Showing posts with label e-Learning Note On Mathematics. Show all posts
Showing posts with label e-Learning Note On Mathematics. Show all posts

### MATHEMATICAL MODELLING & GAME THEORY

|-1| MATHEMATICAL MODELING

|-1.00| INTRODUCTION TO MODELING
Mathematical models describe our beliefs about how the world functions. In mathematical modeling, we translate those beliefs into the language of mathematics.

- Mathematics is a very precise language. This helps us to formulate ideas and identify underlying assumptions.

- Mathematics is a concise language, with well – defined rules for manipulations.

- Computers can be used to perform numerical calculations.

|-1.03| OBJECTIVES OF MODELLING
Mathematical modeling can be used for a number of different purposes. The following are the objectives of modelling :-
-Developing scientific understanding through quantitative expression of current knowledge of a system.

-Test the effect of changes in a system

-Aid decision making including tactical decisions by managers strategic by planners.

|-1.03| VARIABLES IN MODELING
Generally speaking, in any given model or equation, there are two types of variables:
-Independent variables: the values that can be changed in a given model or equation. They provide the “input” which is modified by the model to change the “output”.

-Dependent variables: The values that result from the independent variables.

|-1.04| USING INDEPENDENT AND DEPENDENT VARIABLES
The definition of an independent or dependent variable is more or  less universal in both statistical and scientific experiments and in mathematics; however, the way the variable is used varies slightly between experimental situations and mathematics.

|-1.05| EXAMPLE OF VARIABLES IN SCIENTIFIC EXPERIMENTS
If a scientist conducts an experiment to test the theory that a vitamin could extend a person’s life – expectancy, then:
-The independent variable is the amount of vitamin that is given to the subjects within the experiment. This is controlled by the experimenting scientist.

-The dependent variable, or the variable being affected by the affected by the independent variable, is life span.

-The independent variables and dependent variables can vary from person to person, and the variances are what are being tested; that is, whether the people given the vitamin live longer than the not given the vitamin.

The scientist might then conduct further experiments changing other independent variables gender, ethnicity, overall health, etc in order to evaluate the resulting dependent variables and to narrow down the effects of the vitamin on life span under difference circumstances.

Here are some other examples of independent and dependent variables In scientific experiments:

-A scientist studies the impact of a drug on cancer. The independent variables are the administration of the drug – the dosage and timing. The dependent variable is the impact the drugs have on cancer.

-A scientist studies the impact of withholding affection on rats. The independent variable is the amount of affection. The dependent variable is the reaction of the rats.

-A scientist studies how many days’ people can eat soup until they get sick. The independent variable is the number of days people can eat the soup. The dependent variable is the onset of illness.

|-2| GAME THEORY

|-2.00| INTRODUCTION TO GAME THEORY
Game theory is a study of strategic decision making. It is also the study of mathematical models of conflict and cooperation between intelligent rational decision makers.

Another term suggested game theory as a more descriptive name for the discipline that it is an interactive decision theory. Game theory is mainly used in economics, political science and psychology, as well as logic and biology. The theory first addressed zero sum games, such that one person’s gains exactly equal net losses of the other participant(s).

Today, however, game theory applies to a wide of behavioural relations, and has developed into an umbrella term for the logical side of decision science, to include both human and non-humans, like computers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two person zero-sum games and its proof by a theorist named John Von Neumann.

He used Brownness’s fixed-point theorem on continuous mapping into compact convex sets in his original proof, which became a standard method in a game theory and mathematical economics. The second edition of his book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

This theory was developed extensively in the 1950s by many scholars. Games theory was later applied to biology in the 1970s.

|-2.01| DESCRIPTION OF TYPES OF GAMES
Cooperative or non-cooperative – a game is cooperative if the players are able to form binding commitments. For example the legal system requires them to adhere to their promises.

In non cooperative games this is not possible. Communication among players is allowed in cooperative games, but not in non-cooperative  ones.

