1. Analytical Toolbox Introduction to arithmetic & algebra By Dr J. Whitty

2. 2 Why study mathematics

3. 3 The Laws of Mathematics Asscociative Asscociative 3+(4+5)=(3+4)+5 3+(4+5)=(3+4)+5 a+(b+c)=(a+b)+c a+(b+c)=(a+b)+c a(bc)=(ab)c a(bc)=(ab)c Commutative Commutative 4+5=5+4 4+5=5+4 a+b=b+a a+b=b+a ab+=ba ab+=ba Distributive Distributive 3(4+5)=3x4+4x5 3(4+5)=3x4+4x5 a(b+c)=ab+ac a(b+c)=ab+ac (a+b)/c=a/c+b/c (a+b)/c=a/c+b/c

4. 4 Module 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.

5. 5 Assessment Methods Assignment 1: Assignment 1: Introductory mathematical statistics Introductory mathematical statistics Milestone test 1: Milestone test 1: Mock examination mathematics questions Mock examination mathematics questions Assignment 2: Assignment 2: Introductory mathematical engineering systems modelling Introductory mathematical engineering systems modelling Milestone test 2: Milestone test 2: Three mock-exam science questions Three mock-exam science questions Milestone test 3: Milestone test 3: Computer modelling methods tutorial Computer modelling methods tutorial Examination (informal) Examination (informal)

6. 6 Recommended Reading Engineering mathematics KE Stroud & DJ Booth Engineering mathematics KE Stroud & DJ Booth Palgrave macmillian, Fifth Edition, London, 2001

7. 7 Session Learning objectives After this session you should be able to: After this session you should be able to: Compute numeric expressions using BIDMAS Compute numeric expressions using BIDMAS Evaluate numeric expressions in standard form Evaluate numeric expressions in standard form Transpose simple formulae Transpose simple formulae Solve simple linear equations inc those involving fractions Solve simple linear equations inc those involving fractions Solve elementary non-linear equations inc. squares and square roots Solve elementary non-linear equations inc. squares and square roots

8. 8 BIDMAS Can you remember what this stands for? Can you remember what this stands for? Brackets Brackets Indices Indices Division Division Multiplication Multiplication Additon Additon Subtraction Subtraction 1.2a+5b+3c  2x3 + 5x(-1) +3x2  =7 2.6a-7b  6x3 - 7x(-1) =  18 - -7 =25 3.2a 2 - 3b 2  2xaxa - 3xbxb  2x3x3 -3x(-1)x(-1)  =15 Let a=3, b=-1, c=2

9. 9 Remove 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+30b =8a-12b-5a+30b =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

10. 10 Try some yourselves

11. 11 Indices Consider 2 6 Consider 2 6 The 2 is called the base, the 6 is the power or 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 = 64 The 2 is called the base, the 6 is the power or 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 = 64 On the calculator press 2 x y 6 = On the calculator press 2 x y 6 = 2 6 DOES NOT EQUAL 12 2 6 DOES NOT EQUAL 12

12. 12 Negative Indices Consider 2 - 6 Consider 2 - 6 The rule to eliminate the negative power is : write 1 divided by the old base, to the positive power The rule to eliminate the negative power is : write 1 divided by the old base, to the positive power This is equal to This is equal to

13. 13 Fractional Indices 4 0.5 is the same as the square root of 4 4 0.5 is the same as the square root of 4 4 1/8 is the same as the eighth root of 4 and is again calculated using the x y button on the calculator (= 1.1892) 4 1/8 is the same as the eighth root of 4 and is again calculated using the x y button on the calculator (= 1.1892)

14. 14 Rules of Indices If two powers are multiplied, then if the bases are the same, the powers are ADDED If two powers are multiplied, then if the bases are the same, the powers are ADDED 2 5 x 2 7 = 2 12 2 5 x 2 7 = 2 12 If two powers are divided, then if the bases are the same, the powers are SUBTRACTED If two powers are divided, then if the bases are the same, the powers are SUBTRACTED 3 8  3 5 = 3 3 3 8  3 5 = 3 3

15. 15 Examples on Indices 47  48 =47  48 =47  48 =47  48 = 4 -1 = 1/4 5 -2 x 5 - 4 = 5 -6 = 1/15625 3 -3  3 -1 = 3 -2 =1/9 5 4 x 5 2  5 -3 = 5 9 = 1953125 10 0 = 1 9 0 =1a 0 = 1 3 -7  3 -5 x 27= 3

16. 16 Standard Form Standard Form is used to write very large or very small numbers in a more convenient way Standard Form is used to write very large or very small numbers in a more convenient way A number in standard form is written: a x 10 n A number in standard form is written: a x 10 n where a is a number between 1 and 10, and n is an integer, positive or negative where a is a number between 1 and 10, and n is an integer, positive or negative

17. 17 Standard & Engineering Form examples 2,000,000 = 2x10 6 35,800 = 3.58x10 4 (35.8 x10 3 ) 5,927,000,000 = 5.927x10 9 (5.9x10 9 ) 43 = 4.3 x 10 1 (43) 0.000 004 = 4x10 -6 (4x10 -6 ) 0.000 458 = 4.58x10 - 4 (458x10 -6 ) 0.000 000 000 021 = 2.1x10 -11 (21x10 -12 ) 0.35 = 3.5x10 -1 (350x10 -3 )

18. 18 Using a calculator To calculate 3.98 x 10 12 x 4.2 x 10 11 press the following keys: 3.98 exp 12 x 4.2 exp 11 = Do NOT type x10, the exp button does this automatically. The answer should appear as 1.6716 24 on screen. The correct answer is then 1.6716 x 10 24

19. 19 Evaluate, in standard form 3.2 x 10 6 x 4 x 10 4 = 1.28x10 11 = 1.28x10 11 4.5 x 10 16 + 4 x 10 15 =4.9x10 16 2.8 x10 27 x 3.5x 10 -25 = 980 = 980 = 9.8x10 2 4.4 x10 -10 x 5.2x10 -4 = 2.288x10 -13 5 x 10 -8  2 x 10 -7 = 0.25 2.5x10 -1

20. 20 Rules of Indices

21. 21 Corollaries

22. 22 Let’s use them:

23. 23 They still work algebraically!

