; 1.

E6$!(%"

0* K 12* .-0 , 1 -0U )o&er.

-3. Another &a; that )ercenta!es are used? is to determine a number &hen on(;

a )ortion o# the number is 9no&n. For e,am)(e? i# 410r)m is 30U o# the ma,imum

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s)eed? &hat is the ma,imum s)eedJ To determine this? ;ou must diide the 9no&n7uantit;? 410r)m? b; the decima( e7uia(ent o# the )ercenta!e.

E6$!(%"

410 K .30 11?r)m ma,imum

-4. A common mista9e made on this t;)e o# )rob(em is mu(ti)(;in! b; the)ercenta!e instead o# diidin!. ne &a; o# aoidin! ma9in! this error is to (oo9 atthe )rob(em and determine &hat e,act(; is bein! as9ed. In the )rob(em aboe? i# 410r)m is 30U o# the ma,imum then the ma,imum must be !reater than 410. Theon(; &a; to !et an ans&er that meets this criterion is to diide b; .30.

Ar"$ $nd @o%u!"

Ar"$

-*. 5e are a(read; #ami(iar &ith the conce)t o# (en!th? e.!. the distance bet&een2 )oints? &e e,)ress (en!th in some chosen unit? e.!. in meters. I# &e &ant to #it a)icture>rai( a(on! a &a((? a(( &e need to 9no&n is the (en!th o# the &a((? so that &e canorder su##icient rai(. +ut i# &e &ish to #it a car)et to the room #(oor? the (en!th o# theroom is insu##icient. bious(; &e a(so need to 9no& the &idth. This t&o>dimensiona(conce)t o# si=e is termed Ar"$.

R"ct$n&u%$r Ar"$

-- Consider a room 4m b; 3m as sho&n aboe. C(ear(; it can be diided u) into

12 e7ua( s7uares? each measurin! 1m b; 1m. Each s7uare has an area o# 1 s7uaremeter. ence? the tota( area is 12 s7uare meters "usua((; &ritten as 12m2 #or conenience$. So? to ca(cu(ate the area o# a rectan!(e? mu(ti)(; (en!th o# one side b;the (en!th o# the other side.

4m , 3m 12m2 "Dont #or!et the m2$.

Ar"$ o' Tri$n&%"#

-/. This conce)t can be e,tended to inc(ude non>rectan!u(ar sha)es.

-0. Consider the trian!(es A+C and ADC &hich to!ether #orm a rectan!(e A+CD.

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-. Ins)ection reea(s the 2 trian!(es are con!ruent. ence their areas are e7ua(

/. I# &e consider this dia!ram? the area o# the trian!(e can be seen to e7ua(

/1. This is true #or an; trian!(e? but remember its the )er)endicu(ar hei!ht. Notea!ain that base "in meters$ , hei!ht "in meters$ !ies m2.

/2. A theorem e,ists statin! that trian!(es &ith the same base and dra&n bet&eenthe same )ara((e(s &i(( hae the same area.

Ar"$ o' Circu%$r S$("#

/3. The area o# a circ(e is !ien b; the #ormu(a

A Vr2 "&here r radius$ or

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Ar"$ o' Ot"r S$("#

/4. The tab(e be(o& indicates the areas o# man; common sha)es.

C$%cu%$tion o' Ar"$# o' S$("#

/*. Sometimes an area ca(cu(ation must be made &here the obect or sha)e isnot one o# the common sha)es (isted. Sometimes it is made u) #rom a combinationo# sha)es.

E6$!(%" An o##ice 0.*m b; -.3m is to be #itted &ith a car)et? so as to (eae asurround -mm &ide around the car)et. 5hat is the area o# the surroundJ

/-. 5ith a )rob(em (i9e this? it is o#ten he()#u( to s9etch a dia!ram.

//. The area o# the surround o##ice area > car)et area.

"0.* , -.3$ > "0.* > 2 , .-$ "-.3 > 2 , .-$

*3.** > "/.3$ "*.1$

*3.** > 3/.23 1-.32m2

Note that -mm had to be conerted to .-m. Dont #or!et to inc(ude units in the

ans&er e.!. m2.

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5e ma; need to #ind the area o# an obect that is a combination o# sha)es

In this case the sha)e com)rises a rectan!(e and a semi>circ(e.

The rectan!(e has dimensions 1*mm , 1mm

The semi>circ(e has a diameter o# 1mm

Tota( area is the sum o# the t&o indiidua( areas.

Area "1 , 1*$ r 2 1* W 2* 1* /0*4 220*4mm2

@o%u!"#

/0. So%id# are obects that hae three dimensions (en!th? &idth and hei!ht.ain! the abi(it; to ca(cu(ate o(ume enab(es ;ou to determine the ca)acit; o# a #ue(tan9 or reseroir? ca(cu(ate the ca)acit; o# a car!o area or &or9 out the o(ume o# ac;(inder. 6o(umes are ca(cu(ated in cubic units such as cubic centimetres? cubicmetres? cubic inches etc. o&eer? o(umes are easi(; conerted to other terms?such as (itres. For e,am)(e? a cubic metre contains 1 (itres o# (i7uid.

/. Instead o# s7uares? &e no& consider cubes. This is a 3>dimensiona( conce)tand the t;)ica( units o# o(ume are cuic !"tr"# "m3$.

0. I# &e hae a bo,? (en!th 4m? &idth 3m and hei!ht 2m? &e see that the tota(o(ume 24 cubic metres "24m3$.

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01. +asica((;? there#ore? &hen ca(cu(atin! o(ume? it is necessar; to (oo9 #or threedimensions? at X to each other? and then mu(ti)(; them to!ether. For a bo, > t;)esha)e? mu(ti)(;in! (en!th , &idth , hei!ht o(ume.

02. For irre!u(ar or )articu(ar sha)es? di##erent techni7ues or a))ro,imations can

be used? or sometimes a s)eci#ic #ormu(a ma; e,ist.

Note that a(( these #ormu(ae contain 3 dimensions so that &hen mu(ti)(ied? a o(ume&i(( resu(t.

e.!. R2h R , R , h or R3 R , R , R

I' :ou $)" not &ot 3 di!"n#ion#; :ou $)" not &ot $ )o%u!"

03. E,am)(e 5hat is the cubic ca)acit; o# a 2 c;(inder en!ine? &ith a or" o# //mm and a stro9e o# 0mmJ

6o(ume o# 1 c;(inder 41444 mm3

6o(ume o# 2 c;(inders 02000 mm3

Note that in this e,am)(e? the dimensions hae been !ien in mm. The o(ume &ou(dnorma((; be !ien in cm3.

Note? to conert mm3.to cm3.? diide b; "1$3.

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02000 mm3 becomes 020.00 cm3.

04. 5hen ca(cu(atin! areas or o(umes? remember the basic #ormu(as? but beread; to s)ot &hen an area or so(id bod; is a combination o# basic sha)es that canbe added or subtracted.

Po9"r# $nd Root#

Po9"r#

-*. 5hen a number is mu(ti)(ied b; itse(#? it is said to be raised to a !ien )o&er.For e,am)(e? - , - 3-L there#ore -2 3-. The number o# times the $#" nu!"r ismu(ti)(ied b; itse(# is e,)ressed as an "6(on"nt and is &ritten to the ri!ht ands(i!ht(; aboe the base number. A )ositie e,)onent indicates ho& man; times anumber is mu(ti)(ied b; itse(#.

E6$!(%"

32 is read H3 s7uaredH or H3 to the )o&er o# 2H. Its a(ue is #ound b; mu(ti)(;in! 3 b;itse(#.

