School of Information and Communications Technology
ALBUKHARY INTERNATIONAL UNIVERSITY, MALAYSIA
Lecture Notes
CSM1023 MATHEMATICAL TECHNIQUES I
School of ICT, AiU
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2012
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
Phone: 04-774-7421
Consultation Hours:
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Who are U?
• Introduce briefly yourself:
–
–
–
–
Name
Home Country
Ambition
What do you Think about Mathematics?
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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. ?
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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.
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Overview of The Course
• Teaching Mode – Lecture and Practical Lab
• Assessment
– Homework
– Quizzes
– Assignments
5%
5%
20%
– Lab Work
15%
– Midterm Test
– Final Examination
– Total
15%
40%
100%
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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: PrenticeHall.
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
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Rules & Regulation
• All students must abide by 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 all the time e.g. respectful, honest,
responsible, polite, etc.
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Week 1
Session 1
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Why Study Mathematics?
• A significant proportion of computer science concepts
cannot
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
science depends on mathematics.
[Source: http://users.dickinson.edu/~jmac/selectedtalks/math-and-cs-talk.pdf]
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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.
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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.
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Computer Number Systems
• Expand the decimal number 645810:
6458=(6 x 103)+(4 x 102)+(5 x 101)+(8 x 100)
series of additions.
• Decimal place values
105
104
103
102
101
100
100,000
10,000
1,000
100
10
1
Tens
Ones
Hundred
Ten
Thousands Hundreds
Thousands Thousands
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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.
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Computer Number Systems
• Expand the binary 10102:
10102=( 1 x 23 )+( 0 x 22 )+( 1 x 21 )+( 0 x 20 )
series of additions.
• Binary place values
211
210
29
28
2048 1024 512 256
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26
128
64
15
25
24
23
22
21 20
32
16
8
4
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1
Computer Number Systems
• The octal system is a base-8 system.
• There are 8 distinct digits (0-7) to
represent any quantity.
• To express any number in base-8 we
use powers much like our own
decimal system.
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Computer Number Systems
• Expand the hexadecimal 1278:
1278=( 1 x 82 )+( 2 x 81 )+( 7 x 80 ) series of
additions.
• Octal place values
86
85
84
83
82
81
80
32,768
4096
512
64
8
1
262,144
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Computer Number Systems
• The hexadecimal system is a base-16
system.
• 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.
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Computer Number Systems
• Expand the hexadecimal 5AE716:
5AE716=( 5 x 163 )+( 10 x 162 )+( 14 x 161 )+
(7 x 160 ) series of additions.
• Hexadecimal place values
166
165
164
163
1,048,576
65536
4096
162 161 160
16,777,216
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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 +
=
1710
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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 )+ ( 1 x 24 )+ ( 1 x 23 )+ (0 x 22 ) +
( 0 x 21 )+ ( 1 x 20 )]
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Number Systems Conversion
• Converting octal to decimal, e.g. 5268
5268=( 5 x 82 )+( 2 x 81 ) )+( 6 x 80 )
= 320 + 16
+
6
= 3428
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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) + ( 7 x 81 )+ ( 1 x 80)]
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Number Systems Conversion
• Converting hexadecimal to decimal,
e.g. 70F16
70F16=( 7 x 162 )+( 0 x 161 ) )+( 15 x 160 )
= 1792 + 0
+
15
= 180710
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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) + ( 2 x 161 )+ ( 12 x 160)]
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Number Systems Conversion
• Converting binary to hexadecimal,
e.g. 1011012.
0010
11012
2
13
= 2D16
Hence, 1011012 = 2D16
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Number Systems Conversion
• Converting hexadecimal to binary, e.g.
8F16.
8
F16
1000
1111
= 100011112
Hence, 8F16 = 100011112
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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]
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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.
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Exponential Functions and
Inverse Relations
• The exponential functions 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.
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Exponential Functions and
Inverse Relations
• Problem: Graph f(x) = 2x.
• Solution: Plug in numbers for x and 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.
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Exponential Functions and
Inverse Relations
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Exponential Functions and
Inverse Relations
• The inverse functions of exponential functions is
called logarithmic functions.
• For example, the inverse of y = ax is 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.
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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.
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Exponential Functions and
Inverse Relations
• The general conversion between exponential
and logarithmic (including natural logarithmic,
ln) functions:
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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.
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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 + log2 6.
Solution: log2 (x * 6).
2. For any positive numbers x and p,
logaxp = p * loga x.
Example 5: Simplify: logb9-x.
Solution: -x * logb9.
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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 2nd rule.)
Use the 3rd rule in reverse yields loga (x/y5).
4. loga a = 1.
5. loga 1 = 0.
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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.
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Exponential Functions and
Inverse Relations
• To solve logarithmic equations, you convert them
to exponential form and solve for x.
• Example 8: Solve log3 (5x + 7) = 2 for x.
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.
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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.
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Binary and Natural Logarithms
• Plot of log2 n.
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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).
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Binary and Natural Logarithms
• Plot of logen = ln(n).
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Binary and Natural Logarithms
• The conversion of natural log into binary log.
• 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.
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THANK YOU
Q&A
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