analytical toolbox introduction to arithmetic & algebra by dr j. whitty
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Analytical ToolboxAnalytical Toolbox
Introduction to arithmetic & algebraIntroduction to arithmetic & algebraByBy
Dr J. WhittyDr J. Whitty
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Why study mathematicsWhy study mathematics
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The Laws of MathematicsThe Laws of Mathematics
AsscociativeAsscociative 3+(4+5)=(3+4)+53+(4+5)=(3+4)+5 a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c a(bc)=(ab)ca(bc)=(ab)c
CommutativeCommutative 4+5=5+44+5=5+4 a+b=b+aa+b=b+a ab+=baab+=ba
DistributiveDistributive 3(4+5)=3x4+4x53(4+5)=3x4+4x5 a(b+c)=ab+aca(b+c)=ab+ac (a+b)/c=a/c+b/c(a+b)/c=a/c+b/c
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Module Leaning ObjectivesModule Leaning Objectives
To achieve this unit a learner must:
1. Determine the fundamental algebraic laws and apply algebraic manipulation techniques to the solution of problems involving algebraic functions, formulae and graphs
2. Use trigonometric ratios, trigonometric techniques and graphical methods to solve simple problems involving areas, volumes and sinusoidal functions
3. Use statistical methods to gather, manipulate and display scientific and engineering data
4. Use the elementary rules of calculus arithmetic to solve problems that involve differentiation and integration of simple algebraic and trigonometric functions.
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Assessment MethodsAssessment Methods
Assignment 1: Assignment 1: Introductory mathematical statisticsIntroductory mathematical statistics
Milestone test 1: Milestone test 1: Mock examination mathematics questionsMock examination mathematics questions
Assignment 2: Assignment 2: Introductory mathematical engineering systems modellingIntroductory mathematical engineering systems modelling
Milestone test 2:Milestone test 2: Three mock-exam science questionsThree mock-exam science questions
Milestone test 3: Milestone test 3: Computer modelling methods tutorialComputer modelling methods tutorial
ExaminationExamination (informal) (informal)
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Recommended ReadingRecommended Reading
Engineering Engineering mathematics KE mathematics KE Stroud & DJ BoothStroud & DJ Booth
Palgrave Palgrave macmillian,macmillian,
Fifth Edition, Fifth Edition, London, 2001London, 2001
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Session Learning objectivesSession Learning objectives
After this session you should be able to:After this session you should be able to: Compute numeric expressions using BIDMASCompute numeric expressions using BIDMAS Evaluate numeric expressions in standard form Evaluate numeric expressions in standard form Transpose Transpose simplesimple formulae formulae Solve simple linear equations inc those involving Solve simple linear equations inc those involving
fractionsfractions Solve elementary non-linear equations inc. squares Solve elementary non-linear equations inc. squares
and square rootsand square roots
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BIDMAS BIDMAS Can you remember Can you remember
what this stands for?what this stands for? BracketsBrackets IndicesIndices DivisionDivision MultiplicationMultiplication AdditonAdditon SubtractionSubtraction
1.1. 2a+5b+3c2a+5b+3c 2x3 + 5x(-1) +3x22x3 + 5x(-1) +3x2 =7=7
2.2. 6a-7b6a-7b 6x3 - 7x(-1) =6x3 - 7x(-1) = 18 - -718 - -7 =25=25
3.3. 2a2a22 - 3b - 3b22
2xaxa - 3xbxb2xaxa - 3xbxb 2x3x3 -3x(-1)x(-1)2x3x3 -3x(-1)x(-1) =15=15
Let a=3, b=-1, c=2Let a=3, b=-1, c=2
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Remove the bracketsRemove the brackets
3(a+b)3(a+b) =3a + 3b=3a + 3b
5(2a-5c)5(2a-5c) =10a-25c=10a-25c
4(2a-3b)-5(a-6b)4(2a-3b)-5(a-6b) =8a-12b-5a=8a-12b-5a++30b30b =3a+18b=3a+18b
6c-(3a+2b-5c)6c-(3a+2b-5c) =6c-3a-3b+5c=6c-3a-3b+5c =11c-3a-3b=11c-3a-3b
