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Mathematics Ability

CHENG Kai Ming

Department of Physics, CUHK

Time Allocation: 6 hours

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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）

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• 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）

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• Appendix– Functions and Inverse Functions（函數與反函數）

– Logistic Function（邏輯函數）– Surface Area & Volume of Solids

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Laws of Indices

p q p qa a a

p q p qa a a

qp pqa a

p p pab a b

p p pa b a b

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Some Properties of Indices

0 1a

1p pa a

1 qqa a

qp q pa a

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Proof of the properties

Let 0 .a c

Then 0p p pa c a a a

0 1.a

To prove: 0 1a

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Proof of the properties

Let .pa d

Then 0 1p p pa d a a a

1 .p pa a

To prove: 1p pa a

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Proof of the properties

Let 1 .qa f

Then 1 1qq qf a a a

1 .qqa a

To prove: 1 qqa a

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Proof of the properties

1.

q qp q p pa a a

To prove: qp q pa a

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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 ,ya x y a x

where a > 0 and a 1, x > 0.

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Particular Cases:

log 1 ?a

log ?ma a

log 1 0a

log ma a m

Common Logarithm（常用對數） (a = 10):

10log x log xor simply

Natural Logarithm（自然對數） (a = e = 2.71828…):

loge x ln xor simply

and

log ?a ba loga ba b

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Some Properties of Logarithms

log log loga a axy x y

log log loga a ax y x y

log logna ax n x

loglog

logb

ab

xx

a

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Proof of the properties

log log loga a axy x y

Let

Then

u v u vxy a a a

To prove:

loga x u and loga y v

ua x andva y

log log log logu va a a axy a u v x y

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Proof of the properties

Let

Then

nn u nux a a

To prove:

loga x u

ua x

log log logn nua a ax a nu n x

log logna ax n x

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Proof of the properties

To prove: log log loga a ax y x y

1log log log 1 log log

log loga a a a a

a a

x y x y x y

x y

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Proof of the properties

Let

Then

log logub ba x

To prove:

loga x u

ua x

loglog log

logb

b bb

xu a x u

a

loglog

logb

ab

xx

a

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Exercises

1. If , then what is ?2log 12 3.58 2

1log

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2. Compute . 5

5

log 1000

log 100

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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.

0 0

10 10

1 1 1 1

10 2 10 2

1 1log log 1 0.301

10 2

3.322

n n

N N

n n

n

So it takes 5n = 16.6 years to remain one-tenth of its radioactive elements.

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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 .

5 5

5 5

0.85 1.23log 1.23log0.85

1.23

log 0.85 1.23log

5 5 5

0.25

X X

X

C C

X

D C

D D

D C

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

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14 48

14 48 0.

x x

x x

Quadratic Equation

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Quadratic Equations

Now consider the quadratic equation:

2 14 48 0.x x

We can factorize the LHS as

2 14 48 6 8 .x x x x

Therefore, x = 6 or x = 8. What should the true value of x be?

6 and 8 are roots of the quadratic equation

Factorization（因子分解）

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Exercises

Find the roots of the following quadratic equations:

1. x2 + 2x − 15 = 0;

2. 9x2 − 6x + 1 = 0;

3. 2x2 + 3x − 4 = 0.

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Quadratic formula

The roots（根） of the quadratic equation ax2 + bx + c = 0 (a 0) are given by

2 4.

2

b b ac

a

If b = 2d, the formula reduces to

2

,d d ac

a

which is useful especially when b is an even number.

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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

xO

y

xO

y

xO

y

Consider: y = ax2 + bx + c, we have the following geometric pictures for different values of :

> 0 = 0 < 0

Two real roots (y = 0) One real roots (y = 0) No real roots (y = 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.

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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.

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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, . . . .

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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

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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1 2 1

.n

n

y yyr

y y y

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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)

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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.

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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).

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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.

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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?

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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.

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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.

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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.

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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.

1. Mean（平均值） : (x1 + x2 + . . . + xn)/n.

2. Median（中位數） :• If n is odd, then median:

• If n is even, then median:

3. Mode（眾數） : the value(s) that occur(s) most in x1, x2, . . . , xn.

The followings give different ways to measure the “central tendency"（集中趨勢） of a set of data.

12 2

1

2 n nx x

1

2

nx

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Measure of deviationsGiven a set of numbers x1, x2, . . . , xn. Let be the mean of the numbers. The standard deviation（標準偏離） measures the “average differences" between the numbers and the mean.

Let .

Then .

x

2 2 2

1 2 ns x x x x x x

s

n

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Measure of deviations

1. Mean deviation（平均偏差） :

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.

1 2 nx x x x x x

n

The followings give other ways to measure the “average differences (deviation)" of a set of data.

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Simple Geometry

y

xO

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（笛卡兒坐標） .

P (x, y)

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Equation of Straight Line

y

xO

The equation of a straight line on the 2-dimensioal plane is given by:

y = mx + a

Where a is the y-intercept （ y-截距） as shown and m is the slope（斜率） of the line.

(0, a)

(b, 0)

tana

mb

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Equation of Parabola

xO

y

2 , 0.y ax a

xO

y

2 , 0.y ax a

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Equation of Parabola

xO

y

xO

y

2 , 0.x by b 2 , 0.x by b

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Equation of Parabola

For a parabola with vertex（頂點） at point (, ):

xO

y

(, )

2

2 2

2 2

2

2 .

y a x

y a x x

y ax a x a

Note: The equation of a “tilted” parabola is more complicated.

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Simple Trigonometry

A

B

C

ca

b

Consider the right angle triangle（直角三角形） ABC as shown (angle C = 90).

Since A + B + C = 180, both A and B < 90.

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Trigonometric Functions

A

B

C

ca

b

Definitions:

sin , sin , sin 1.a b

A B Cc c

cos , cos , cos 0.b a

A B Cc c

sintan , .

cos

A aA

A b

（正弦）

（餘弦）

（正切）

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An important identity

2 2cos sin 1

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Functions and Inverse Functions

Appendix

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Exponential（指數） (ex) and Natural Logarithm（自然對數） (lnx)

y=ex y=lnx

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Sine (sinx) and Arcsin（反正弦） (sin-

1x)

0--2 2

1

-1

y = sinx

0

/2

-/2

1-1

y = sin-1x

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Cosine (cosx) and Arccosine（反餘弦） (cos-1x)

0--2 2

1

-1

y = cosx

0

/2

1-1

y = cos-1x

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Tangent (tanx) and Arctangent（反正切） (tan-1x)

y = tanx y = tan-1x

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Logistic Function

0

1 1 rt

MN t

Me

N

with M > N0 and r > 0.

Population Growth

N

t

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Surface Area & Volume of Solids

Cylinder（圓柱體）

Sphere（球形）

Cone（圓錐體）

Area = 2r2+2rh

Area = 4r2

Volume = (4r3)/3

Volume = r2h

Area = r2+ rl

Volume = (r2h)/3

l