3 , 3

23 is read H2 cubedH or H2 to )o&erH. Its a(ue is #ound b; mu(ti)(;in! 2 b; itse(# 3times.

2 , 2 , 2 0

I# the e,)onent is a ne!atie inte!er? the minus si!n indicates the inerse or reci)roca( o# the number &ith its e,)onent made )ositie.

E6$!(%"

--. An; number? e,ce)t =ero? that is raised to the =ero )o&er e7ua(s 1. 5hen anumber is &ritten &ithout an e,)onent? the e,)onent a(ue is assumed to be 1.Furthermore? i# the e,)onent does not hae a si!n? " or >$ )recedin! it? the e,)onentis assumed to be )ositie.

Root#

-/. The root o# a number is that a(ue &hich? &hen mu(ti)(ied b; itse(# a certainnumber o# times? )roduces that number. For e,am)(e? 4 is a root o# 1- because&hen mu(ti)(ied b; itse(#? the )roduct is 1-. o&eer? 4 is a(so a root o# -4 because 4, 4 , 4 -4. The s;mbo( used to indicate a root is the r$dic$% si!n " x $ )(aced oer

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the number. I# on(; the radica( si!n a))ears oer a number? it indicates ;ou are toe,tract the #*u$r" root o# the number under the si!n. The s7uare root o# a number is the root o# that number? &hen mu(ti)(ied b; itse(#? e7ua(s that number. 5hen as9edto e,tract a root other than a s7uare root? an ind"6 nu!"r is )(aced outside theradica( si!n.

For e,am)(e? the cube root o# -4 is e,)ressed as " -4$13

-0. Another &a; o# indicatin! roots is b; sho&in! the root o# a number is b;sho&in! an e,)onent as in )o&ers. In the case o# roots? ho&eer? the e,)onent issho&n as a #raction.

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CHAPTER 2

ALGEBRA

Introduction

1. 6er; o#ten students &i(( c(aim that the; neer hae and neer &i(( understand A(!ebra. The; sa; the; can understand and &or9 &ith numbers? but not &ith (etters?and ;et A(!ebra is desi!ned to ma9e matters sim)(e and c(ear.

2. For e,am)(e? su))ose a room is * metres (on! b; 3 metres &ide and &e needto 9no& ho& much car)et is needed to coer the #(oor. No one &ou(d hae an;hesitation in ca(cu(atin! the ans&er? 1* s7uare metres "m 2$. +ut that ans&er on(;a))(ies to that room. The &"n"r$% ans&er is that the area is #ound b; !u%ti(%:in&%"n&t b; 9idt "or breadth$.

i.e. Area (en!th , breadth.

3. +ut it is easier to &rite A % , b? &here the (etters A? %? b re)resent in thiscase Area? %en!th and breadth? and that is &hat A(!ebra is a(( aboutL (ettersre)resent some ariab(e and on(; &hen )articu(ar a(ues. i.e. numbers are 9no&n?do &e resort to them instead.

4. So &hen usin! A(!ebra? it is im)ortant to state &hat the (etters re)resent.Some (etters are o#ten used? )articu(ar(; , and ;? but ! o#ten re)resents acce(eration

due to !rait;? Y re)resents densit;? and so on. This is &hat A(!ebraic not$tion isabout.

O("r$tion

*. A(!ebraic o)erations are in essence the same as &hen usin! numbers.

So Addin& a and b is &ritten a b

Sutr$ctin& a and b is &ritten a b

Mu%ti(%:in& a and b is &ritten ab

0i)idin& a b; b is &ritten ab

S*u$rin& a a2

5e are not restricted to 2 (etters on(;.

Note a(so that the order in &hich (etters a))ear is basica((; unim)ortant.

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a , b , c , d abcd bdac cadb etc. etc."3 , 4 is obious(; the same as 4 , 3 etc.$

-. 5hen s;mbo(s such as , and ; are mu(ti)(ied to!ether &e do not need to

inc(ude the mu(ti)(ication si!n. This is the same i# a number and a s;mbo( aremu(ti)(ied to!ether.

3 , ;? 4 , =? s , )? a , b? ; , = , m

can a(( be &ritten &ithout the mu(ti)(ication si!n as 3;? 4=? s)? ab and ;=m

The same is not true o# numbers on their o&n

/ , 0? 4 , * and - , / cannot be &ritten as /0? 4* and -/.

Li>" T"r!# are terms com)rised o# the same a(!ebraic 7uantit; > this is im)ortant./,? *, and >3, are a(( terms containin! ,

/a? 4b? 3a and >-b can be s)(it into t&o !rou)s o# (i9e terms? /a and 3a?and 4b and >-b.

I# (i9e terms contain numerica( coe##icients? the; can be sim)(i#ied.

/, *, > 3, "/ * > 3$, ,

/a 3a 4b > -b 1a > 2b.

Terms (i9e ab cb > db ma; be sim)(i#ied as "a c > d$ b."b is a common #actor o# the 3 terms$

/. 5hen dea(in! &ith a(!ebraic terms and e,)ressions the abi(it; to #actorise is a!reat asset. Simi(ar(;? the abi(it; to diide numerator and denominator b; the sameterms "i.e. cance((in! to) and bottom$ a((o&s sim)(i#ication.

B$#ic L$9#

0. A(!ebra obe;s the same (a&s o# )rocedure as Arithmetic? i.e. +D'AS.

Note that +rac9ets a))ear rather more o#ten in A(!ebra? and are on(; remoed &henthere is a !ood reason to do so? #or e,am)(e? &hen #urther o)erations u(timate(; (eadto !reater sim)(i#ication.

"3, /;$ > "4, 3;$ 3, /; > 4, > 3; >, 4;

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Note es)ecia((; that &hen remoin! brac9ets? a(( the terms inside the brac9ets aremu(ti)(ied b; &hat is immediate(; outside the brac9ets. The basic )rocedure is as#o((o&s.

a ", ;$ a, a;

a b ", ;$ a b, b; "both , and ; are mu(ti)(ied b; b$

"a b$ ", ;$ a, a; b, b; ", and ; are mu(ti)(ied b; "ab$

"a b$2 "a b$ "a b$ "a , a$ "a , b$ "b , a$ "b , b$

a2 ab ab b2 a2 2ab b2

. 5hen '$ctori#in&? e,amine each term is order to (oo9 #or common #actors.

the common #actors o# a2b and >2ab2 are a and b "the; a))ear in both$?

hence a2b > 2ab2 can be &ritten "ab$"a > 2b$.

"ab$ and "a > 2b$ are both #actors o# the com)(ete e,)ression a2b and >2ab2.

As a(read; stated? the abi(it; to HseeH #actors is an asset.

a, b, a; b; , "a b$ ; "a b$ ", ;$ "a b$

or a ", ;$ b ", ;$ ", ;$ "a b$

1. A(!ebra can be e,tended to inc(ude #ractions.

E*u$tion#

11. The statement a 4 * is an e7uation. 5hat &e are sa;in! is that anun9no&n 7uantit; minus 4 e7ua(s *. It does not ta9e a !enius to &or9 out that theun9no&n 7uantit; in this case is ? there is on(; one a(ue that &i(( be correct. The

a(ue o# a can be ca(cu(ated usin! !uess&or9 or e(imination. The )rocess o# estab(ishin! that a is ca((ed so(in! the e7uation.

