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Mathematics Ability
CHENG Kai Ming
Department of Physics, CUHK
Time Allocation: 6 hours
1
Content
• Laws of Indices（指數）
– Some Properties of Indices
– Definition of Logarithms（對數）
– Some Properties of Logarithms
• Quadratic Equations（二次方程）
– Quadratic Formula
– Geometrical Interpretation
• Sequences（序列）
– Arithmetic Sequence（算術序列）
– Geometric Sequence（幾何序列）
2
• Probability（概率）and Statistics（統計）
– Measurements of Probability
– Addition Rule of Probability（概率加法定律）
– Multiplication Rule of Probability（概率乘法定律）
– Measure of Central Tendencies（集中趨勢）
– Measure of Deviations（偏差）
• Simple Geometry（幾何）
– Equation of Straight Line（直線方程）
– Equation of Parabola（拋物線方程）
• Simple Trigonometry（三角學）
3
• Appendix
– Functions and Inverse Functions（函數與反函
數）
– Logistic Function（邏輯函數）
– Surface Area & Volume of Solids
4
Laws of Indices
a a a
p
q
pq
a a a
p
a
q

p q
p q
 ab 
p
 a pb p
p
p
a
b

a
b
 
p
 a pq
5
Some Properties of Indices
a0  1
a p  1 a p
1q
 a
p q
 a
a
a
q
q
p
6
Proof of the properties
To prove:
a0  1
Let a 0  c.
Then
a p  c  a p  a0  a p
 a 0  1.
7
Proof of the properties
a p  1 a p
To prove:
Let a  p  d .
Then
a p  d  a p  a p  a0  1
 a p  1 a p .
8
Proof of the properties
 a
q
1q
To prove:
a
Let a1 q  f .
Then
f  a
q
 a
1q

1q q
a a
1
 a.
q
9
Proof of the properties
To prove:
a
p q
a
 a
p q
q
p
 a

p 1q
 ap .
q
10
Definition of Logarithms
The log of x to the base a, written as logax is the
value y such that ay = x, i.e.,
log a x  y  a y  x,
where a > 0 and a  1,
x > 0.
11
Particular Cases:
log a 1  ?
log a 1  0
log a  a m   ?
log a  a m   m
and
a
log a b
a
?
log a b
b
Common Logarithm（常用對數）(a = 10):
log10 x
or simply
log x
Natural Logarithm（自然對數）(a = e = 2.71828…):
loge x
or simply
ln x
12
Some Properties of Logarithms
log a xy  log a x  log a y
loga  x y   loga x  loga y
log a x n  n log a x
logb x
log a x 
logb a
13
Proof of the properties
To prove: log a xy  log a x  log a y
Let
log a x  u and log a y  v
u
v
a

x
a
y
Then
and
 xy  a a  a
u
v
u v
 log a xy  log a a
u v
 u  v  log a x  log a y
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Proof of the properties
To prove: log a x n  n log a x
Let
log a x  u
u
a
x
Then
 x  a
n

u n
a
nu
 log a x  log a a  nu  n log a x
n
nu
15
Proof of the properties
To prove: loga  x y   loga x  log a y
log a  x y   log a x  log a 1 y   log a x  log a y 1
 log a x  log a y
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Proof of the properties
logb x
To prove: log a x 
logb a
Let
log a x  u
u
a
x
Then
 log b a  log b x
u
log b x
 u log b a  log b x  u 
log b a
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Exercises
1
1. If log 2 12  3.58 , then what is log 2
?
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log 5 1000
2. Compute
.
log 5 100
18
Example (Applications)
A radioactive source（放射源）has half of its radioactive elements
（放射性元素）remained after every 5 years. After how many
years will the radioactive source has one-tenth of its radioactive
elements remained?
Let N0 be the amount of radioactive elements at the beginning.
1
1
N0  N0  
10
2
n

