The Beauty of Numbers

 Most students’ introduction to mathematics has been bereft of appreciation of its beauty. This article is written to glass case just a few beautiful examples of mathematical marvels with a view to enable students develop deeper interest in Numbers and Mathematics for their gain:

Example (1):  When we use a calculator to determine the following sum, we find it to be zero.
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123,789 ² + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0 
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This may not be quite appealing at the first glance; since we have the squares of large numbers and they seem to show no particular pattern. Yet when we begin to manipulate these numbers in an orderly manner, the sum amazingly remains equal to zero in all the cases! 
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 In the first case, let us delete the hundred-thousand place (the left-most digit) from each number:
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23,789 ² + 61,945²  + 42,864²  - 42,868² -61,943²  - 23,787² = 0

Let us repeat this process by deleting the left-most digit of each number and look at the results: 
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3,789 ² + 1,945²  + 2,864²  - 2,868² -1,943²  - 3,787² = 0
­789 ² + 945² + 864² - 868² -943² - 787² = 0
­89 ² + 45² + 64² - 68² -43² - 87² = 0
­9 ² + 5² + 4² - 8² -3² - 7² = 0
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Let us now follow a similar process. This time we shall delete the unit’s digit from each of the numbers. Yet again, amazingly, we see each time that the resulting sum is zero: 
123,789 ² + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0
123,78 ² + 561,94²  + 642,86²  - 242,86² -761,94²  - 323,78² = 0
123,7 ² + 561,9²  + 642,8²  - 242,8² -761,9²  - 323,7² = 0
123 ² + 561²  + 642²  - 242² -761²  - 323² = 0
12 ² + 56²  + 64²  - 24² -76²  - 32² = 0


Let us now combine the two types of deletions into one by removing the right and left
digits from each number and, yet again, we retain zero sums!

­123,789 ² + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0
2378 ² + 6194² + 4286² - 4286² -6194² - 2378² = 0
37 ² + 19² + 28² - 28² -19² - 37² = 0

Example (2)
The beauty of mathematics lies in the surprising nature of its numbers. Not many words are needed to demonstrate this appeal. Just look, enjoy, and spread these amazing properties to all.

1 X 1 = 1
11 X 11 = 121
1111 X 1111 = 1234321
11111 X 11111 = 123454321
111111 X 111111 = 12345654321
1111111 X 1111111 = 1234567654321
11111111 X 11111111 = 123456787654321
111111111 X 111111111 = 12345678987654321




Example (3)

1 X 8 + 1 = 9
12 X 8 + 2 = 98
123 X 8 + 3 = 987
1234 X 8 + 4 = 9876
12345 X 8 + 5 = 98765
123456 X 8 + 6 = 987654
1234567 X 8 + 7 = 9876543
12345678 X8 + 8 = 98765432
123456789 X 8 + 9 = 987654321



Example (4)


Another interesting number is 142,857, which is a cyclic number and is obtained as 0.142857142857142857… when we divide 1 by 7. When it is multiplied by the numbers 2 through 8, the results are amazing.

142857 X 2 = 285714
142857 X 3 = 428571
142857 X 4 = 571428
142857 X 5 = 714285
142857 X 6 = 857142

We can see symmetries in the products and also notice that the same digits are used in the product as in the first factor. Further, consider the order of the digits. With the exception of the starting point, they are in the same sequence. Now look at the product, 142857 X 7 = 999999!

It gets even stranger with the product, 142857 X 8 = 1142856. If we remove the millions digit and add it to the units digit, the original number is formed!


Example (5)



Here are some number charmers!


12345679 X 9 = 111111111
12345679 X 18 = 222222222
12345679 X 27 = 333333333
12345679 X 36 = 444444444
12345679 X 45 = 555555555
12345679 X 54 = 666666666
12345679 X 63 = 777777777
12345679 X 72 = 888888888
12345679 X 81 = 999999999

In the following pattern, notice that the first and last digits of the
products are the digits of the multiples of 9.

987654321 X 9 = 08 888 888 889
987654321 X 18 = 17 777 777 778
987654321 X 27 = 26 666 666 667
987654321 X 36 = 35 555 555 556
987654321 X 45 = 44 444 444 445
987654321X 54 = 53 333 333 334
987654321 X 63 = 62 222 222 223
987654321 X 72 = 71 111 111 112
987654321 X 81 = 80 000 000 001


Example (6)


0 X 9 + 1 = 1
1 X 9 + 2 = 11
12 X 9 + 3 = 111
123 X 9 + 4 = 1111
1234 X 9 + 5 = 11111
12345 X 9 + 6 = 111111
123456 X 9 + 7 = 1111111
1234567 X 9 + 8 = 11111111
12_345_678 _ 9 + 9 = 111111111