So%)in& Lin"$r E*u$tion#

12. A (inear e7uation is one containin! on(; the #irst )o&er o# the un9no&n7uantit;.

*; * 3; or *"m 2$ 1*

13. These are both (inear e7uations.

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14. 5hen &e so(e (inear e7uations? the a))earance o# the e7uation ma; chan!e.For e,am)(e? the #irst e7uation cou(d be re>&ritten as *; 3; * and the secondas *m 1 1*. +oth o# these (oo9 di##erent #rom the ori!ina( #orm? but e7ua(it; hasbeen maintained and the; are there#ore the same.

1*. The !enera( ru(e #or a(( e7uations is

=$t")"r :ou do to on" #id" o' t" "*u$tion; :ou !u#t do t" #$!"to t" ot"r #id".

1-. +; conention &e name each side o# the e7uation %e#t and Side "%S$ or Ri!ht and Side "RS$

E*u$tion# R"*uirin& Mu%ti(%ic$tion or 0i)i#ion

E*u$tion# R"*uirin& Addition or Sutr$ction

The sim)(est t;)e o# (inear e7uation is o# this t;)e

, > -

1/. To so(e a(( e7uations &e must mani)u(ate the e7uation to !et the un9no&n

on one side and the 9no&n a(ues on the other side. In this case &e must e(iminatethe a(ue o# >- #rom the %S.

This can be done b; addin! - to the %S? but &e must a(so add - to the RS.

So the e7uation becomes , > - - -

5e then Si!(%i': the e7uation to obtain , - 1*

So the so(ution is , 1*

10. A sim)(er &a; o# so(in! this t;)e o# e7uation is to s&itch a(ues #rom one sideto another. 5hen &e do this? &e must? ho&eer chan!e the si!n.

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E6$!(%" So(e ; 4 14

I# &e s&itch the 4 to the RS and chan!e the si!n it becomes

O 14 4 So the so(ution is ; 1

1. I# the e7uation has mu(ti)(es o# the un9no&n 7uantit;? such as

So(e *, 12 3

the #irst sta!e is the same? i.e. *, 3 12

So *, 1*

It seems obious that , 3? but ho& mathematica((; is this achieedJ

I# &e diide both sides b; * &e &i(( !et the so(ution , 3.

E*u$tion# Cont$inin& Dn>no9n# on ot Sid"#

2. In e7uations o# this t;)e &e shou(d !rou) the un9no&n 7uantities on one sideand the other terms on the other side.

For e,am)(e? so(e 0; 4 *; 22

I# &e subtract 4 #rom both sides? and a(so subtract *; #rom both sides &e &i(( !et

3; 10 The so(ution can then be obtained b; diidin! each side b; 3.

Note As in a(( cases o# so(in! e7uations? 9" c$n $nd #ou%d c"c> our #o%ution is correct b; substitutin! the so(ution in the ori!ina( e7uation.

i.e. %S "0 , +$ 4 40 4 *2 RS "* , +$ 22 3 22 *2

E*u$tion# Cont$inin& Br$c>"t#

a. The #irst ste) is to remoe the brac9ets and then so(e as norma(

3"2; 3$ 21 #irst e,)and the brac9ets to obtain

-; 21 then subtract #rom both sides

-; 12 then diide both sides b; -

The so(ution is ; 2

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21. To chec9 the so(ution is correct? &e substitute ; 2 in the ori!ina( e7uation.

%S 3"2 , 2 3$ 3"4 3$ 3 , / 21

RS 21

E*u$tion# Cont$inin& 8r$ction#

22. In this case &e must mu(ti)(; each term b; the %C' o# the denominators.

23. The %C' o# the denominators 4? * and 2 is 2? so &e must mu(ti)(; each termin the e7uation b; 2

Note this is e,act(; the same as 2*; *2 . This can be )roed b; ta9in! the e7uation− 2*; −*2 and addin! 2*; to both sides? and then addin! *2 to both sides.

24. The %C' o# 3 and 2 is -? so &e mu(ti)(; a(( o# the terms b; -

Tr$n#(o#ition In E*u$tion#

2*. Consider a #ormu(a "e7uation$ !ien in a certain #orm.

-a 11 2* a

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2-. This contains one a(!ebraic 7uantit;? HaH? &ithin an e7uation. Thin9 o# ane7uation as a statement o# ba(ance. In this one? -a 11 on the %S e7ua(s? or ba(ances? 2* > a on the RS.

2/. As &e hae one e7uation and one un9no&n a? there is on(; one numerica(

a(ue &hich can )roduce a ba(ance. 5hat is itJ

20. +; mani)u(atin! "tr$n#(o#in& is the &ord$ the e7uation? it is )ossib(e toiso(ate the a on the %S and ba(ance it &ith an actua( number on the RS. This &i((then be the uni7ue a(ue o# a. %oo9 a!ain at the e7uation.

-a 11 2* a

To remoe the a on the RS? &e must add a to both sides.

-a 11 a 2* > a a

there#ore /a 11 2*

To remoe 11? &e must subtract 11 #rom both sides

/a 11 11 2* 11

So /a 14

and i# /a 14 then a 2

2. 5e hae #ound that a 2. This is the uni7ue a(ue &hich satis#ies -a 11 2* > a.

3. Stud; it a!ain to see ho& &e &or9ed to iso(ate the re7uired term a on oneside? and r"!"!"r ? &hat ;ou do to one side o# an e7uation? ;ou must do to theother side i# the ba(ance is to be maintained.

31. ere is another a #ormu(a ino(in! seera( a(!ebraic s;mbo(s.

32. Remember? &e &ant N on one side b; itse(#. It is im)ortant to !et a #ee( #or the #orm o# the e7uation. To he()? &e &i(( )ut brac9ets around "N > n$.

33. To remoe the 2) &e must mu(ti)(; both sides b; 2)

&hich !ies 2C) "N n$2

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To remoe the >n? &e must add n to both sides

2C) n "N n$ n N

Thats it? N 2C) n

eres another e,am)(e.

R"!"!"r ? to #ind r? ta9e the s7uare root o# r 2 and do the same to both sides.

34. This is &hat trans)osition is a(( about. 5e are re>arran!in! #ormu(ase,)ressed as e7uations? &hich then a((o&s us to #ind a )articu(ar numerica( a(ue #or one"un9no&n$ 7uantit; i# the other numerica( a(ues are !ien.

3*. ne im)ortant )oint? it is on(; )ossib(e to #ind an un9no&n 7uantit; i# a(( theother a(ues are 9no&n. This is 9no&n as so(in! an e7uation.

3-. The ru(e is?

"a$ ne un9no&n 7uantit; can be deduced #rom one e7uation?

"b$ T&o un9no&ns re7uire t&o di##erent e7uations?"c$ Three un9no&ns re7uired three di##erent e7uations?"d$ and so on.

Con#truction O' E*u$tion#

3/. As a(read; stated? 'aths seres as a Htoo(H #or En!ineers at the desi!n sta!e.Desi!n is the creation o# a com)onent or mechanism on ($("r ? i.e. be#ore it ta9esha)e in meta( or )(astic. The desi!n en!ineer ho)e#u((; ma9es it stron! enou!h > his9no&(ed!e o# materia(s and their stren!ths a((o& him to do this b; ca(cu(ation. euses #ormu(as and e7uations.

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30. To do this? he must a((ocate (etters to re)resent some ariab(e or 9no&n7uantit;. e can then construct a #ormu(a or e7uation b; usin! the (etters &ithinsome reasonab(e statement about the situation. e studies the situation and thenma9es the statement.

