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School of Information and Communications Technology

School of Information and Communications Technology ALBUKHARY INTERNATIONAL UNIVERSITY, MALAYSIA. Lecture Notes CSM1023 MATHEMATICAL TECHNIQUES I. KNAR, October 2012. Who am I?. Mr. Khairul Najmy Abdul Rani Room 09-4-12, Level 4, School of ICT Building Email: najmy@aiu.edu.my

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School of Information and Communications Technology

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  1. School of Information and Communications Technology ALBUKHARY INTERNATIONAL UNIVERSITY, MALAYSIA Lecture Notes CSM1023 MATHEMATICAL TECHNIQUES I KNAR, October 2012

  2. Who am I? Mr. Khairul Najmy Abdul Rani Room 09-4-12, Level 4, School of ICT Building Email: najmy@aiu.edu.my Phone: 04-774-7421 Consultation Hours: KNAR, September 2012 KNAR, October 2012

  3. Who are U? • Introduce briefly yourself: • Name • Home Country • Ambition • What do you Think about Mathematics? KNAR, September 2012 KNAR, October 2012

  4. Students Mathematics and Related Software Knowledge • Please raise your hand, who has learnt about linear algebra or calculus? • Please raise your hand, who used to study probability and statistics? • Please raise your hand, who has learnt to use any mathematical software e.g. MATLAB® or SIMULINK® or SCILAB®or MATHEMATICA®, or MATHCAD® or SPSS® etc. ? KNAR, September 2012 KNAR, October 2012

  5. Overview of the Course • Course Name: Mathematical Techniques I • Course Learning Outcome At the end of the course, the student should be able to:- LO1: describe the application of given mathematical concepts or structures. LO2: demonstrate the basic principle of probability and statistics. LO3: illustrate the basics of linear algebra, number theory and their applications. LO4: use MATLAB® or equivalent tools for mathematical computation and solving problems. KNAR, September 2012 KNAR, October 2012

  6. Overview of The Course • Teaching Mode – Lecture and Practical Lab • Assessment • Homework 5% • Quizzes 5% • Assignments 20% • Lab Work 15% • Midterm Test 15% • Final Examination 40% • Total 100% KNAR, September 2012 KNAR, October 2012

  7. Reference Main references supporting the course 1. Fleming, W., Varberg, D., & Kasube, H. (1992). Algebra and Trigonometry: A Problem Solving Approach, (4th Ed). New Jersey: Prentice-Hall. 2. Patterson, B. A.S. (1991). Computer Related Mathematics. Oxford: NCC Blackwell Limited. 3. Kolman, B., & Hill, D. R. (2001). Introductory Linear Algebra with Applications, (7th Ed). New Jersey: Prentice-Hall. 4. Freund, J. E., & Perles, B. M. (2004).Statistics – A First Course, (8th Ed). New Jersey: Prentice-Hall. 5. Stark, H. M. (1978). An Introduction to Number Theory. Cambridge: MIT Press. Additional text references supporting the course 6. Montgomery, D. C., & Runger, G. C. (1999). Applied Statistics and Probability for Engineers, (2nd Ed). New York: John Wiley & Sons. 7. Munro, J. E. (1992). Discrete Mathematics. New Jersey: Prentice-Hall. 8. Perry, W. L. (1988). Elementary Linear Algebra. New York: McGraw-Hill. 9. Strang, G. (2006). Linear Algebra and its Applications, 4th Ed). Florence: Thomson Brooks/Cole. Not limited to the above list only KNAR, September 2012 KNAR, October 2012

  8. Rules & Regulation • All students must abideby the Student Code of Conduct as stipulated in Student Handbook • Specifically on : discipline, attendance/punctuality, plagiarism, and examination • Merit/demerit point system still applies • Late submissions of assignments/projects without a valid reason will be penalized Tests and Final Examination are compulsory • Being ethical allthe time e.g. respectful, honest, responsible, polite, etc. KNAR, September 2012 KNAR, October 2012

  9. Week 1Session 1 KNAR, September 2012 KNAR, October 2012

  10. Why Study Mathematics? • Asignificant proportion of computer science conceptscannot be described without sophisticated mathematics. • Studying mathematics is probably the best way of learning how to think logically and clearly, and this kind of thinking helps immensely when doing computer science. • In short, a lot of computing study, e.g. computer sciencedepends on mathematics. [Source: http://users.dickinson.edu/~jmac/selected-talks/math-and-cs-talk.pdf] KNAR, September 2012 KNAR, October 2012

  11. Conventional Number Systems • A number system is a set of rules and symbols used to represent a number. • Additive: Numbers have intrinsic value:, e.g.: Roman numerals: LVIII = 50 + 5 + 1 + 1 + 1 = 58 • Positional: Value depends on position: e.g.: Decimal system: 55 = 5 x 10 + 5 x 1 • Additive number systems are not used much anymore: • Awkward to use. • Prone to errors. KNAR, September 2012 KNAR, October 2012