Of the two types of games, non-cooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the games focus at large. Hybrid games coalitions of players are formed in a cooperative game, but these play in a non-cooperative fashion.

|-2.02| SYMMETRIC AND ASYMMETRIC
A symmetric game is a game where the pay offs for playing a particular strategy depend only on the other strategies employed, not on who is playing them.

If the identities of the players can be changed without changing the payoff to the strategies, then the game is symmetric. The standard representation of chicken, the prisoner’s dilemma, and the stag hunt are all symmetric games. Many of the commonly studied 2 x 2 games are symmetric.

Asymmetric games are game where there are identical strategy sets for both players. For examples, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible for a game to have identical strategies for both players, yet be asymmetric.

|-2.03| ZERO-SUM AND NON-SUM
Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources.

In zero-sum games, the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Some theorists exemplifies a zero sum game (ignoring the possibility of the house’scut), because one wins exactly the amount one’s opponents lose.

Other zero-sum games include matching amount one’s opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and Chess.

Many games studied by game theorists including the infamous prisoner’s dilemma are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero sum games, a gain by one player does not necessarily correspond with a loss by another.

|-2.04| SIMULTANEOUS AND SEQUENTIAL
Simultaneous games are games where both players move simultaneously or if they do not move simultaneously, the later players are unaware of the earlier players’ actions (making them effectively simultaneously).

Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need be have some knowledge about earlier actions. This need be perfect information about every action of earlier players, it might be very little knowledge, for example, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.

### ELEMENTARY TREATMENT OF SEQUENCE & SERIES

SEQUENCE
A sequence is an ordered list of numbers whose subsequent values are formed based on a definite rule. The numbers in the sequence are called terms and these terms are normally separated from each other by commas.

Examples:
2,  4, 6,  8, 10,……
Rule: Addition of 2 for subsequent terms.
70,  66, 62,  58, 54,……
Rule: Subtraction of 4 for subsequent   terms.
3,  -6, 12,  -24,……
Rule: Multiply each term by –2.

Sequences are either finite or infinite
A finite sequence is a sequence whose terms can be counted. i.e. it has an end. These types of sequences are usually terminated with a full stop. e.g.   (i) 3,5,7,9,11,13. (ii) -7,-10,-13,-16,-19,-21.

If however, the terms in the sequence have no end, the sequence is said to be infinite. These types of sequences are usually ended with three dots, showing that it is continuous. e.g.
(i)    5,8,11,14,17,20…
(ii) -35,-33,-31,-29,-27,…

TYPES OF SEQUENCE
We two types of sequences. They are:
(i) Arithmetic progression
(ii) Geometric progression

ARITHMETIC PROGRESSION (AP){LINEAR SEQUENCE}
If in a sequence of terms T1, T2, T3, ...Tn-1, Tn the difference between any term and the one preceding it is constant, then the sequence is said to be in arithmetic progression (A.P) and the difference is known as the common difference, denoted by d.
d = Tn – Tn-1, where n = 1, 2, 3, 4, …
i.e d = T2 – T1 = T3 – T2 = T4 – T3 and so on.

Examples of A.P
(i) 1, 3, 5,7, 9, …
Tn – Tn-1Þ 5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
d = 2
The difference is common, hence it is an A.P.
(ii) 2, 4, 8, 16, 32, …
Tn – Tn-1Þ 4 - 2 = 2
8 - 4 = 4
16 - 8 = 8
32 - 16 = 16
The difference is NOT common; therefore it is not an A.P.
(iii) 70, 66, 62, 58, 54, …
Tn – Tn-1Þ 66 - 70 = -4
62 - 66 = -4
58 - 62 = -4
d = -4
The difference is common; hence it is an A.P.
(iv) –2, -5, -8, -11, …
Tn – Tn-1Þ (-5) - (-
2) = -5 + 2  = -3
(-8) - (-5) = -8 + 5  = -3
(-11) - (-8) = -11 + 8 = -3.
The difference is common; hence it is an A.P.