4(5a-b)-2(3a-b-c)4(5a-b)-2(3a-b-c) =20a- 4b-6a+2b+2c=20a- 4b-6a+2b+2c =14a-2b+2c=14a-2b+2c
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Try some yourselvesTry some yourselves
4);2)(2)(( xxxa
10,98,7
13;2)(
gl
g
lb
5.1,3);23)(12)(( xxxc
1,3;))(( 2 qpqpd
yxyxyxe 3;32)32)(( 2222
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IndicesIndices
Consider 2Consider 26 6
The 2 is called the base, the 6 is the power or The 2 is called the base, the 6 is the power or the index. The above is said “2 to the power the index. The above is said “2 to the power of 6” and is calculated 2 x 2 x 2 x 2 x 2 x 2 = of 6” and is calculated 2 x 2 x 2 x 2 x 2 x 2 = 6464
On the calculator press 2 x On the calculator press 2 x y y 6 =6 = 226 6 DOES NOT EQUAL 12DOES NOT EQUAL 12
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Negative IndicesNegative Indices
Consider 2 Consider 2 -- 66
The rule to eliminate the negative power is : The rule to eliminate the negative power is : write 1 divided by the old base, to the positive write 1 divided by the old base, to the positive powerpower
This is equal to This is equal to
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Fractional IndicesFractional Indices
440.50.5 is the same as the square root of 4 is the same as the square root of 4 4 4 1/8 1/8 is the same as the eighth root of 4 and is is the same as the eighth root of 4 and is
again calculated using the x again calculated using the x y y button on the button on the calculator (= 1.1892)calculator (= 1.1892)
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Rules of IndicesRules of Indices
If two powers are multiplied, then if the bases If two powers are multiplied, then if the bases are the same, the powers are ADDEDare the same, the powers are ADDED
2255 x 2 x 277 = 2 = 21212
If two powers are divided, then if the bases If two powers are divided, then if the bases are the same, the powers are SUBTRACTEDare the same, the powers are SUBTRACTED
3388 3 355 = 3 = 333
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Examples on IndicesExamples on Indices
4477 4 488 ==
44-1-1 = 1/4 = 1/4
55-2-2 x 5 x 5- 4- 4 = =
5 5 -6-6 = =
1/156251/15625
33-3-3 3 3-1-1 = =
33-2-2=1/9=1/9
5544 x 5 x 522 5 5-3-3 = =
5599 = 1953125 = 1953125
101000==
11
9900==11 aa00 = = 11
33-7-7 3 3-5-5 x 27= x 27=
33
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Standard FormStandard Form
Standard Form is used to write very large or Standard Form is used to write very large or very small numbers in a more convenient wayvery small numbers in a more convenient way
A number in standard form is written: A number in standard form is written: a x 10 a x 10 nn
where a is a number between 1 and 10, and where a is a number between 1 and 10, and n is an integer, positive or negativen is an integer, positive or negative
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Standard & Engineering Form examplesStandard & Engineering Form examples
2,000,000 2,000,000
= 2x10 = 2x10 66
35,800 35,800
= 3.58x10 = 3.58x10 4 4 (35.8 x10(35.8 x1033))
5,927,000,000 5,927,000,000
= 5.927x10= 5.927x1099 (5.9x10(5.9x1099))
43 43
= 4.3 x 10 = 4.3 x 10 11 (43)(43)
0.000 0040.000 004
= 4x10= 4x10-6-6 (4x10(4x10-6-6))
0.000 4580.000 458
= 4.58x10= 4.58x10- 4- 4 (458x10(458x10-6-6))
0.000 000 000 0210.000 000 000 021
= 2.1x10= 2.1x10-11-11 (21x10(21x10-12-12))
0.350.35
= 3.5x10= 3.5x10-1 -1 (350x10(350x10-3-3))
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Using a calculatorUsing a calculator
To calculate 3.98 x 10 To calculate 3.98 x 10 1212 x 4.2 x 10 x 4.2 x 10 1111
press the following keys:press the following keys:
3.98 exp 12 x 4.2 exp 11 =3.98 exp 12 x 4.2 exp 11 =
Do NOT type Do NOT type x10x10, the exp button does this , the exp button does this automatically. automatically.
The answer should appear as The answer should appear as 1.6716 1.6716 2424 on on screen.screen.