3. o& do &e construct e7uations #rom the #acts contained &ithin a scenarioJ

E6$!(%" 1

Thin9 o# a number? doub(e it? add - and diide the resu(t b; 3. 5hat is the ans&erJ

%et the number ;ou thin9 o# be A. Doub(in! this number !ies 2A.

I# - is then added? &e hae 2A -? &hich must then be diided b; 3? ma9in!

%et the number ;ou thin9 o# be A. Doub(in! this number !ies 2A.

I# - is then added? &e hae 2A -? &hich must then be diided b; 3? ma9in!

the ans&er This #ormu(a can be used to ca(cu(ate the ans&er no matter &hat number ;ou thin9 o#.

E6$!(%" 2

I# one side o# a rectan!u(ar #ie(d is t&ice as (on! as the other? and the short side is1m. Ca(cu(ate the area o# the #ie(d.

%et the short side o# the #ie(d be %. The (on! side is there#ore 2 , % or 2%.

To ca(cu(ate the area &e mu(ti)(; one side b; the other? so

Area 2% , % 2%2 &here % e7ua(s 1m

Area 2"1$2 2m2

E6$!(%" 3

A certain t;)e o# motor car cost seen times as much as a certain ma9e o# motor c;c(e. I# t&o cars and three motor c;c(es cost 0*? #ind the cost o# each ehic(e.

%et the cost o# a car be C "at )resent C is an un9no&n$.%et the cost o# a motor c;c(e be ' "another un9no&n$.

5e 9no& that 2C 3' 0* "this has t&o un9no&ns &ithin one e7uation$.

+ut &e a(so 9no& that C / , '? there#ore? &e can substitute #or C in the #irste7uation.

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The cost o# a motor c;c(e is there#ore *? and the cost o# a car must be / Z * 3*.

ere 2 e7uations &ere constructed #rom the #acts? and then combined to a((o& aso(ution to be #ound.

In the ne,t e,am)(e? &e #orm e7uations #rom the #acts? and then trans)ose to)roduce a so(ution.

E6$!(%" 4

Three e(ectric radiators and #ie conector heaters to!ether cost /4. A conector cost 2 more than a radiator. Find the cost o# each.H

%et R re)resent the cost o# a radiator? and C re)resent the cost o# a conector.

Then 3R *C /4

And C R 2

∴ 3R * "R 2$ 3R *R 1 /4

∴ 0R /4 > 1 -4

R 0 "the cost o# a radiator$

and C 0 2 1 "the cost o# a conector$

Si!u%t$n"ou# E*u$tion#

4. Consider the e7uation 4, > 3; 1. There are 2 un9no&ns ", and ;$ in one

e7uation? and so the e7uation cannot be so(ed to !ie a sin!(e a(ue #or , and ;.There are an in#inite number o# a(ues o# , #or &hich there are corres)ondin! a(ueso# ;. For e,am)(e

i# , 4? then ; * i# , /? then ; i# , 1? then ; 1

41. o&eer? i# a second e7uation e,ists? #or e,am)(e , 3; 1? then these t&oe7uations can be ea(uated simu(taneous(; to !ie sin!(e a(ues #or , and ;.

42. The )rocess is sim)(e and ino(es modi#;in! the e7uations? &hi(st sti(()reserin! the e7ua(ities.

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"a$ 4, 3; 1 "1$"b$ , 3; 1 "2$

43. The method o# so(ution o# a(( simu(taneous e7uations is to

"a$ #irst mani)u(ate one or both o# the e7uations so that the coe##icient o# one o#

"b$ the un9no&ns is the same in both e7uations.

"c$ then add or subtract one o# the e7uations #rom the other to )roduce athird"d$ e7uation &ith on(; one un9no&n. The other hain! become =ero.

"e$ so(e the ne& e7uation to #ind the un9no&n.

"#$ )ut the so(ution into one o# the ori!ina( e7uations to #ind the other un9no&n.

"!$ )ut both so(utions into the e7uation not used in the sta!e aboe tochec9 ;our ans&ers.

44. 8sin! the t&o e7uations aboe as an e,am)(e

4*. 5e do not need to mani)u(ate either o# the e7uations because the co>e##iciento# ; is the same in both e7uations. There#ore? &e can e(iminate the ;B a(ue sim)(;b; addin! the t&o e7uations. The resu(t is

*, 2 So 6 7 4

I# &e then substitute , 4 in the second e7uation &e !et

4 3; 1 So 3; 1 > 4 1* So : 7 5

ur so(utions are , 4 and ; *

E6$!(%" 1

2, 3; 0 "1$

3, *; 11 "2$

4-. 'u(ti)(; e7uation "1$ b; the coe##icient o# , in e7uation "2$.

"2, 3; 0$ , 3 -, ; 24

4/. 'u(ti)(; e7uation "2$ b; the coe##icient o# , in e7uation "1$.

"2, *; 0$ , 2 -, 1; 22

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So -, ; 24 "3$

-, 1; 22 "4$

Subtract e7uation "4$ #rom "3$

, > 1; 2.

So >; 2 and ; >2

substitute ; > 2. in either e7uation "1$ or "2$ to so(e #or ,. I hae se(ected "1$.

2, 3">2$ 0 there#ore 2, 14 and , /

40. Chec9 ;our ans&er b; substitutin! both a(ues in e7uation "2$. Do not usee7uation "1$ because it &i(( not hi!h(i!ht an error. I# ;ou had used e7uation "2$ to #ind,? then the chec9 shou(d be carried usin! e7uation "1$.

3, *; 11

3"/$ *">2$ 11 there#ore 21 ">1$ 11 correct

4. The same resu(t &ou(d be #ound i# ; &as e(iminated as sho&n be(o&.

u$dr$tic E*u$tion#

*. An; e7uation o# the #orm ; a,2 b, c? &here a? b and c are numbers? is9no&n as a 7uadratic e7uation. An e7uation o# this t;)e &i(( )roduce a cure ca((ed a)arabo(a. The actua( a(ue #or coe##icients a? b and c &i(( determine the e,act sha)eand )osition o# the cure.

*1. It &i(( be noted that one o# the cures cuts the ,>a,is at )oints P and S.

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*2. P and S are 9no&n as the root# o# the e7uation. A(ternatie(;? P and S are thea(ues o# , &hich satis#; the condition ; a,2 b, c o.

*3. It can be sho&n that the Roots are #ound to be e7ua( to

*4. This e7uation !ies t&o a(ues? one #or P the other #or S.

E6$!(%" Find the roots o# ; -,2 > *, > - "a -? b >*? c >-$

Note > de)endin! on a? b and c? it is )ossib(e that b 2 > 4ac resu(ts in a n"&$ti)")$%u". It has been considered im)ossib(e to #ind the s7uare root o# a ne!atie a(ue.The e7uation concerned is then said to hae no r"$% root#. 5hen b2 > 4ac isne!atie? the e7uation is said to hae co!(%"6 root#? &here the roots com)rise

both a rea( and ima!inar; com)onent. This conce)t is not considered in these notes.

NDMBERS

Indic"# $nd Po9"r#

**. It is o#ten to necessar; to mu(ti)(; a number b; itse(# once? t&ice or seera(times. To indicate this? a method o# notation has eo(ed? &hich is both conenientand ca)ab(e o# bein! e,tended to introduce other conce)ts.

3 , 3 is &ritten as 32

2 , 2 , 2 , 2 , 2 is &ritten as 25

4 , 4 , 4 is &ritten as 43 etc? etc.