EXERCISE
Which of the following are arithmetic progressing sequence
1. 4,6,8,10,…
2. 3,7,9,11,..
3. 1,6,11,16,21,26…
4. 100,96,92,88,84,…
5. 20,17,15,11,…
6. 45,42,39,36,…

THE nth TERM OF AN A.P
If the first term of an A.P is 3 and the common difference is 2. The terms of the sequence are formed as follows.

1st term   = 3
2nd term = 3+2                 = 3 + (1)2
3rd term = 3+2+2             = 3 + (2)2
4th term = 3+2+2+2         = 3 + (3)2
5th term = 3+2+2+2+2     = 3 + (4)2
nth term = 3+2+2+2+ …  = 3 + (n - 1)2
Hence, the nth term (Tn) of an A.P whose first term is “a” and the common difference is “d”

Example :
Find the21st term of the A.P 3, 5, 7, 9, …

Solution
a = 3
d = 2
n = 21
Tn  = a + (n  – 1)d
T21 = 3 + (21 – 1)2
= 3 + 20 x 2
= 3 + 40
= 43.

Example :
Find the 27th term of the A.P
100, 96, 92, 88, …

Solution
a = 100
d = -4
n = 27
Tn   = a + (n – 1)d
T27  = 100 + (27 – 1)(-4)
= 100 + 26 x –4
= 100 – 104
= -4.

HOW TO SOLVE FOR ‘n’, ‘a’ AND ‘d’
Example :
Find the value of n given that 77 is the nth term of an A.P 3½, 7, 10½, …

Solution:
a = 3½
d = 7 - 3½
d = 3½
Tn= 77
Tn = a + (n – 1)d
77 = 3½ + (n – 1)3½
77 = 3½ + 3½n – 3½
77 = 3½n
77 = 7/2n
7n = 77 x 2
n = 77 x 2
7
n = 11 x 2
n = 22.

Example 4:
The question in example 1can be reframed as follows to find a.
“What is the first term of an A.P whose 21st term is 43 and the common difference is 2   ?”

Solution:
T21 = 43
n = 21
d = 2
Tn = a + (n –1)d
43 = a + 20 x 2
43 = a + 40
a = 43 – 40
a = 3

Example :
The same example in 4 above can be reframed also as follows to find d.
“Find the common difference of an A.P given that 43 is the 21st term of the sequence and the first term is 3”.
Solution:
a = 3
T21 = 43
n = 21
Tn = a + (n – 1)d
43 = 3 + (21 – 1) d
43 = 3 + 20d
43 –3 = 20d
20d = 40
d = 40
20
d = 2.

EXERCISES
1. Find the 31st term of the sequence       –7, -10, -13, -16, …
2.  What is the 26th term of the A.P 5, 10, 15, 20, …?
3. Find the 20th term of the sequence 27, 24, 21, 18, …
4. Find the 18th term of the A. P  6, 12, 18, 24, …
5 .Find the 26th term of the A.P –16, -13, -10, -7, …
6. Find the 29th term of the A.P 43, 39, 35, 31, …
7. Find n given that 697 is the nth term of the A.P –3, 4, 11, 18, …
8. Find n given that –8 is the nth term     of the A.P 82, 79, 76, 73, 70, …
9. Find n given that 63 is the nth term of the sequence –17, -13, -9, -5, …
10. Find the number of terms in an A.P given that 147 is the last term of the A.P whose first term is 6 and common difference is 3.
11. Find the first term of an A.P given that 124 is the 41st term of the A.P and 3 is the common difference.
13. Find the first term of an A.P given that –5 is the 26th term of the sequence and –3 is the common difference.
14. Given that 57 is the 27th term of an A.P whose common difference is 2, find the first term.
15. Given that 76 is the 21st term of an A.P whose first term is 16. Find the common difference.
16. Given that 67 is the 21st term of an A.P whose first term is 7, find the common difference.
17. Find the common difference of an A.P, given that the 27th term is 90 and the first term is –14.