The correct answer is then The correct answer is then 1.6716 x 10 1.6716 x 10 2424
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Evaluate, in standard formEvaluate, in standard form
3.2 x 10 3.2 x 10 66 x 4 x 10 x 4 x 10 44
= 1.28x10 = 1.28x10 1111
4.5 x 10 4.5 x 10 1616 + 4 x 10 + 4 x 10 1515
=4.9x10 =4.9x10 1616
2.8 x10 2.8 x10 2727 x 3.5x 10x 3.5x 10-25-25
= 980= 980
= 9.8x10 = 9.8x10 22
4.4 x10 4.4 x10 -10-10 x 5.2x10 x 5.2x10 -4-4
= 2.288x10 = 2.288x10 -13-13
5 x 10 5 x 10 -8 -8 2 x 10 2 x 10 -7-7
= 0.25= 0.25
2.5x10 2.5x10 -1-1
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Rules of IndicesRules of Indices
mnmn xxx
mnmnm
n
xxxx
x /
nmmn xx
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CorollariesCorollaries
11 0 xxx
x nnn
n
mnm n xx /
10 x
mn
m xxx nm n 1
nn
xx
1
nnnn
xx
xxx
100
111
1 :NB xxx
xxxx nn
n
n
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Let’s use them:Let’s use them:
4
32
3
333)(
a )2)(6)(( 0xb
21
)36)(( c 43
)16)(( d
4
3223
3
32)(
e
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They still work algebraically!They still work algebraically!
yx
yxa
2
23
4
12)(
2
2
4
423
)(c
a
bca
cbab
202323)( acabcbc
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Recap: Make x the subjectRecap: Make x the subject
Equation:Equation:
3x+2 = 233x+2 = 23
3x+23x+2 - 2= 23= 23 -2
3x = 23 - 23x = 23 - 2
x = (23-2)/3x = (23-2)/3
x=7x=7
Formula:Formula:
gx + h = kgx + h = k
gx + hgx + h - h = k= k - h
gx = k- hgx = k- h
x = (k - h)/gx = (k - h)/g
Last lecture we examined the differences between equations and formulae and their subsequent solution protocols:
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Transposition Of FormulaeTransposition Of Formulae
The rules are exactly the same as for The rules are exactly the same as for algebra, except the final result is an algebraic algebra, except the final result is an algebraic expression instead of a numerical answer.expression instead of a numerical answer.
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Simple TranspositionSimple Transposition
In the Science units you In the Science units you will come across very will come across very simple formulae for simple formulae for instance instance DensityDensity Newton’s second law Newton’s second law
(mechanics)(mechanics) Electrical chargeElectrical charge Ohms LawOhms Law
maF
It
Q
IRV
V
m
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Simple TranspositionSimple Transposition
Here the same Here the same rules apply as the rules apply as the letters in the letters in the formulae are just formulae are just numbers in numbers in disguise disguise
maF
am
F m
a
F
Fm a
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ActivityActivity
In groups use the systematic transposition (or In groups use the systematic transposition (or otherwise) approach to develop calculation otherwise) approach to develop calculation transposition triangles for the formulae, describing:transposition triangles for the formulae, describing: DensityDensity Electrical ChargeElectrical Charge VoltageVoltageIn each case define each of the variables you have In each case define each of the variables you have
usedused
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Solution of Linear EquationsSolution of Linear Equations
MathematicalMathematical 6x+1=2x+96x+1=2x+9
6x+16x+1-1-1=2x+9=2x+9-1-1 6x=2x+86x=2x+8 6x6x-2x-2x =2x =2x-2x-2x+8+8 4x=84x=8 4x4x/4/4=8=8/4 /4 =2=2
SystematicSystematic 6x+1=2x+96x+1=2x+9 6x6x-2x-2x= +9= +9-1-1 4x=84x=8 x=8x=8/4/4=2=2
Note: The approaches are exactly the same however in the systematic approach we MOVE numbers & variables
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What about brackets?What about brackets?
The rule with The rule with brackets is simply brackets is simply MULTIPLY (that’s all MULTIPLY (that’s all they mean).they mean).
MULTIPLY MULTIPLY everything inside the everything inside the bracket by everything bracket by everything outside the bracket.outside the bracket.