  12. Computer Number Systems • The decimal system is a base-10 system. • There are 10 distinct digits (0 to 9) to represent any quantity. • For an n-digit number, the value that each digit represents depends on its weight or position. • The weights are based on powers of 10. KNAR, September 2012 KNAR, October 2012

  13. Computer Number Systems • Expand the decimal number 645810: 6458=(6x 103)+(4 x 102)+(5x 101)+(8x 100) series of additions. • Decimal place values KNAR, September 2012 KNAR, October 2012

  14. Computer Number Systems • The binary system is a base-2 system. • There are 2 distinct digits (0 and 1) to represent any quantity. • To express any number in base-2 we use powers much like our own decimal system. KNAR, September 2012 KNAR, October 2012

  15. Computer Number Systems • Expand the binary 10102: 10102=( 1 x 23)+( 0x 22)+( 1x 21)+( 0 x 20 ) series of additions. • Binary place values KNAR, September 2012 KNAR, October 2012

  16. Computer Number Systems • The octalsystem is a base-8system. • There are 8distinct digits (0-7) to represent any quantity. • To express any number in base-8 we use powers much like our own decimal system. KNAR, September 2012 KNAR, October 2012

  17. Computer Number Systems • Expand the hexadecimal 1278: 1278=( 1x 82 )+( 2x 81 )+( 7x 80 ) series of additions. • Octal place values KNAR, September 2012 KNAR, October 2012

  18. Computer Number Systems • The hexadecimalsystem is a base-16system. • There are 16 distinct digits (0-9, A(10)-F(15)) to represent any quantity. • To express any number in base-16 we use powers much like our own decimal system. KNAR, September 2012 KNAR, October 2012

  19. Computer Number Systems • Expand the hexadecimal 5AE716: 5AE716=( 5x 163 )+( 10x 162 )+( 14x 161 )+ (7 x 160 ) series of additions. • Hexadecimal place values KNAR, September 2012 KNAR, October 2012

  20. Number Systems Conversion • Converting binary to decimal, e.g. 100012. 100012 = (1 x 24) + ( 0 x 23 )+ ( 0 x 22 )+ ( 0 x 21 )+ ( 1 x 20 ) = 16 + 0 + 0 + 0 + 1 = 1710 KNAR, September 2012 KNAR, October 2012

  21. Number Systems Conversion • Converting decimal to binary, e.g. 8910. 89/2 = 44 remainder 1 44/2 = 22 remainder 0 22/2 = 11 remainder 0 11/2 = 5 remainder 1 5/2 = 2 remainder 1 2/2 = 1 remainder 0 Hence, 8910=10110012 [(1 x 26) + ( 0 x 25)+ ( 1x 24)+ ( 1x 23 )+ (0x 22 ) + ( 0x 21)+ ( 1 x 20 )] KNAR, September 2012 KNAR, October 2012

  22. Number Systems Conversion • Converting octal to decimal, e.g. 5268 5268=( 5x 82 )+( 2x 81 ) )+( 6x 80 ) = 320 + 16 + 6 = 3428 KNAR, September 2012

  23. Number Systems Conversion • Converting decimal to octal, e.g. 50510. 505/8 = 63 remainder 1 63/8 = 7 remainder 7 Hence, 50510=7718 [(7 x 82) + ( 7x 81)+ ( 1x 80)] KNAR, September 2012 KNAR, October 2012

  24. Number Systems Conversion • Converting hexadecimal to decimal, e.g. 70F16 70F16=( 7 x 162)+( 0x 161) )+( 15x 160 ) = 1792 + 0 + 15 = 180710 KNAR, September 2012

  25. Number Systems Conversion • Converting decimal to hexadecimal, e.g. 30010. 300/16 = 18 remainder 12 = C 18/16 = 1 remainder 2 Hence, 30010=12C16 [(1 x 162) + ( 2x 161)+ ( 12x 160)] KNAR, September 2012 KNAR, October 2012

  26. Number Systems Conversion • Converting binary to hexadecimal, e.g. 1011012. 0010 11012 213 = 2D16 Hence, 1011012=2D16 KNAR, September 2012 KNAR, October 2012

  27. Number Systems Conversion • Converting hexadecimal to binary, e.g. 8F16. 8 F16 10001111 = 100011112 Hence, 8F16=100011112 KNAR, September 2012 KNAR, October 2012

  28. Exponential Functions and Inverse Relations • The inverse of a function is the same as reflecting a function across the line y = x • Firstly, interchange x and y and then solve for y. • The inverse of f(x) is denoted by f-1(x). [Source: http://library.thinkquest.org/20991/alg2/log.html] KNAR, September 2012 KNAR, October 2012

  29. Exponential Functions and Inverse Relations • Example: Find f-1(x) of 3x + 1. • Solution:The equation is y = 3x + 1. Interchange x and y: x = 3y + 1. Solve for y: x - 1 = 3y. (x - 1)/3 = y. Then,f-1(x) = (x - 1)/3. KNAR, September 2012 KNAR, October 2012