1 1
 
10  2 
n
1
1
 log10
 n log10     1  0.301n
10
2
 n  3.322
So it takes 5n = 16.6 years to remain one-tenth of its radioactive
elements.
19
Example (Applications)
For a certain chemical reaction（化學反應）, one uses a graph to
record the data where y-axis shows log5(D) where D is the reaction
rate（反應速率）while x-axis shows log5(CX ) where CX is the
concentration of the reactant（反應物）X and the resultant graph is a
straight line. From the line one calculates that the slope（斜率）of it
equals to 1.23 and the y-intercept equals to −0.85. Derive an
equation describing the relationship between the reaction rate D and
the concentration CX .
log5  D   0.85  1.23log5  C X 
 D  50.851.23log5 CX 
 D  50.85  51.23log5 CX 
 D  0.25  C X 
1.23
Slope - independent of the base used!
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Applications
Logarithm is used in the following measurements:
1 Strength of earthquakes (Ritcher scale - 黎克特制);
2 Loudness of sound (Decibel - 分貝);
3 Brightness of a star (Apparent magnitude - 視星等 ).
In the above, we are measuring the intensity（強度）of the
“waves” （波動）.
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Quadratic Equations
Consider a question:
There is a rectangular playground. The perimeter（周界）of the
playground is 28m and the area of it is 48m2. What is the dimension
(length and width) of this playground?
Let the length of the playground be x m. The width of the playground
is (14 − x)m and the area of it is x(14 − x)m2, which means
x 14  x   48
 x 2  14 x  48  0.
Quadratic Equation
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Quadratic Equations
Now consider the quadratic equation:
x  14 x  48  0.
2
We can factorize the LHS as
Factorization（因子分解）
x 14x  48   x  6 x  8 .
2
Therefore, x = 6 or x = 8. What should the true value of
x be?
6 and 8 are roots of the quadratic equation
23
Exercises
Find the roots of the following quadratic equations:
1. x2 + 2x − 15 = 0;
2. 9x2 − 6x + 1 = 0;
3. 2x2 + 3x − 4 = 0.
24
Quadratic formula
The roots（根）of the quadratic equation ax2 + bx + c = 0
(a  0) are given by
b  b2  4ac
.
2a
If b = 2d, the formula reduces to
d  d 2  ac
,
a
which is useful especially when b is an even number.
25
Nature of roots of quadratic equations
Consider the quadratic equation ax2 + bx + c = 0
and let  = b2 − 4ac.
1. When  > 0, there will be two real roots（實根）.
2. When  = 0, there will be two repeated real roots
（重覆實根）.
3. When  < 0, there will be no real roots.
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Geometrical interpretation
Consider: y = ax2 + bx + c, we have the following
geometric pictures for different values of :
y
y
O
x
y
O
x
Two real roots (y = 0)
One real roots (y = 0)
>0
=0
O
x
No real roots (y = 0)
<0
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Exercises
Find the roots of the following quadratic equations:
1. 2x2 + 3x − 4 = 0;
2. x2 − 8x + 16 = 0;
3. 9x2 − x + 1 = 0.
28
Sequences
What should the values x be in the following?
1. 20, 25, 30, 35, x, . . . ;
2. 100, 81, x, 43, 24, . . . .
It is easy to see that there are common differences（公差）
between the successive terms in both sequences. For sequence 1,
the common difference is 5 and hence x = 40. For sequence 2, the
common difference is -19 and hence x = 62.
29
Arithmetic Sequence
Definition: A sequence of numbers x1, x2, . . . , xn is said
to be an arithmetic sequence if
x2 − x1 = x3 − x2 = . . . = xn − xn−1 = d .
1. The value d is called the common difference of the
arithmetic sequence.
2. The sequence can then regarded as x1, x1 + d, x1 + 2d, . . . .
30
Sequences
What should the values y be in the following?
1. 16, 24, 36, 54, y, . . .;
2. 64, 32, y, 8, 4, . . ..
It is easy to see that there are common ratios（公比）between the
successive terms in both sequences. For sequence 1, the common
ratio is 3/2 and hence y = 81. For sequence 2, the common ratio is
1/2 and hence y = 16.
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Geometric Sequence
Definition: A sequence of numbers y1, y2, . . . , yn is said
to be a geometric sequence if
y2 y3


y1 y2
yn

 r.
yn 1
1. The value r is called the common ratio of the geometric
sequence.
2. The sequence can then regarded as y1, y1r, y1r2, . . . .
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But NOT every sequence is either
arithmetic or geometric!
What should the values x be in the following?
1) 4, 9, 16, 25, x, . . . ;
(x = 36)
2) 10, 17, 31, x, 115 . . . ;
(x = 59)
3) 1, 3, 4, 7, 11, 18, x, 47, . . . ;
(x = 29)
4) 8, 2, 24, 6, 40, x, 56, 14, . . . ;
(x = 10)
33
Probability
The probability of an event E describes the likelihood that
the event will occur, it is a number (denoted by P(E))
between 0 and 1 inclusively such that
1. P(E) = 0 if E is impossible to occur;
2. P(E) = 1 if E always occurs;
3. when P(E1) > P(E2), E1 is more likely to occur than E2.
34
Measurements of probability
Theoretical probability（理論概率）of an event
is the number of ways that the event can occur,
divided by the total number of outcomes.
Empirical probability（實驗概率∕經驗概論） is
an estimate that an event will happen based on how
often the event occurs after collecting data or
running an experiment (in a large number of trials).
35
Examples
1. Students who studied before examination have their
examinations passed with probability 7/8.
2. A pregnant woman gives birth to a boy with a
probability of 1/2.
3. To throw a dice, the number 1 is faced up with a
probability of 1/6.
4. P(the sun rises from the east) = 1.
36
Addition rule of probability
There are two events, with probabilities p1 and p2
respectively, that CANNOT happen together. The
probability that either of the two events happen is p1 + p2.
When throwing a dice, the probability of having 3 or
having a red number is 1/6 + 1/3 = 1/2.
What happens if the above question asks the probability of
having 4 or having a red number?
37
Addition rule of probability
Definition: Two events are said to be exclusive（不包含）if
the two events CANNOT happen together.
Which of the following events are exclusive?
1. Throw a dice: the event of having a red number and the
event of having an even number.
2. Choose a card from a deck: the event of getting a heart
and the event of getting a queen.
3. The event of the sun rises and the event of moon rises.
38
Multiplication rule of probability
There are two events, with probabilities p1 and p2
respectively, that the occurrence of the second event does
not depend on the occurrence of the first one. The
probability that the first event occurs and is followed by
the second event is p1p2.
There are 3 red balls, 5 blue balls and 2 purple balls in a
box. Now a girl draws a ball from the box, records the
colour and putting back the ball into the box, and draws a
ball again. The probability that she first draws a red ball
and then a blue one is 3/10 · 1/2 = 3/20.
39
Multiplication rule of probability
Definition: Two events are said to be independent（獨立）
if the occurrence of the first event does not affect the
probability of occurrence of the second event, and vice versa.
Which of the following events are independent?
1. Throw two dice separately, the first dice gives 5 while
the second dice gives a black number.
2. The event that today is cloudy and the event that today
is rainy.
3. A married couple first gives birth to a boy and then a
girl.
40
Measure of central tendencies
Given a set of numbers x1, x2, . . . , xn. We want to use a
value x to “characterize" or “represent" these numbers.
The followings give different ways to measure the “central
tendency"（集中趨勢）of a set of data.
1. Mean（平均值）: (x1 + x2 + . . . + xn)/n.
2. Median（中位數）:
x n 1
• If n is odd, then median: 2