FURTHER EXAMPLES
Example :
The first three terms of an A.P are
x, 3x + 1, and (7x - 4). Find the
(i) Value of x
(ii) 10th term

Solution:
(i) Recall that given an A.P T1, T2, T3
T2 – T1 = T3 – T2
Hence for,   x, (3x + 1), (7x - 4)
(3x + 1) – x = (7x - 4) – (3x + 1)
3x+1 – x = 7x – 4 – 3x - 1
2x + 1 = 4x – 5
1+ 5 = 4x – 2x
2x = 6
x = 6/2
x = 3.

The sequence x, (3x + 1), (7x - 4) is
= 3, (3x3 + 1), (7x3 - 4)
= 3, 10, 17.
(ii)  a = 3
n = 10
d = 7
Tn = a + (n – 1)d
T10 = 3 + (10 – 1)7
= 3 + 9x7
= 3 + 63
= 66.
Example :
The 6th term of an A.P is –10 and the 9th term is –28.
Find the (i) Common difference
(ii) First term
(iii) 26th term of the sequence.
Solution:
(i)     T6 = -10       Tn = a + (n - 1)d
n = 6          -10 = a + (6 - 1)d
-10 = a + 5d ----------- (1)
T9 = -28      -28 = a + (9 - 1)d
n = 9           -28 = a + 8d ---------- (2)
Solve equation (1) and (2) simultaneously.
Eqn. (1): -10 = a + 5d
Eqn. (2): -28 = a + 8d
18 = -3d
d =18/-3
d = -6.
(ii)      Put d = -6 in equation (1)
-10 = a + 5(-6)
-10 = a - 30
-10+30 = a
a = 20.
(iii) To find the 26th term of the sequence.
a = 20
d = -6
n = 26
Tn= a + (n - 1)d
T26 = 20 + (26 - 1)(-6)
= 20 + 25(-6)
= 20 - 150
= -130.
EXERCISE
(1) The 6th term of an AP is –10 and the 9th term is 18 less than the 6th term. Find the
(a)common difference
(b) first term
(c) 26th term of the sequence.
(2) The 7th term of an AP is 17 and the 13th term is 12 more than the 7th term. Find the
(i) common difference
(ii) first term
(iii) 21st term of the AP.
(3) The 6th term of an A.P is 26 and the 11th   term is 46. Find the
(i) Common difference
(ii) First term
(iii) 25th term of the A.P
(4) The 5th term of an A.P is 11 and the 9th term is 19. Find the (i) common difference
(ii) First term
(iii) 21st term of the A.P
(5) The fourth term of an A.P is 37 and the     6th term is 12 more than the fourth term. Find the first and seventh terms.
(6) The first three terms of an A.P are (x+2) ,(2x-5) and (4x+1). Find the
(i) Value of x
(ii) 7th term.
(7) The first three terms of an A.P are x,    (2x-5) and (x+6). Find the
(i) Value of x
(ii) 21st term.
(8) If the first three terms of an A.P are  (4x+1), (2x-5) and (x+3). Find the
(i) Value of x
(ii) Sequence
(iii) 11th term of the sequence
(9) Given that 9, x, y, 24 are in A.P, find the values of x and y.
(10) If –5, a, b, 16 are in A.P, find the values    of a and b.