  30. Exponential Functions and Inverse Relations • The exponentialfunctions are generally depicted as f(x) = ax + B where a is any real constant and B is any expression. • An example of an exponential function: f(x) = e-x - 1 • To graph exponential functions, remember that unless they are transformed, the graph will always pass through (0,1) and will approach, but not touch or cross the x-axis. KNAR, September 2012 KNAR, October 2012

  31. Exponential Functions and Inverse Relations • Problem: Graph f(x) = 2x. • Solution: Plug in numbers for xand find values for y, as done in the table below: ________________ | x:| 0 | 1 | 2 | 3 | --------------------------- | y:| 1 | 2 | 4 | 8 | --------------------------- • Now plot the points and draw the graph. KNAR, September 2012 KNAR, October 2012

  32. Exponential Functions and Inverse Relations KNAR, September 2012 KNAR, October 2012

  33. Exponential Functions and Inverse Relations • The inverse functions of exponential functions is called logarithmic functions. • For example, the inverse of y = axis y = logax, which is the same as x = ay. • Logarithms written without a base are understood to be base 10. • This definition is explained by knowing how to convert exponential equations to logarithmic form, and vice versa. KNAR, September 2012 KNAR, October 2012

  34. Exponential Functions and Inverse Relations • Examples 1: Convert to logarithmic form: 8 = 2x. Solution: Remember that the logarithm is the exponent: x = log28. • Example 2: Convert to exponential form: y = log35. Solution: Remember that the logarithm is the exponent: 3y= 5. KNAR, September 2012 KNAR, October 2012

  35. Exponential Functions and Inverse Relations • The general conversion between exponential and logarithmic (including natural logarithmic, ln) functions: KNAR, September 2012 KNAR, October 2012

  36. Exponential Functions and Inverse Relations • Sometimes you can solve equations containing logarithms by changing everything in logarithmic form to exponential form. • Example 3: Solve log2x = -3. Solution: Convert the logarithm to exponential form: 2-3= x, then, x = 1/23= 1/8. KNAR, September 2012 KNAR, October 2012

  37. Exponential Functions and Inverse Relations • There are five special rules that you ought to always have in mind when working with logarithms. 1.  For any positive numbers x and y, loga(x * y) = loga x + loga y when a <> 1.  Example 4: Simplify: log2 x + log26. Solution: log2 (x * 6). 2.  For any positive numbers x and p, logaxp= p * loga x.  Example 5: Simplify: logb9-x. Solution: -x * logb9. KNAR, September 2012 KNAR, October 2012

  38. Exponential Functions and Inverse Relations 3.  For any positive numbers x and y, loga(x/y) = logax - logay.   Example 6: Express as a single logarithm: logax - 5logay Solution: logax - logay5(Using the 2ndrule.) Use the 3rdrule in reverse yields loga (x/y5). 4.  loga a = 1. 5.  loga 1 = 0. KNAR, September 2012 KNAR, October 2012

  39. Exponential Functions and Inverse Relations • To solve for an exponential equation, take logarithms of both sides and use the listed five rules. • Example 7: Solve for x: 3x = 8. Solution: Take the logarithm of both sides: log 3x = log 8. Use theorem 2: x * log 3 = log 8. Solve for x by dividing each side by log 3: x = (log 8/log 3). A decimal approximation may be found if desired: x = 1.8929. KNAR, September 2012 KNAR, October 2012

  40. Exponential Functions and Inverse Relations • To solve logarithmic equations, you convert them to exponential form and solve for x. • Example 8: Solvelog3 (5x + 7) = 2 forx. Solution: Write an equivalent exponential expression: 5x + 7 = 32. 5x + 7 = 9. Solve for x: 5x = 9 – 7 = 2. Then, x = (2/5) = 0.4. KNAR, September 2012 KNAR, October 2012

  41. Binary and Natural Logarithms • The binary logarithm (log2 n) is the logarithm to the base 2. • Its inverse function: n ↦ 2n. • The binary logarithm of n is the power to which the number 2 must be raised to obtain the value n. • This makes the binary logarithm useful for anything involving powers of 2, i.e. doubling. KNAR, September 2012 KNAR, October 2012

  42. Binary and Natural Logarithms • Plot of log2n. KNAR, September 2012 KNAR, October 2012

  43. Binary and Natural Logarithms • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828. • The natural logarithm is generally written as ln(x), loge(x) or sometimes, if the base of e is implicit, as simply log(x). KNAR, September 2012 KNAR, October 2012

  44. Binary and Natural Logarithms • Plot of logen = ln(n). KNAR, September 2012 KNAR, October 2012

  45. Binary and Natural Logarithms • The conversion of natural log into binarylog. • Example 9: log2x = log x / log 2.On the right side, you can use logarithm in any base (calculators usually provide base-10 and base-e), just be sure to use the same base in both cases. Thus:log2x = logex / loge2 = ln x / ln 2. or:log2x = log10x / log102. KNAR, September 2012 KNAR, October 2012

  46. THANK YOU Q & A KNAR, September 2012

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