Introduction9 (nine) is the natural number following 8 and preceding 10. The ordinal adjective is

ninth.

Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and

hence the third square number. 9 is a Motzkin number. It is the first composite lucky

number.

Nine is the highest single-digit number in the decimal system. It is the second non-

unitary square prime of the form (p2) and the first that is odd. All subsequent squares

of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. 9 is;

and can be, the only square prime with an aliquot sum of the same form. The aliquot

sequence of 9 has 5 members (9,4,3,1,0) this number being the second composite

member of the 3-aliquot tree.

There are nine Heegner numbers. Since , 9 is an exponential factorial. 8 and

9 form a Ruth-Aaron pair under the second definition that counts repeated prime

factors as often as they occur. A polygon with nine sides is called a nonagon or

enneagon. A group of nine of anything is called an ennead.

In base 10 a number is evenly divisible by nine if and only if its digital root is 9.

That is, if you multiply nine by any natural number, and repeatedly add the digits of

the answer until it is just one digit, you will end up with nine:

2 × 9 = 18 (1 + 8 = 9)

3 × 9 = 27 (2 + 7 = 9)

9 × 9 = 81 (8 + 1 = 9)

121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)

234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)

578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27 (2 + 7 = 9))

482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 +

4 + 0 + 9 = 45 (4 + 5 = 9)

1

The only other number with this property is three. In base N, the divisors of N − 1

have this property. Another consequence of 9 being 10 − 1, is that it is also a

Kaprekar number.

The golden ratio is a very interesting thing to know. There are so many

secret behind the golden ratio. Golden ratio can be find or it is find by Fibonacci

sequence. What must you know? Firstly, to know what is golden ratio, defined what

is Fibonacci sequence. This I stated the Fibonacci sequence ;

1,1,2,3,5,8,13,......

2

3.0 TASK 1

3.1 The Magic of 9

A. EXPLORING MAGIC OF 9

(a) 1 x 9 = 9

2 x 9 = 18

3 x 9 = 27

4 x 9 = 36

5 x 9 = 45

6 x 9 = 54

7 x 9 = 63

8 x 9 = 72

9 x 9 = 81

i. The pattern that i can saw after I complete the multiplication fact for 9 is:

a) The first digit of 2-digit product of multiple of 9 is ascending order from 1 to

8.

b) The second digit of 2-digit product of multiple of 9 is descending order from

8 to 1.

ii. The use of this pattern is:

a) Easier to remember

b) much interesting to learn

c) increase the curiosity of the people on learning mathematics

d) much more fun to learn.

e) nice to see,

iii. The rule for multiplying of 9 is

a) some of the digit equal to 9n where n = 1,2,3,4,….

(b) Multiplying 2-digit number by 9:

11 x 9 = 99

12 x 9 = 108

13 x 9 = 117

14 x 9 = 126

15 x 9 = 135

16 x 9 = 144

17 x 9 = 153

18 x 9 = 162

19 x 9 = 171

20 x 9 = 180

21 x 9 = 189

3

22 x 9 = 198

23 x 9 = 207

24 x 9 = 216

25 x 9 = 225

26 x 9 = 234

27 x 9 = 243

28 x 9 = 252

29 x 9 = 261

30 x 9 = 270

31 x 9 = 279

32 x 9 = 288

33 x 9 = 297

34 x 9 = 306

35 x 9 = 315

36 x 9 = 324

37 x 9 = 333

38 x 9 = 342

39 x 9 = 351

40 x 9 = 360

41 x 9 = 369

42 x 9 = 378

43 x 9 = 387

44 x 9 = 396

45 x 9 = 405

46 x 9 = 414

47 x 9 = 423

48 x 9 = 432

49 x 9 = 441

50 x 9 = 450

51 x 9 = 459

52 x 9 = 468

53 x 9 = 477

54 x 9 = 486

55 x 9 = 495

4

i. Answer 1

a) One of the pattern is the same as the multiply of 1-digit number by 9, that

is the third digit number of the product of the multiply of 2-digit number by

9. The third digit of the product is in descending order from 9 to 0.

b) There are two more pattern that we can find from the multiply of 2-digid by

9 that is:

I. The first digit of the product of multiply of 2-digit number by 9 will

repeat until 11 times. The next number will be bigger than it

previous number and it also repeated until 11 times and this

pattern will continued

II. Another pattern is the second number of the product of the multiple

of 9, the number is in ascending order from 0 to 9 and one of

number between 0 to 9 will repeat 2 times one by one as the

number goes on.

ii. answer 2

The rule for finding the product of multiplying a 2-digit number by 9 is

iii. answer 3

The rule for finding the product of multiplying a 3-digit number by 9 is

b. 1. i. (0 x 9) + 1 = 1

ii. (1 x 9) + 2 = 11

iii. (12 x 9) + 3 = 111

iv (123 x 9) + 4 = 1111

v. (1234 x 9) + 5 = 11111

2. i. (1 x 9) -1 = 8

ii. (21 x 9) -1 = 188

iii. (321 x 9) -1 = 2888

iv. (4321 x 9) -1 = 38888

v. (54321 x 9) -1 = 488888

TASK 2

(a)

1 1 2 3 5 8 13………….,

(T1) (T2) (T3) (T4) (T5) (T6) (T7)………..,

i. We assumed the first number as T1, the second number as T2 and so on.