Example (7)


0 X 9 + 8 = 8
9 X 9 + 7 = 88
98 X 9 + 6 = 888
987 X 9 + 5 = 8888
9876 X 9 + 4 = 88888
98765 X 9 + 3 = 888888
987654 X 9 + 2 = 8888888
9876543 X 9 + 1 = 88888888
98765432 X 9 + 0 = 888888888

Example (8)


1 X 8 = 8
11 X 88 = 968
111 X 888 = 98568
1111 X 8888 = 9874568
11111 X 88888 = 987634568
111111 X 888888 = 98765234568
1111111 X 8888888 = 9876541234568
11111111 X 88888888 = 987654301234568
111111111 X 888888888 = 98765431901234568
1111111111 X 8888888888 = 987654321791234568


Example (9)


Numbers form beautiful relationships! There is much more to numbers than meets the eye.


135 = 1¹ + 3² + 5³
175 = 1¹ + 7² + 5³
518 = 5¹ + 1² + 8³
598 = 5¹ + 9² + 8³

Now, taken one place further, we may get:

1306 = 1¹ + 3² + 0³ + 6⁴
1676 = 1¹ + 6² + 7³ + 6⁴
2427 = 2¹ + 4² + 2³ + 7⁴

The next ones are equally amazing.

3435 = 33 + 44 + 33 + 55
438579088 = 4⁴ + 3³ + 8⁸ + 5⁵ + 7⁷ + 9⁹ + 0⁰ + 8⁸ + 8⁸

 (For convenience and for the sake of amusement, 0⁰   has been taken as 0, though in fact, it is indeterminate.)


Example (10)


1 = 1!
2 = 2!
145 = 1! + 4! + 5!
40_585 = 4! + 0! + 5! + 8! + 5!


Example (11)


There are times when the numbers speak more effectively than any explanation.

Here is one such case

1¹ + 6¹ + 8¹ = 15 = 2¹ + 4¹ + 9¹
1² + 6² + 8² = 101 = 2² + 4² + 9²

1¹ + 5¹ + 8¹ + 12¹ = 26 = 2¹ + 3¹ + 10¹ + 11¹
1² + 5² + 8² + 12² = 234 = 2² + 3² + 10² + 11²
1³ + 5³ + 8³ + 12³ = 2366 = 2³ + 3³ + 10³ + 11³

Golden Ratio

 Golden Ratio, Divine Ratio ((√5+1)/2)

There are a very few numbers which are as fascinating and intriguing as Golden Ratio. Mathematicians, Scientists, Engineers, Architects, Philosophers, Biologists, Artists, Musicians, Historians, Psychologists, numerologists and even Mystics have all been charmed by this number for more than 2000 years.

Before delving deep into the characteristics, properties, ubiquity and profundity of the number let us first examine what this number is all about.

What is Golden Ratio?

Let us take a line AB. Suppose if we choose a point C on the line AB such that   AC: CB = AB: AC.

Let AC =a, CB=b and AB=a+b. Here a>b.

A___________a_____________C________b_______B

   

    

Now a/b= (a+b)/a

Simplifying the above equation, we get

axa  = axb + bxb

a² = ab + b²

Dividing the equation by b²

We get

(a/b)²  = a/b + 1

Let us assume a/b = x .

Now ,   x²= x+1 

Solving the equation, we find the value of x = (√5+1)/2.Thus a/b= (√5+1)/2.

This ratio of a and b is known as Golden Ratio or Golden Section. This ratio is known to mankind for thousands of years. It is generally denoted by φ.

Φ = (√5+1)/2 

The approximate value of Φ is 1.6180339887…

This ratio is also known as Divine Ratio.

What is so golden/divine about this Golden/Divine Ratio? 

This is a ratio which pops up in different fields of Mathematics like Geometry, Algebra, Permutation and Combination, Number Theory, Trigonometry, Progressions etc so often that many mathematicians attached a lot of mystical properties to this number in addition to genuine mathematical properties..

Further occurrence of the number in the fields of biology, cosmology, religion, art also attracted other professionals to this widely found number in diverse fields.

The modern history of the Golden Ratio starts with Luca Pacioli (Italian Mathematician) in   1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

Over a period of time, people also started referring to it as Divine ratio considering its ubiquitous presence as a ratio in human and animal bodies, leaves and flowers, and also about galaxies among other fields as mentioned above.