1
• If n is even, then median:  xn  xn 
1
2 2
2 
3. Mode（眾數）: the value(s) that occur(s) most in x1, x2, . . . , 41xn.
Measure of deviations
Given a set of numbers x1, x2, . . . , xn. Let x be the mean
of the numbers. The standard deviation（標準偏離）
measures the “average differences" between the numbers
and the mean.
Let
s   x  x1    x  x2  
2
2
  x  xn .
2
s
Then  
.
n
42
Measure of deviations
The followings give other ways to measure the “average
differences (deviation)" of a set of data.
1. Mean deviation（平均偏差）:
x  x1  x  x2 
n
 x  xn
2. n-th tile（位數）(Suppose there are 100 values x1, x2, . . . , x100
in ascending/descending order):
• Quartile（四分位數）(Qm): Q1 = x25(?),
Q2 = median, Q3 = x75(?)
• Percentile（百分位數）(Pn): P10 = x10, P25 = x25 = Q1, …
3. Variance: 2.
43
Simple Geometry
y
Any point on the 2-dimensioal
plane（二維平面）can be
labeled by 2 real numbers as (x,
y). The ordered pair（序偶）
(x, y) are the coordinates（坐標）
of the point in the given
Cartesian coordinates（笛卡兒
O
坐標）.
P (x, y)
x
44
Equation of Straight Line
The equation of a straight line y
on the 2-dimensioal plane is
given by:
(0, a)
y = mx + a
Where a is the y-intercept（y截距）as shown and m is the
slope（斜率）of the line.
a
m  tan   
b

O
(b, 0)
x
45
Equation of Parabola
y
y
O
O
x
y  ax , a  0.
2
x
y  ax , a  0.
2
46
Equation of Parabola
y
y
O
O
x
x  by , b  0.
2
x
x  by , b  0.
2
47
Equation of Parabola
y
For a parabola with vertex
（頂點）at point (, ):
 y     a  x  
2
(, )

y    a  x  2 x  

y  ax  2a x  a   .
2
2
2

O
x
2
Note: The equation of a “tilted” parabola is more complicated.
48
Simple Trigonometry
Consider the right angle
triangle（直角三角形）
ABC as shown (angle
C = 90).
B
c
a
Since A + B + C = 180,
both A and B < 90.
C
A
b
49
Trigonometric Functions
B
Definitions:
a
b
sin A  , sin B  , sin C  1.
c
c
（正弦）
c
a
b
a
cos A  , cos B  , cos C  0.
c
（餘弦） c
sin A a
tan A 
 ,
（正切） cos A b
.
C
A
b
50
An important identity
cos   sin   1
2
2
51
Appendix
Functions and Inverse Functions
52
Exponential（指數）(ex) and
Natural Logarithm（自然對數）(lnx)
y=ex
y=lnx
53
Sine (sinx) and Arcsin（反正弦）(sin-1x)
y = sin-1x
y = sinx
/2
1
-2
-
0
-1

2
-1
0
1
-/2
54
Cosine (cosx) and Arccosine（反餘弦）
(cos-1x)
y = cos-1x
y = cosx

1
/2
-2
-
0
-1

2
-1
0
1
55
Tangent (tanx) and Arctangent（反正切）
(tan-1x)
y = tanx
y = tan-1x
56
Logistic Function
N t  
M
 M   rt
1  1 
e
 N0 
N
with M > N0 and r > 0.
t
Population Growth
57
Surface Area & Volume of Solids
Area = 4r2
Area =
Volume = (4r3)/3
2r2+2rh
Volume = r2h
Sphere（球形）
Area = r2+ rl
l
Cylinder（圓柱體）
Volume = (r2h)/3
Cone（圓錐體）
58