*-. In the aboe e,am)(es? the number bein! mu(ti)(ied b; itse(# is 9no&n as the$#" and the number o# times it is mu(ti)(ied b; itse(# is 9no&n as the (o9"r or ind"6. A(ternatie(;? the number 2 has been raised to )o&er *.

*/. Po&er 2 and )o&er 3 are !enera((; re#erred to as the s7uare and the cube.

3 , 3 32 is the s7uare o# 3 or 3 s7uared e7ua(s

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4 , 4 , 4 43 -4 -4 is the HcubeH o# 4. or 4 cubed e7ua(s -4

*0. +ut )ut another &a;? 3 is said to be the #*u$r" root o# ? 4 is the cu" rooto# -4 and 2 is the 'i't root o# 32.

*. The method o# notation used is that

-. It is )ossib(e to re>&rite the aboe? so that 3 .*? 2 32.2 and 4

-4.333. 5here the )o&er is e,)ressed as a decima(? instead o# a #raction.

-1. To a((o& the use o# numbers ino(in! )o&ers and indices? some ru(es haeeo(ed? &hich are re)roduced? usin! the s;mbo( N to re)resent an; base number.

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St$nd$rd 8or!

-2. I# the number 0.34/ is mu(ti)(ied b; 1? then the )roduct is 034/. Thisca(cu(ation can be &ritten as 0.34/ , 14 034/.

-3. 5hen 034/ is &ritten as 0.34/ , 14? it is 9no&n as Standard Form.

-4. A number in standard #orm has t&o )arts. The #irst )art is a number bet&een 1and 1 "but does not e7ua( 1$? and the second )art is 1 raised to some &ho(enumber )o&er. The #irst )art is ca((ed the M$nti##$? the second )art the E6(on"nt.

-*. To e,)ress a number in standard #orm? moe the decima( )oint (e#t or ri!ht tocreate a number bet&een 1 and 1 "the mantissa$? and then create the e,)onent.

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The a(ue o# &hich e7ua(s the number o# )(aces b; &hich the decima( )oint has beenmoed. I# the )oint &as moed %e#t? the )o&er is )ositie? i# the )oint &as moedRi!ht? it is ne!atie.

E6$!(%"# *2- *.2- , 12

.3/1- 3./1- , 1>1

.2 2. , 1>3

Nu!"rin& S:#t"!#

--. The most &ide(; used s;stem o# numbers is the decima( s;stem? based on thehindu>arabic s;mbo(s ? 1? 2? 3 etc but roman s;mbo(s such as 6? Z? % and C area(so &e(( 9no&n and understood. To>da;? the )ractice o# en!ineerin! re7uires ameasure o# com)etence in hand(in! seera( di##erent s;stems o# numera(s.

-/. In !enera( a s;stem o# numeration consists o# a set o# s;mbo(s to!ether &ith aru(e b; &hich the s;mbo(s can be combined to!ether.

-0. Nu!"r is the )ro)ert; associated &ith a set or co((ection o# thin!s. It isinde)endent o# the nature o# the indiidua( items in the set. The number #ourteenma; be &ritten as 1* or ZI6. In this case the number is the same but the s;stem or numeration is di##erent.

0"ci!$% S:#t"! O' Nu!"r$tion

-. In the decima( s;stem? the s;mbo(s are combined b; arran!in! them in ahori=onta( (ine? the contribution that each di!it ma9es bein! !oerned b; its )osition. A decima( )oint enab(es numbers (ess than one to be re)resented.

E6$!(%" 1

Decima( 3-0 is rea((;

"3 W 12$ "- W 11$ "0 W 1/$

or in co(umn #orm

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E6$!(%" 2

Decima( 4*2.-4 is rea((;

"4 W 12$ "* W 11$ "2 W 1$ "- W 1>1$ "4 W 1>2$

or in co(umn #orm

/. Ten is 9no&n as the $#" or r$di6 o# the decima( s;stem. The ind"6 indicatesthe )o&er to &hich the base is raised.

/1. The base? and the )articu(ar inde, to &hich it is raised is ca((ed the 9"i&t.

e.!. (east si!ni#icant &ei!ht 1 1

ne,t most si!ni#icant &ei!ht 11 1

/2. The numbers b; &hich &ei!ht is mu(ti)(ied are ca((ed di&it#. In )ractice on(;the di!its o# the s;stem are &ritten? the &ei!ht bein! im)(ied e.!. 3-0? *3.24.

Not" is counted as a di!it? so that there are ten di!its in the decima( s;stem? to inc(usie.

Bin$r: S:#t"! O' Nu!"r$tion

/3. n(; the s;mbo(s and 1 are used and the base is t&o? other&ise the s;stemo# numeration is the same as be#ore. The t&o di!its and 1 are re#erred to as it#?an abbreiation o# binar; di!its.

E6$!(%" 1

1111 is rea((;

"1 W 2*$ " W 24$ "1 W 23$ "1 W 22$ " W 21$ "1 W 2$ or in co(umn#orm

" 4* in decima($

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E6$!(%" 2

11.11 is rea((;

"1 W 22$ "1 W 21$ " W 2$ "1 W 2>1$ "1 W 2>2$ or in co(umn #orm

" -./* in decima($

Not" A(( di!its to the ri!ht o# the in$r: (oint re#er to ne!atie )o&ers.

/4. The binar; s;stem is er; suitab(e #or use &ith e(ectrica( s&itchin! circuits. As&itch is either o## or on corres)ondin!? #or e,am)(e? to and 1 res)ectie(;.There is no ambi!uit;.

Oct$% S:#t"! O' Nu!"r$tion

/*. In the octa( s;stem o# numeration the s;mbo(s to / are used and the base is0. A!ain the s;stem o# numeration is the same as that used #or decima( and binar;? &itheach co(umn increasin! b; a )o&er o# one as ;ou moe #rom ri!ht to (e#t.

E6$!(%" 1

3/-0 is rea((;

"3 W 02$ "/ W 01$ "- W 0$

or in co(umn #orm

" 2*4 in decima($

E6$!(%" 2

3/M13 is rea((;

"3 W 01$ "/ W 0/$ "1 W 0>1$ "3 W 0>2$or in co(umn #orm

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in decima( "3 , 0$ "/ , 1$ "1 , M12*$ "3 , M1*-2*$

31M14-2*

Not" A(( di!its to the ri!ht o# the oct$% (oint re#er to ne!atie )o&ers.

Con)"r#ion To Ot"r B$#"#

/-. Conersion #rom d"ci!$% to $n: ot"r $#" can be achieed b; diidin! thedecima( number re)eated(; b; the ne& base and recordin! the remainder. Theremainder !ies the number in the ne& base and shou(d be read #rom bottom to to).

E,am)(e conert 21 to binar;.

E,am)(e 2 conert */1 to octa(

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E,am)(e 3 conert -31 to he,adecima(

//. To conert in$r: nu!"r# to d"ci!$%.

/0. The easiest &a; to conert #rom binar; to decima( is to remember the&ei!htin!s? or i# necessar; &rite the &ei!htin!s aboe each binar; di!it? and addthem u).

E6$!(%" 1 conert 1 1 1 1 to decima(.

/. An a(ternatie method #or (on! binar; numbers is to ta9e the (e#t>hand di!it?doub(e it and add the resu(t to the ne,t di!it to the ri!ht as sho&n be(o& "doub(e andadd to ne,t di!it to the ri!ht$.