It is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Mathematical Properties of Golden Ratio:

The Golden Ratio, Φ, is an irrational number that has the following unique properties:

1. Taking the reciprocal of Φ and adding one yields Φ. phi=1/phi+1, or Φ=(1/Φ)+1.
                             Φ =    (1/Φ)+1 = 0.618…
2. Φ squared equals itself plus one. In other words, Φ² = Φ+1. These characteristics are indeed very interesting; it is the only number in the world which has such properties.
                            Φ² = Φ+1= 2.618….
3. If we convert the equation Φ² = Φ+1 into the equation Φ²-Φ-1=0 which is in the format ax² + bx + c = 0, we can solve using the quadratic equation formula, x= (-b ± √(b² - 4ac))/(2a). Solving the equation we get x = (1 ± √5)/2.
   Together, these two solutions are known as Phi (1.618033989) and phi (0.618033989). Phi and phi are reciprocals.
4. Another way of representing Golden  Ratio is                           

                                         

Solving the above expression we get the same value of Golden Ratio.

Golden Ratio and its connection with Fibonacci series:  

Leonardo Fibonacci discovered in 12^th Century, a simple Mathematical series that is the basis for a mind-boggling mathematical relationship with Golden Ratio.

Starting with 0 and 1, each new number in the series is simply the sum of the two terms before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

The ratio of each successive pair of numbers in the series approximates phi (1.618. . .), as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.

The ratios of the successive numbers in the Fibonacci series quickly converge on Phi.  After the 40th number in the series, the ratio is accurate to 15 decimal places.

1.618033988749895 . 

Mysterious Examples of Golden Ratio in Nature:

(1)    Shells: This shape, a rectangle in which the ratio of the sides a/b is equal to the Golden Ratio can result in a process that can be repeated into infinity — and which takes on the form of a spiral. It's called the logarithmic spiral that abounds in nature and is associated with Golden Ratio.

(2)    Spiral Galaxies: Spiral galaxies follow the same Fibonacci pattern which in turn is related to Golden Ratio. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees showing the ubiquity of Golden Ratio.

(3)    Hurricanes: Like Spiral Galaxies, Hurricanes also show the pattern of logarithmic spiral associated with the Golden Ratio.

(4)    Branches of trees: The Golden Ratio through Fibonacci sequence can be eerily seen in the way trees branch out or split. The main trunks of various trees grow vertically until they produce a branch, which creates two nodal points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems evincing Fibonacci pattern. A good example is the sneezewort. Root systems and algae also exhibit this pattern.

(5)    Pinecones: The seed pods of a pinecones are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards in opposing directions. The number of steps almost always matches a pair of consecutive Fibonacci numbers. 

(6)    Flower Petals: The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals; buttercups have five, the chicory's 21, the daisy's 34.All Fibonacci numbers.

(7)    Seed Heads of flowers: The heads of flowers also follow the Fibonacci pattern. Generally the seeds are produced at the center, and then move towards the outside to fill the vacant space in predetermined mathematical spiraling way in line with the Fibonacci series and Golden Ratio. Sunflowers provide a classic example of such logarithmic spirals.

(8)    Faces of humans and animals: Faces, both human and nonhuman, evince examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).It's worth noting that every person's body is different, but that averages across populations tend towards phi.

(9)     Fingers, Teeth, Arms, Legs: The length of our fingers, each section — from the tip of the base to the wrist — is larger than the preceding one by roughly the ratio of phi.Same relationship can be found in the teeth , arms and legs of human beings.

(10) DNA Molecules: In a recent study, it has been found that The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

(11) Reproduction of Honey Bees: Honey bees follow Fibonacci sequence in two interesting ways. The most beautiful example is by dividing the number of females in a colony by the number of males. The answer is generally very close to 1.618. Additionally, the family tree of honey bees also follows the Fibonacci pattern.

Counter Point about Golden Ratio: There are some unsubstantiated claims about occurrence of Golden ration in Egyptian Pyramids. Further there are Pseudo Numerologists who want to point out presence of Golden Ratio in even stock Markets for ulterior motives.

There is a trend to show presence of Golden Ratio in every conceivable situation. The mysticism about phi is actually a classic obsession for numerologists and Sacred Geometry. In many instances, attempts to find Golden ratio in different situations are no better than the manifestation of numerological mysticism. The readers are advised to be beware of such futile numerological mysticism.

Fermat's Last Theorem

                                           
                            The holy grail of mathematics.


Background of Fermat’s Last Theorem:

Let us first look at positive Integral Solutions of some basic equations. For example let us take an equation a+b=c.  How many positive Integral Solutions does this equation have? Obviously Infinite. We can see that 2+3= 5 or 5+10= 15 and like that infinite solutions.