Consider:Consider: 3(x-2)=93(x-2)=9
3x-6=93x-6=9 Solve as beforeSolve as before
4(2r-3)-2(r-4)=3(r-3)-14(2r-3)-2(r-4)=3(r-3)-1 8r-12-2r8r-12-2r++8=3r-9-18=3r-9-1 Solve as beforeSolve as before
ExerciseExercise
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Class Discussion/ExerciseClass Discussion/Exercise
2x+5=72x+5=7 2c/3-1=32c/3-1=3 7-4p=2p-37-4p=2p-3 8-3t=28-3t=2 2x-1=5x+112x-1=5x+11 2a+6-5a=02a+6-5a=0 3x-2-5x=2x-43x-2-5x=2x-4 20d-3+3d=11d+5-820d-3+3d=11d+5-8
2(x-1)=42(x-1)=4 16=4(t+2)16=4(t+2) 5(f-2)=3(2f+5)-155(f-2)=3(2f+5)-15 2x=4(x-3)2x=4(x-3) 6(2-3y)-42=-2(y-1)6(2-3y)-42=-2(y-1) 2(g-5)-5=02(g-5)-5=0 4(3x+1)=7(x+4)-2(x+5)4(3x+1)=7(x+4)-2(x+5)
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Linear Equations with FractionsLinear Equations with Fractions
The systematic method The systematic method is especially useful with is especially useful with fractional coefficients:fractional coefficients:
e.g.e.g.
65
2x
65
2x
562 x
152
56
x
5.262
56
2
5
5
2
xMathematical
Approach
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Linear Equations with FractionsLinear Equations with Fractions
There are several There are several methods to attempt methods to attempt such problems but by such problems but by far the far the bestbest is to is to attempt to clear the attempt to clear the fractions (some how) fractions (some how) in order to reduce the in order to reduce the equation to something equation to something simpler which can be simpler which can be solvedsolved
Consider: Consider: (math)(math)
2
3
20
15
4
3
5
2 yy
2
320
20
120520
4
320
5
220
yy
yy 301100158
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Linear Equations with FractionsLinear Equations with Fractions
This is the so called This is the so called mathematical mathematical approachapproach
There is the analogous There is the analogous systematic approach, systematic approach, which most engineers which most engineers find a little easier to find a little easier to applyapply
Thus:Thus:
2
3201520
4
320
5
220 yy
2
3
20
15
4
3
5
2 yy
100151308 yy
11438 y
3y
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Class ExerciseClass Exercise
1.1.
2.2.
3.3.
4.4.
5.5.
6.6.
6
5
2
31
4
32 yy
2
1514
3
1x
2
1
53xx
2
1312
4
1x
2
3
5
6
4
xxx
343/ d
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Squares and their rootsSquares and their roots
Proceed as before but at the end to get rid of Proceed as before but at the end to get rid of a square root simply square everythinga square root simply square everything
2x 22 x
44 22 xx55 33 xx
Likewise:82 xWhat About:
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Squares and their rootsSquares and their roots
A similar approach can be applied to A similar approach can be applied to squares and powers, thussquares and powers, thus
252 x3
2
4
152
tWhat about
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ExamplesExamples
52 y
61
3
x
x
1
2510
x
916
2t
3
26 a
a
2
85
2
11
x
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SummarySummary
Have we met our learning objectives, Have we met our learning objectives, specifically, are you specifically, are you able to:able to: Compute numeric expressions using BIDMASCompute numeric expressions using BIDMAS Evaluate numeric expressions in standard formEvaluate numeric expressions in standard form Derive the rules of indices from first principlesDerive the rules of indices from first principles Evaluate and simplify mathematical expressions Evaluate and simplify mathematical expressions
using the rules of indices.using the rules of indices.
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HomeworkHomework
Evaluate Evaluate or or simplify the followingsimplify the following
3
42
7
77)(
a
2
2
6
66)(
b 2
1
5.0)( c1
7
3)(
d 02876)(e 9
3
6
30)(x
xf
26 12/36)( xxg 26 34)( xxh 26 34)( xxi
063)( xk 7
7
10
5)(x
xl 255)( xn
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More homeworkMore homework
1.1. 2.2.
3.3.
4.4.
5.5.
6.6.
5
4
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Further StudyFurther Study Foundation topics Foundation topics
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