0. To conert in$r: to oct$% or )ic" )"r#$.

01. Each octa( di!it can be re)resented b; 3 binar; di!its. There#ore? to conert#rom binar; to octa(

i. s)(it the binar; number into !rou)s o# 3 di!its startin! #rom the ri!ht.

ii. &ei!ht the numbers in each !rou) 4 2 1

iii. #ind the tota( o# each !rou) o# 3 di!its? the resu(t is the octa( a(ue.

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E6$!(%" 1 conert 1 1 1 1 1 to octa(

02. The reerse )rocess shou(d be used to conert octa( to binar;. Conert eachdi!it into a 3 di!it binar; number 9ee)in! the order o# di!its the same. 5or9 #rom thebottom to the to) o# the tab(e sho&n aboe to conert 2/10 to binar;.

03. To conert in$r: to "6$d"ci!$% or )ic" )"r#$.

04. The )rocess #or conertin! a binar; number to a he,adecima( one? is thesame as that used to conert binar; numbers to octa(. Each he,adecima( di!it canbe re)resented b; 4 binar; di!its? there#ore the binar; number is s)(it into !rou)s o# 4 di!its startin! #rom the ri!ht. The &ei!htin!s this time are 0 4 2 1.

0*. A!ain? the reerse )rocess is used to conert #rom he,adecima( to binar;.Conert each he,adecima( di!it into its binar; e7uia(ent 9ee)in! the order thesame.

E,am)(e 1 conert A/1- to binar;.

Lo&$rit!#

0-. %o!arithms are a mathematica( conce)t that &as dee(o)ed to sim)(i#;mu(ti)(ication and diision o# (ar!e numbers. %o!arithms enab(e mu(ti)(ication anddiision to be )er#ormed usin! addition and subtraction. The use o# (o!arithms is no(on!er so &ides)read as the e(ectronic ca(cu(ator has become so readi(; aai(ab(e.

0/. Rememberin! that &hen? #or e,am)(e? 2* is &ritten as *2? * is 9no&n as the$#" and 2 as the (o9"r ? then the %o&$rit! o# 2* can be e,)ressed as 2? to thebase *.

The &"n"r$% d"'inition is? that i# : 7 $6 then 6 7 %o&$ :

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00. So (o!arithms can be ca(cu(ated #or an; base a? but !enera((; on(; (o!arithmsto the base o# 1 or e "2./1$ are used? and are common(; aai(ab(e in tabu(ar #orm.o&eer? (o!arithms are more easi(; obtained #rom the ca(cu(ator.

0. An e,am)(e o# the #unction o# (o!arithms is sho&n be(o&.

E6$!(%" Ca(cu(ate -.412 , 23.1-2

From the ca(cu(ator the (o!1 o# -.412 is .0- and the (o!1 o# 23.1-2 is1.3-4/0.

So -.412 , 23.1-2

1/.-/+ , 11.3+4,-

and usin! the (a&s o# indices

-.412 , 23.1-2 1/.-/++ 1.3+4,-

12.1,1,,

. It is no& necessar; to #ind the base 1 number &hose (o!arithm is 2.1/1//.The ca(cu(ator sho&s this to be 140.*14/4 "this is the $nti%o& o# 2.1/1//$. I# theca(cu(ator is used to so(e -.412 , 23.1-2? the )roduct is 140.*14/4.

1. It is im)ortant to rea(ise that this e,am)(e sho&s ho& (o!arithms c$n be used?in )ractice? the ca(cu(ator is used as norma(. I# a diision is to be )er#ormed? the)o&ers o# (o!s are #utr$ct"d.

2. It is the conc"(t o# a (o!arithm that is im)ortant at this sta!e? because itrea))ears (ater.

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CHAPTER 3

GEOMETRY

An&u%$r M"$#ur"!"nt

0*. +ut ho& are an!(es e,)ressed or measured. Consider a sin!(e (ine? androtate it throu!h a com)(ete reo(ution.

Note that ha(# a reo(ution is there#ore 10X and a ri!ht an!(e "[ o# a reo(ution$ is

X.

Note that 1 de!ree can be sub>diided into - minutes and 1 minute can besubdiided into - seconds "er; sma(($.

A #e& de#initions are inc(uded here

An Acute an!(e > (ess than X

An btuse an!(e > bet&een X and 10X

A Re#(e, an!(e > !reater than 10X

Com)(ementar; an!(es > their sum is X

Su))(ementar; an!(es > their sum is 10X

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An&%"# A##oci$t"d =it P$r$%%"% Lin"#

0-. No& consider 2 )ara((e( (ines? cut b; a transersa(.

A C? + D "the; are o))osite and e7ua($? simi(ar(; % P? and ' :.

A(so A %? D :? etc. etc. "the; are corr"#(ondin& an!(es$

G"o!"tric Con#truction#

0/. There are man; di##erent sha)es associated &ith !eometr;. The morecommon ones are described in the #o((o&in! te,t.

Tri$n&%"

00. A trian!(e obious(; has 3 sides and 3 "interna($ an!(es. The sides are o#tenre)resented b; the 3 "sma(($ (etters a? b and cL the an!(es b; the "(ar!e$ (etters A? +

and C.

The 3 an!(es add u) to 10X.

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0. The construction o# a dotted (ine )ara((e( to A+ and an e,tension o# +C )roesthis.

The area o# a trian!(e \ base , ertica( hei!ht

Tri$n&%" T:("#

. There are man; di##erent t;)es o# trian!(e. The main t;)es and #eatures aresummarised as #o((o&s

1. Acut"$n&%"d trian!(e has a(( o# its an!(es (ess than X.

2. Otuc"$n&%"d trian!(e has one an!(e !reater than X.

3. Sc$%"n" trian!(e has three sides o# di##erent (en!ths.

4. Ri&t$n&%"d trian!(e has one o# its an!(es e7ua( to X. The (on!est side iso))osite the X an!(e "ri!ht>an!(e$ and is ca((ed the h;)otenuse.

*. I#o#c"%"# trian!(e has t&o sides and t&o an!(es e7ua(. The e7ua( an!(es (ieo))osite to the e7ua( sides.

-. E*ui%$t"r$% trian!(e has a(( its sides and an!(es e7ua(.

Si!i%$r < Con&ru"nt Tri$n&%"#

/. Oou ma; stud; t&o trian!u(ar sha)es and estimate &hether the; are the sameor not. 5e need to be more )recise.

0. I# the; hae the same sha)e? &e are rea((; sa;in! that their $n&%"# are the

same? the; are then described as #i!i%$r tri$n&%"#. Simi(ar trian!(es do not hae tobe the same si=e. ne trian!(e ma; hae sides t&ice or ten times as (ar!e asanother trian!(e and sti(( be c(assi#ied as simi(ar.

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. I# the; are e,act(; the same sha)e and #iF"? their #id"# are the same (en!th?then the; are described as Con&ru"nt tri$n&%"#.

1. It is sometimes necessar; to determine &hether trian!(es are Con!ruent. Asim)(e criteria e,ists to assist us. T&o trian!(es are con!ruent i#

11. Their corres)ondin! sides are o# e7ua( (en!th. "side? side? side$

12. The; hae t&o an!(es and the common side e7ua(. "an!(e? side? an!(e$

13. The; hae t&o sides and the inc(uded an!(e is e7ua(. "side? an!(e? side$

14. The h;)otenuse and one side o# a ri!ht>an!(ed trian!(e are e7ua( to theh;)otenuse and the corres)ondin! side o# another ri!ht>an!(ed trian!(e.