The next example is     a²+b²=c².  How many positive Integral Solutions does this equation have? All Pythagorean triplets like 3-4-5, 5-12-13, 8-15-17, 7-24-25… and their multiples like 6-8-10 etc; all are solutions of this equation. So, this equation too has infinite solutions. This particular equation a²+b²=c² and its solutions are known to mankind for the past two thousand years.  

Such polynomial equations like a²+b²=c² in which the solutions must be positive integers are known as Diophantine equations. Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solutions. Diophantus's major work is the Arithmetica which was written in the third century and remained the source for such equations for next several centuries.

Claude Bachet, a Mathematician, wrote a translation  of Arithmetica in 1621. Pierre de Fermat, a legal adviser and an amateur Mathematician was deeply involved in solving Diophantine equations and got a copy of this book. 

Pierre de Fermat thought about the next higher level Diophantine equations like:

a³+b³=c³
a⁴+b⁴=c⁴
a⁵+b⁵=c^{5
…………………..}
aⁿ+bⁿ=cⁿ

He spent several years on the above equations and could not find any integral solutions for the above equations. Finally he was convinced that 

aⁿ+bⁿ≠cⁿ

Where a,b, c and n are natural Numbers and n>2.

And the above is what is known as Fermat’s Last Theorem.

Pierre de Fermat wrote many theorems on number Theory which all were proved subsequently except the one as mentioned above. However, around 1637, Fermat wrote his above–mentioned most famous Theorem in the margin of his copy of the Arithmetica next to Diophantus' sum-of-squares problem:

 “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”.

The Search for the proof of Fermat’s Last theorem:

Those were the most mysterious words ever written by Pierre de Fermat. Decades and centuries went by but nobody could locate the proof written by him. It was lost for ever. It sort of became a big mathematical challenge to prove the theorem. Mathematicians started considering it as a holy grail of Mathematics. Hundreds of Mathematicians including Euler, Gauss, Newton and their worthy successors worked on the problem but could never prove the theorem in the next three and a half centuries till 1995.Many Mathematicians devoted their lives just on this theorem but could not succeed. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years.

That abnormally long time-span of more than 350 years accentuates the significance of this Conjecture- Fermat’s last theorem. It is hard to conceive of any problem, so simply and clearly stated that could have withstood the test of advancing knowledge for so long. 

Let us consider the progress in physics, chemistry, biology, medicine and engineering that have taken place since the seventeenth century. All the subjects have changed beyond recognition. The Physics became quantum Physics. Medicine metamorphosed into genetics. Modern Chemistry makes the chemistry of 17^th century look like alchemy. There is no comparison of the primitive technology of the seventeenth century to today’s space age.  But the Fermat’s last theorem (FLT) could not be conquered till 1995.

Mount Everest could be conquered in two to three decades but the FLT took more than 350 yrs. Fermat’s Last Theorem was the Himalayan peak of number theory. And Fermat was the father of modern number theory.

Prizes Declared on the Proof of Fermat’s last theorem:

In 1816 and in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. Another prize was offered in 1883 by the Academy of Brussels.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 marks to the Göttingen Academy of Sciences to be offered as a prize for a complete proof of Fermat's Last Theorem.

Finally Andrew Wiles a British Mathematician proves the Theorem in 1995:

Sir Andrew John Wiles, a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory ultimately proved the Fermat’s last theorem in 1995 with the help of Computers and latest research in the field of Number Theory.

Wiles came to know about Fermat’s Last Theorem when he was 10 years old. He was Surprised by the fact that the statement of the theorem was so easy that he at the age of 10 could understand it.He aspired to be the first person to prove it. However, he soon realized that his knowledge of mathematics was too inadequate, so he abandoned his childhood dream, until 1986, when he heard that Ribet, a famous Mathematician, had established a link between Fermat's Last Theorem and the Taniyama–Shimura conjecture about elliptical curves and modular functions which are intricately linked to each other.

Based on successive progress of the previous few years of renowned Mathematicians like Gerhard Frey, Jean-Pierre Serre and Ken Ribet; he dedicated all of his research time to this problem in relative secrecy. In 1993, he presented his proof to the public for the first time at a conference in Cambridge. However certain errors were noticed by Mathematicians in the Proof.

Wiles tried to correct the mistakes but found out that the gap he had made was significant. The essential idea bypassing, rather than closing this gap, came to him in September 1994. Together with his former student Richard Taylor, he published a second paper which circumvented the gap and thus completed the proof. Both papers were published in 1995 in a special volume of the Annals of Mathematics.

Finally Andrew Wiles collected the well deserved Wolfskehl prize money, then worth $50,000, on 27 June 1997.

Wiles’ proof is well over 100 pages long, and involves some of the most advanced Mathematics of its time, and so the question does linger, “Is there a ‘simple’ proof of the Theorem as originally claimed by Fermat?”