24. 24 Recap: Make x the subject Equation: 3x+2 = 23 3x+2 = 23 3x+2 - 2= 23 -2 3x = 23 - 2 x = (23-2)/3 x=7 Formula: gx + h = k gx + h = k gx + h - h = k - h gx = k- h x = (k - h)/g Last lecture we examined the differences between equations and formulae and their subsequent solution protocols:

25. 25 Transposition Of Formulae The rules are exactly the same as for algebra, except the final result is an algebraic expression instead of a numerical answer. The rules are exactly the same as for algebra, except the final result is an algebraic expression instead of a numerical answer.

26. 26 Simple Transposition In the Science units you will come across very simple formulae for instance In the Science units you will come across very simple formulae for instance Density Density Newton’s second law (mechanics) Newton’s second law (mechanics) Electrical charge Electrical charge Ohms Law Ohms Law

27. 27 Simple Transposition Here the same rules apply as the letters in the formulae are just numbers in disguise Here the same rules apply as the letters in the formulae are just numbers in disguise

28. 28 Activity In groups use the systematic transposition (or otherwise) approach to develop calculation transposition triangles for the formulae, describing: In groups use the systematic transposition (or otherwise) approach to develop calculation transposition triangles for the formulae, describing: Density Density Electrical Charge Electrical Charge Voltage Voltage In each case define each of the variables you have used

29. 29 Solution of Linear Equations Mathematical Mathematical 6x+1=2x+9 6x+1=2x+9 6x+1-1=2x+9-1 6x+1-1=2x+9-1 6x=2x+8 6x=2x+8 6x-2x =2x-2x+8 6x-2x =2x-2x+8 4x=8 4x=8 4x/4=8/4 =2 4x/4=8/4 =2 Systematic Systematic 6x+1=2x+9 6x+1=2x+9 6x-2x= +9-1 6x-2x= +9-1 4x=8 4x=8 x=8/4=2 x=8/4=2 Note: The approaches are exactly the same however in the systematic approach we MOVE numbers & variables

30. 30 What about brackets? The rule with brackets is simply MULTIPLY (that’s all they mean). The rule with brackets is simply MULTIPLY (that’s all they mean). MULTIPLY everything inside the bracket by everything outside the bracket. MULTIPLY everything inside the bracket by everything outside the bracket. Consider: Consider: 3(x-2)=9 3(x-2)=9 3x-6=9 3x-6=9 Solve as before Solve as before 4(2r-3)-2(r-4)=3(r-3)-1 4(2r-3)-2(r-4)=3(r-3)-1 8r-12-2r+8=3r-9-1 8r-12-2r+8=3r-9-1 Solve as before Solve as before Exercise Exercise

31. 31 Class Discussion/Exercise 2x+5=7 2x+5=7 2c/3-1=3 2c/3-1=3 7-4p=2p-3 7-4p=2p-3 8-3t=2 8-3t=2 2x-1=5x+11 2x-1=5x+11 2a+6-5a=0 2a+6-5a=0 3x-2-5x=2x-4 3x-2-5x=2x-4 20d-3+3d=11d+5-8 20d-3+3d=11d+5-8 2(x-1)=4 2(x-1)=4 16=4(t+2) 16=4(t+2) 5(f-2)=3(2f+5)-15 5(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=0 2(g-5)-5=0 4(3x+1)=7(x+4)-2(x+5) 4(3x+1)=7(x+4)-2(x+5)

32. 32 Linear Equations with Fractions The systematic method is especially useful with fractional coefficients: The systematic method is especially useful with fractional coefficients: e.g. e.g. Mathematical Approach

33. 33 Linear Equations with Fractions There are several methods to attempt such problems but by far the best is to attempt to clear the fractions (some how) in order to reduce the equation to something simpler which can be solved There are several methods to attempt such problems but by far the best is to attempt to clear the fractions (some how) in order to reduce the equation to something simpler which can be solved Consider: (math) Consider: (math)

34. 34 Linear Equations with Fractions This is the so called mathematical approach This is the so called mathematical approach There is the analogous systematic approach, which most engineers find a little easier to apply There is the analogous systematic approach, which most engineers find a little easier to apply Thus: Thus:

35. 35 Class Exercise 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6.

36. 36 Squares and their roots Proceed as before but at the end to get rid of a square root simply square everything Proceed as before but at the end to get rid of a square root simply square everything Likewise: What About:

37. 37 Squares and their roots A similar approach can be applied to squares and powers, thus A similar approach can be applied to squares and powers, thus What about

38. 38 Examples

39. 39 Summary Have we met our learning objectives, specifically, are you able to: Have we met our learning objectives, specifically, are you able to: Compute numeric expressions using BIDMAS Compute numeric expressions using BIDMAS Evaluate numeric expressions in standard form Evaluate numeric expressions in standard form Derive the rules of indices from first principles Derive the rules of indices from first principles Evaluate and simplify mathematical expressions using the rules of indices. Evaluate and simplify mathematical expressions using the rules of indices.

40. 40 Homework Evaluate or simplify the following Evaluate or simplify the following

41. 41 More homework 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6.

42. 42 Further Study Foundation topics Foundation topics F1 Arithmetic F1 Arithmetic F2 Introduction to algebra F2 Introduction to algebra