Po%:&on

1*. A )o(;!on is a !eometric c(osed #i!ure bounded b; strai!ht (ines. The term)o(; means mu(ti. A trian!(e has the (east number o# sides. ther mu(ti>sided #i!ureshae names indicatin! the number o# sides. ence

1-. Penta!on * sided? e,a!on - sided? cta!on 0 sided

u$dri%$t"r$%#

1/. A 7uadri(atera( is an; #our>sided sha)e. There are arious t;)es? some arecommon and ;ou are )robab(; #ami(iar &ith their names. Some are not so common.

10. Since a 7uadri(atera( has #our sides? it can be diided into t&o trian!(es. Thesum o# its an!(es must there#ore be 3-X.

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P$r$%%"%o&r$!

1. A )ara((e(o!ram has both )airs o# o))osite sides )ara((e(. The #o((o&in!)ro)erties a))(; to )ara((e(o!rams

11. Each )air o# o))osite sides is e7ua( in (en!th.

111. Each )air o# o))osite an!(es are e7ua(

112. The dia!ona(s bisect each other

113. The dia!ona(s bisect the )ara((e(o!ram and #orm t&o con!ruent trian!(es

R"ct$n&%"

114. A rectan!(e is a )ara((e(o!ram &ith its an!(e e7ua( to X. It has the same)ro)erties as a )ara((e(o!ram &ith the addition that the dia!ona(s are e7ua( in (en!th.

Ro!u#

11*. A rhombus is a )ara((e(o!ram &ith a(( o# its sides e7ua( in (en!th. It a(so hasa(( o# the )ro)erties o# a )ara((e(o!ram and the #o((o&in! additiona( )ro)erties

11-. The dia!ona(s bisect at ri!ht an!(es

S*u$r"

11/. A s7uare is a rectan!(e &ith a(( the sides e7ua( in (en!th. It has a(( the)ro)erties

o# a )ara((e(o!ram? rectan!(e and rhombus.

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Tr$("Fiu!

110. A tra)e=ium is a 7uadri(atera( &ith one )air o# sides )ara((e(.

Circ%"#

11. Circ(es are not ust )articu(ar mathematica( sha)es but are ino(ed in our eer;da; (i#e? #or e,am)(e? &hee(s are circ(es? !ears are basica((; circu(ar and sha#tsreo(e in a circu(ar #ashion. ence? &e must be a&are o# some im)ortant de#initionsand )ro)erties.

12. I# the (ine P is #i,ed at and rotated around ? the )oint P traces a )ath&hich is circu(ar > it #orms a circ(e.

121. The (en!th P is the R$diu# o# the circ(e. Note that P A + and thatthe (en!th o# the (ine A+ is c(ear(; e7ua( to t&ice the radius. A+ 2P. A+ is the0i$!"t"r o# the circ(e "D 2R$.

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122. 5e a(read; 9no& that i# P is rotated throu!h 1 com)(ete reo(ution? it &i((hae rotated throu!h 3- de!rees? but &hat is the distance trae((ed b; P in tracin!this circu(ar )athJ Put another &a;? ho& #ar &i(( a &hee( &hose radius is R? ro(( a(on!a sur#ace? durin! one reo(utionJ

123. The distance? 9no&n as the Circum#erence is obious(; de)endent on the(en!th o# the (en!th o# the diameter? but can be ca(cu(ated )recise(; #rom thee7uation C VD " 2VR$. The a(ue V is actua((; the ratio bet&een thecircum#erence o# a circ(e and its diameter.

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12-. Consider a circ(e o# radius R and consider an arc A+? &here (en!th is a(soe7ua( to R. The an!(e at the centre o# the circ(e? A+ is then e7ua( to I Radian.

12/. It can be deduced that I reo(ution is e7uia(ent to 2V Radians? i.e. I re -.2032 rads.

120. There#ore 3-X 2V rads? and &e can derie conersion #actors? as thatL

12. ne #ina( and use#u( )oint concernin! radian measure.

13. I# an arc o# a circ(e? radius r? subtends an an!(e? e7ua( to Radians? the (en!tho# the arc is r..

Note a(so that i# a )oint P is moin! &ith s)eed N? then the rotationa( s)eed ise7ua( to Nr "N r.$.

is e,)ressed in Radians )er second.

Gr$(#

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!ra)h is better. I# &e are )(ottin! tem)erature &ith res)ect to time then a continuous(ine !ra)h is better?

Con#truction

132. In order to construct !ra)hs e##ectie(;? some sim)(e ru(es shou(d be #o((o&ed.

133. First o# a((? )resent the data in a c(ear? tabu(ar #orm. The data &i(( data &i((!enera((; com)rise 2 ariab(es? one that is bein! aried? the ind"("nd"nt ariab(e?and the one that chan!es as a resu(t o# the ariation? the d"("nd"nt ariab(e "itsa(ue de)ends on the a(ue o# the other$.

134. For e,am)(e? an e,)eriment &as conducted? &here a o(ume o# !as &asheated. As the tem)erature o# the !as increased? it &as noted that the !ase,)anded its o(ume increased. The #irst 7uantit;? the tem)erature? is theind"("nd"nt ariab(e and the second 7uantit;? the o(ume? is the d"("nd"ntariab(e.

13*. The ne,t sta!e is to )(an the use o# the !ra)h>)a)er so as to )resent the!ra)h in the c(earest manner )ossib(e.

13-. The !ra)h constructed b; )(ottin! a series o# )oints? each one re)resentin! a)articu(ar a(ue o# the inde)endent and corres)ondin! de)endent ariab(e. So the!ra)h must be dra&n so that each a(ue a))ears "or #its$ on the )a)er.

13/. +e#ore )(ottin!B the )oints? the t&o a,es must be dra&n? and the sca(eschosen. The hori=onta( ",>a,is$ &i(( re)resent the inde)endent ariab(e and theertica( ";a,is$ the de)endent ariab(e. The sca(es cross at the ori!in .

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130. There is no merit in dra&in! sma(( !ra)hs. Choose sca(es so that com)(eted!ra)h #its the sheet o# !ra)h )a)er.

13. %oo9 at the (ar!est ri!ht>hand? and the sma((est (e#t>hand a(ues that &i(( be)(otted a(on! the ,>a,is. Subtract the % a(ue #rom the R a(ue to !ie a ran!e o# a(ues " some number o# units$. Stud; the !ra)h )a)er to #ind ho& man; (ar!es7uares there are #rom (e#t to ri!ht.

14. No& diide the a(ue #ound b; the subtraction? b; the number o# (ar!es7uares. This shou(d !ie an idea o# a suitab(e sca(e. That is? so man; units shou(dbe re)resented b; 1 (ar!e s7uare a(on! the ,>a,is. The most use#u( sca(es are 1? 2?*? 1? 2? * units etc. etc to 1 (ar!e s7uare.

141. The same )rocedure is used #or the ;>a,is. Subtract the sma((est "(o&er$ a(ue#rom the (ar!est "u))er a(ue$ to !ie the ran!e? diide b; the number o# (ar!es7uares bet&een to) and bottom o# the )a)er.

142. ain! done this? dra& the 2 a,es? and mar9 o## the units? usin! ;our chosensca(es.

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143. The !ra)h )a)er has no& been )re)ared #or the obect o# the e,ercise? i.e. totrans#er the data #rom the tab(e to the !ra)h.

144. The trans#er is er; sim)(e? ta9e one a(ue o# the inde)endent ariab(e anddra&s a "#aint$ (ine to coincide &ith its a(ue a(on! the ,>a,is so as to intersect &ith asimi(ar (ine dra&n #rom the ;>a,is #or its corres)ondin! de)endent a(ue.