Arithmetic Series
These are series formed from an arithmetic progression. e.g.
1 + 4 + 7 + 10 + …
In general, if Sn is the sum of n terms of an arithmetic series then
Sn = a + (a + d) + (a + 2d) + … + (l - d) +
l -- (1)
Where l is the nth term, a is the first term and d is the common difference. Rewriting the series above starting with the nth term, we have.
Sn = l +(l - d) + (l - 2d) +… + (a + d) + a ----- (2)
Adding equation (1) and (2) we have
2Sn = (a + l) + (a + l) + … + (a + l) + (a + l) in n places
2Sn = n(a + l)
Sn = n/2(a + l)
But l is the nth term i.e a + (n - 1) d
Sn=  n/2{a +a + (n - 1)d}
Sn = n/2 {2a + (n - 1)d}

Example :
Find the sum of the first 20 terms of the series 3+5+7+9+ …

Solution:
a = 3
d = 2
n = 20
Sn = n/2 {2a +(n - 1)d}
S20 = 20/2 {2x3 +(20 - 1)2}
S20 = 10{6 + 19 x 2}
= 10{6 + 38}
= 10{44}
S20 = 440

Example :
Find the sum of the first 28 terms of the series –17 + (-14) + (-11) + (-8) + …
Solution:
a = -17
d = 3
n = 28
Sn = n/2{2a + (n - 1)d}
S28 = 28/2 {2 (-17) + (28 - 1) 3}
= 14 {-34 + 27 x 3}
= 14 { -34 + 81}
= 14 {47}
S28  = 658

### NUMBER BASES

The decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.
 CONVERSION FROM OTHER BASE LESS THAN TEN TO BASE TEN
Firstly, we shall consider conversion of numbers in a base less than ten to a base ten number. A number in base ten is known as a decimal or denary number. All numbers in a given n can be written using only the  following digits 0, 1, 2 , ….., n -1. For instance in base two, the only digits that can be used are only 0 and 1. In base three, you can only use digits 0, 1 or 2.

Generally our normal counting is done in base ten when doing this, the base is normally indicated.

E.g in the denary number 546 is 546ten.  The first digit from the right towards left is 6 and is called the unit digit. The next digit is 4 and is called the tens digit and has value 4x10.

The next digit 5 is called the hundred digit i.e 500. Hence 52 41 60ten= 5 4 6 = 5x102+4x101+6x100
= 500 + 400+ 6
= 546
Notice that the digits listed on top of the denary numbers 546 are the powers of the base.

SAMPLE 1
Convert 1203 in base five to a denary number.
Solution
Circle the digits of the number 1203 from the last to the first, beginning at zero
i.e203five
Therefore expand number raising the base five to the grades listed on the top of the number as shown below;
2 0 3 five= 1x53+2x52+0x51+3x50
= 1x125+2x25+0x5+3x1
= 125+50+0+3
= 178
The new number is in base ten i.e 1203five=  178ten
This expansion method can be used in converting from any base to base ten.

EXERCISE 1
Convert the following to denary numbers
1. 10.2three
2. 214seven
3. 780nine
4.Find if 200x + 144nine = 14Btwelve.
5.Solve for x and y if 32x + 53y + 61nine
24x + 35y = 45ten

 CONVERSION FROM OTHER BASE GREATER THAN TEN TO BASE  TEN
Expansion method can be used to convert numbers in base say base thirteen to base ten, Remember in base thirteen kthe digits we have are 0,  1, 2, 3,4,5, 6,7,8,9, A, B, C. where A represents ten B represents eleven and C represents twelve. Letters are used for two- digits numbers less than the base thirteen.

SAMPLE 2
Convert 1B9thirteen to denary number
Solution
1B9thirteen= 1x132+Bx131+9x130
= 1x169+11x13+9x1
= 169+143+9
= 321ten

SAMPLE 3
Convert 20Cfifteen to a denary number
Solution
20Cfifteen= 2x152+0x151+12x150
= 2x225+0x15+12x1
= 450+0x15+12x1
= 462ten

EXERCISE 2
1. Convert the following to denary numbers
[I] 1024eleven
[II] 2059twelve
[III] 51Cfourteen

2. Convert the following numbers to denary numbers
[I] 10011two
[II] 768nine
[III] 10Aeleven
[IV] B12twelve
[V] 7B3Atwelve
[VI] 6D4Fsixteen