From examination, I have found that the sequence number above has a

pattern. The first number of the sequence is 1 that is T1. To get the

second term, the T1 must add with the number before it that might be

zero. So, the second term is also 1. To get the third term, T2 must add

with T1. We can conclude that ;

Tn = Tn-1 + Tn-2

ii. (1) T8 = T7 + T6 (2) T9 = T8 + T7 (3) T10

= T9 + T8

T8 = 13 + 8 T9 = 21 + 13 T10 = 34

+ 21

T8 = 21 T9 = 34 T10 = 55

(4) T11 = T10 + T9 (5) T12 = T11 + T10 (6) T13 = T12

+ T11

T11 = 55 + 34 T12 = 89 + 55 T13

= 144 + 89

T11 = 89 T12 = 144 T13 = 233

iii. The general formula that can be used to generate the nth number in the

series is ;

Un = Un-1+Un-2

iv. The series above is known by Fibonacci sequence.

(b)

Ratios of consecutive

numbers in the series

Result after dividing

second number by number

1 : 1 1.00

2 : 1 2.00

3 : 2 1.50

5 : 3 1.67

8 : 5 1.60

13 : 8 1.63

21 : 13 1.62

34 : 21 1.62

55 : 34 1.62

89 : 55 1.62

144 : 89 1.62

233 : 144 1.62

i. I noticed that when each time when I dividing second number by number,

the result will produced a number that nearer to 1.62. The bigger the

number, it will produced the answer nearer to 1.62. When the number 21

is divided by 13, it start to produced answer 1.62. When it comes to this,

the number divided after the number will also produced 1.62.

ii. I think if I keep dividing in the sequence, the result will remain the answer

1.62.

(c) EXISTENCES OF NUMBER SEQUENCE

There are some of the list most common patterns and how they are made.

1. SEQUENCE OF ODD NUMBER

2. SEQUENCE OF EVEN NUMBER

3. SEQUENCE OF SQUARES OF NATURAL NUMBERS

4. A FIBONACCI SEQUENCE

5. ARITHMETIC SEQUENCE

6. GEOMETRIC SEQUENCE

There are many more sequence but we will explain and give the example about

the sequence ;

Arithmetic Sequences

An Arithmetic Sequence is made by adding some value each time.

Examples:

This sequence has a difference of 3 between each number.

The pattern is continued by adding 3 to the last number each time.

The value added each time is called the "common difference"  

Geometric Sequences

A Geometric Sequence is made by multiplying by some value each time.

Examples:

2, 4, 8, 16, 32, 64, 128, 256, ...

This sequence has a factor of 2 between each number.

The pattern is continued by multiplying the last number by 2 each time.

Special Sequences

Square Numbers

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

1, 4, 9, 16, 25, 36, 49, 64, 81, ...

The next number is made by squaring where it is in the pattern.

The second number is 2 squared (22 or 2×2) .

Cube Numbers

1, 8, 27, 64, 125, 216, 343, 512, 729, ...

The next number is made by cubing where it is in the pattern.

The second number is 2 cubed (23 or 2×2×2) .

Fibonacci Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding the two numbers before it together.

The 2 is found by adding the two numbers in front of it (1+1)

Fibonacci numbers have an interesting property. When you divide one number in

the sequence by the number before it, you obtain numbers very close to one

another. In fact, this number is fixed after the 13th in the series. This number is

known as the "golden ratio."

I have always been fascinated by ancient mathematical rules and how they have

been applied in design. The "golden rectangle" or "golden mean" is one such rule

if you like, that has often sneaked its way into my design work - sometimes

planned but more often than not it just seems to happen.

Let me explain. The Golden Mean, just like PI (3.14) is another of those strange

numbers that we seldom question and very often take for granted. This number is

represented by the Greek letter PHI, but dissimilar to PI, the golden mean goes

very much unnoticed in our everyday life in such things as buildings, plants and

even in living creatures - yet we find these things strangely pleasing on the eye.

This is the magical number 1.618.

So how is this number found? An ancient mathematician by the name of

Fibonacci discovered that if you start with the numbers 0 and 1 then add them

together you get a new number - in this case 1. Easy enough but what if you add

the last number and the new number together? You get another new number,

2(See figure below). Keep doing this and you will end up with a very long list of

uniquenumbers.

If you keep going you will see that the decimal figure will revolve around the

magic number 1.618. OK, I here you ask, but what is the point? Well lets look at

the example of how the golden mean occurs in nature. Take a look at the

diagram below. Notice that it is made up solely of squares, yet the overall image

is a rectangle. This rectangle, if you measure it, has the magic ratio of 1.618.