14*. The intersection re)resents one )(otted )oint o# the !ra)h.

14-. The )rocedure is re)eated #or each )air o# a(ues in turn. 5hen a(( the )ointshae been )(otted? a continuous (ine is dra&n throu!h the )oints.

14/. The &a; in &hich the (ine is dra&n de)ends on the nature o# the data. It is)robab(; true to sa; that most mathematica( or scienti#ic data chan!e !radua((; or

)ro!ressie(; > the; ma; #orm a de#inite re(ationshi). In this case? do not oin the)oints &ith a series o# strai!ht (ines.

140. +ut tr; to dra& a continuous #!oot (ine.

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14. This )robab(; means that the (ine on(; !oes throu!h some "not a(($ o# the)oints > dont &orr;L e,)erimenta( or )(ottin! errors can occur. There shou(d berou!h(; the same number o# )oints on both sides o# the smooth cure. Sometimes? itis #air(; obious that a strai!ht (ine is the "most$ reasonab(e #it to the )oint? and thisis o#ten the case #or sim)(e scienti#ic e,)eriments.

Gr$(# $nd M$t"!$tic$% 8or!u%$"

1*. This course is desi!ned #or en!ineers? not mathematicians and so maths is

ie&ed as a serant? not a master.

1*1. %ater? it &i(( be seen that one )h;sica( 7uantit; &i(( ar; as another 7uantit;aries? &ith the t&o (in9ed b; some mathematica( (a& or e7uation. An e,am)(e is thatthe dra! #orce "D$ aries accordin! to the s7uare o# the airs)eed "6$.

E,)ressed as a #ormu(a D 9 62

1*2. This re(ationshi) can be )(otted in !ra)hica( #orm? and it is reasonab(e to)resume that it &ou(d be o# the same #orm as the maths re(ationshi) o# ; ,2 &here; is considered as a #unction o# , ; #",$

1*3. There are man; mathematica( #unctions? e,am)(es mi!ht be

; m,? ; ,2? ; ,3? ; sin ,; e6? ; cos , etc. etc.

1*4. This to)ic (oo9s at the sha)e and characteristics o# these #unctions &hene,)ressed !ra)hica((;? so that a sim)(e (in9 can be made &ith )h;sica( )henomena?&hich demonstrates simi(ar characteristics.

1**. 5hen a mathematica( #unction is )(otted? certain sha)es eo(e characteristic

o# that #unction. I#? #o((o&in! an e,)eriment durin! &hich data is !athered? that datacreates simi(ar sha)es? then a )resum)tion (in9in! #ormu(a and e,)eriment ma;made.

8unction $nd S$("

1*-. The ariab(e ; is o#ten described as a #unction o# ,. ere seera( di##erent#unctions are considered !ra)hica((;.

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1*/. Function : 7 !6 &here m is some constant coe##icient.

; m, !ies a strai!ht (ine? )assin! throu!h the ori!in .

m is the s(o)e o# the !ra)h "and tan $ the !reater the a(ue o# m? the stee)er thes(o)e. bious(; #or a strai!ht (ine? the s(o)e is constant #or a constant a(ue o# m.

I# m is >e? the (ine s(o)es as sho&n. "i# m ? the (ine O coincides &ith the,>a,is$.

1*0. Function : 7 !6 c

1*. This is a ariation o# ; m,.

1-. C is a constant? and is c(ear(; the a(ue o# ; &hen , . "; m. c C$.This a(ue o# C measured a(on! the ; a,is is 9no&n as the interce)t.

1-1. Function : 7 >62 &here 9 is some constant.

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1-2. This !ies a cure? 9no&n as a )arabo(a. As 9 increases the a(ue o# 9,2 a(soincreases. Note that the s(o)e is no (on!er constant. This is a #unction &hich iscommon(; #ound in )h;sica( situations.

1-3. Function : 7 >63 etc.

1-4. This is the characteristic sha)e. Note that the !ra)h has Turnin! )oints? &herethe #%o(" c$n&"# #rom e to e and ice ersa.

1-*. Functions &ithin this #ami(; are (ess (i9e(; to be encountered durin! thiscourse.

1--. Function : 7 #in 6 and : 7 co# 6.

1-/. +oth o# these #unctions are re)etitie but the &ord used to describe suchbehaiour is )eriodic "in this case? the )eriod is 3-X or 2 radians$.

Note that the cosine !ra)h (eads the sine !ra)h b; X &hen suchbehaiour occurs? it is o#ten re#erred to a )hase di##erence.

These !ra)hs are o#ten #ound? )articu(ar(; in e(ectrica( &or9.

Function : 7 "6? : 7 "6? : 7 1 "6

; e, is 9no&n as the E,)onentia( #unction. It is a(so o#ten #ound in En!ineerin!a))(ications. Some ariations on the basic #unction are a(so sho&n.

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1-0. Re#erence has a(read; been made to the s(o)e o# a !ra)h. Strai!ht (ines haea constant s(o)e. Cures hae ariab(e s(o)es? and o#ten inc(ude turnin! )oints"o#ten termed ma,ima and minima$. 'athematicians determine s(o)es b; usin! abranch o# mathematics ca((ed ca(cu(us a (ater to)ic. En!ineers are o#ten interestedin s(o)e? because de)endin! on the ariab(es? the s(o)e itse(# re)resents a )h;sica(7uantit; more about this in the Ph;sics modu(e.

1-. The area under a !ra)h is a(so o#ten use#u( and ma; re)resents a )h;sica(7uantit;.The

1/. The area can be ca(cu(ated b;

"a$ Considerin! sim)(e sha)es and a))ro,imatin!

"b$ Countin! s7uares.

"c$ 8sin! ca(cu(us

No!o&r$(#

1/1. The need to sho& ho& t&o or more ariab(es a##ect a a(ue is common in themaintenance o# aircra#t. No!o&r$(# are a s)ecia( t;)e o# !ra)h that enab(e ;ou toso(e com)(e, )rob(ems ino(in! more than one ariab(e.

1/2. 'ost nomo!ra)hs contain a !reat dea( o# in#ormation and re7uire the use o# sca(es on three sides o# the chart? as &e(( as dia!ona( (ines.

1/3. In #act? some charts contain so much in#ormation? that it can be er; im)ortant#or ;ou to care#u((; read the instructions be#ore usin! the chart and to sho& care

&hen readin! in#ormation #rom the chart itse(#.

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1/4. I((ustrated is a #air(; t;)ica( !ra)h o# three ariab(es? distance? s)eed and time.I# an; t&o o# the three ariab(es is 9no&n? the a))ro,imate a(ue o# the third can be7uic9(; determined. In this e,am)(e? the dotted (ine indicates a 9no&n s)eed andtime. The resu(tin! distance trae((ed can be e,tracted #rom the !ra)h at the )oint&here these t&o dashed (ines meet.

1/*. 5hi(st this nomo!ra)h is much too sma(( #or accurate com)utation? it can beseen that &hen trae((in! at around 2* 9nots #or three and a ha(# hours? ;ou &ou(dtrae( a (itt(e (ess than 1 nautica( mi(es.

Tri&ono!"tr:

1/-. +asic tri!onometr; ino(es e,)ressin! the an!(es o# a ri!ht>an!(ed trian!(e inre(ation to (en!ths o# the sides o# the trian!(e.

1//. The ratio o# the o))osite side (en!th to the h;)otenuse (en!th in the dia!ramis termed the HsineH o# the an!(e J.

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