Also if you look at the curved lines within each of the squares you will notice that

these are infect quarter circles, but, as a whole you would be forgiven for thinking

that they look like the cross section of a sea shell.

The existence of the number sequence and Golden Ratio in nature and everyday

life.

FIBONACCI numbers in nature.

Fibonacci sequences appear in biological settings, in two consecutive Fibonacci

numbers, such as branching in trees, arrangement of leaves on a stem, the fruit

lets of a pineapple, the flowering of artichoke, an uncurling fern and the

arrangement of a pine cone. In addition, numerous poorly substantiated claims of

Fibonacci numbers or golden sections in nature are found in popular sources,

e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of

waves. The Fibonacci numbers are also found in the family tree of honeybees.

An interesting phenomena of nature is the sunflower. If you count the spirals you

will see that there are 55 with either 34 or 89 on either side going in an anti-

clockwise direction

Sunflower head displaying florets in spirals 34 and 55 around the outside.

And you'd be right, for this is the same as the growth rate of the beautiful Nautilus

Sea Shell - i.e. 1.618.

Przemysław Prusinkiewicz advanced the idea that real instances can in part be

understood as the expression of certain algebraic constraints on free groups,

specifically as certain Lindenmayer grammars.

A model for the pattern of florets in the head of a sunflower was proposed by H.

Vogel in 1979. This has the form

where n is the index number of the floret and c is a constant scaling factor;

the florets thus lie on Fermat's spiral. The divergence angle, approximately

137.51°, is the golden, dividing the circle in the golden ratio. Because this

ratio is irrational, no floret has a neighbour at exactly the same angle from

the centre, so the florets pack efficiently. Because the rational

approximations to the golden ratio are of the form F(j):F(j + 1), the nearest

neighbours of floret number n are those at n ± F(j) for some index j which

depends on r, the distance from the centre. It is often said that sunflowers

and similar arrangements have 55 spirals in one direction and 89 in the other

(or some other pair of adjacent Fibonacci numbers), but this is true only of

one range of radii, typically the outermost and thus most conspicuous.

The bee ancestry code

Fibonacci numbers also appear in the description of the reproduction of a

population of idealized bees, according to the following rules:

If an egg is laid by an unmated female, it hatches a male.

If, however, an egg was fertilized by a male, it hatches a female.

Thus, a male bee will always have one parent, and a female bee will have

two.

If one traces the ancestry of any male bee (1 bee), he has 1 female parent

(1 bee). This female had 2 parents, a male and a female (2 bees). The

female had two parents, a male and a female, and the male had one female

(3 bees). Those two females each had two parents, and the male had one (5

bees). This sequence of numbers of parents is the Fibonacci sequence.

This is an idealization that does not describe actual bee ancestries. In

reality, some ancestors of a particular bee will always be sisters or brothers,

thus breaking the lineage of distinct parents.

Architecture

Some studies of the Acropolis, including the Parthenon, conclude that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere are said to be circumscribed by golden rectangles.[18] To the extent that classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the golden ratio and consciously employed it in their designs. Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio. On the other hand, such retrospective analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and that these choices affect the proportions observed.

Some scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."[19] And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value."[20] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[21] It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis.

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."[22]

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[23]

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[24]

In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.[25]

[edit] Painting

Painting

Leonardo Da Vinci's illustration from De Divina Proportione applies geometric proportions to the human face.

Leonardo da Vinci's illustrations in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own paintings. Some suggest that his Mona Lisa, for example, employs the golden ratio in its geometric equivalents.[26] Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate. The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive[citation needed].

Salvador Dalí explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[2][27]

Mondrian used the golden section extensively in his geometrical paintings.[28]

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[29] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.[30]

[edit] Book design

Main article: Canons of page construction

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."[31]

According to Jan Tschichold,[32] "There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimetre."

[edit] Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[2][33]

[edit] Music

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[34] though other music scholars reject that analysis.[2] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[35] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix.

The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."[35]

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.[36] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[37] Also, many works of Chopin, mainly Etudes

(studies) and Nocturnes, are formally based on the golden ratio. This results in the biggest climax of both musical expression and technical difficulty after about 2/3 of the piece.[citation needed]

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[38]

In the opinion of author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio."[39]

Nature

See also: History of aesthetics (pre-20th-century)

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[40] In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form."[41][Need

quotation on talk to verify]

Bibliography

Tannenbaum P. 2007. Excursions in Modern Mathematics with Mini- Excursion

Sixth Edition, Pearson Education Inc.

Posamentier A.S and Stepelman J. 1990. Teaching Secondary School

Mathematics Techniques and Enrichment Units Third Edition, Merril Publishing

Company Columbus, Ohio, 43216.

http://mathworld.wolfram.com/FibonacciNumber.htm. Accessed on 20 September

2009.

http://www.friesion.com. Accessed on 29 September 2009.