RAMANUJAM

"An equation for me has no meaning unless it expresses a thought of God."- RAMANUJAM

Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. His most famous work was on the number p(n) of partitions of an integer n into summands.

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on Advanced Trigonometry written by S. L. Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.

When he was 16, Ramanujan came across the book "A Synopsis of Elementary Results in Pure and Applied Mathematics"  by George S. Carr. This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.

Ramanujan, with the help of Ramaswami Iyer(founder member of the Indian Mathematical Society) , had his work published in the Journal of Indian Mathematical Society.

In January 1913 Ramanujan wrote to G .H. Hardy having seen a copy of his  book Orders of infinity. Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter.

Hardy wrote back to Ramanujan and in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.

On 6 December 1917, he was elected to the London Mathematical Society. 

In 1918, he became a Fellow of the Royal Society , and he was the youngest Fellow in the entire history of the Royal Society.

On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.


TAXICAB NUMBER:
The number derives its name from the following story:
G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
1729 is the second taxicab number (the first is 2= 1^3 + 1^3). The number was also found in one of Ramanujan's notebooks dated years before the incident.

Every positive integer is one of Ramanujan's personal friends" - John Littlewood, on hearing of the taxicab incident

Ramanujan had problems settling in London. He was an orthodox Brahmin and right from the beginning he had problems with his diet. Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died on April 6, 1920.

SUM OF ADJACENT INTEGERS

                                               4 + 5 + 6    =   7 + 8



                                  9 + 10 + 11 + 12    =   13 + 14 + 15


                        16 + 17 + 18 + 19 + 20    =   21 + 22 + 23 + 24


                25 + 26 + 27 + 28 + 29 + 30   =   31 + 32 + 33 + 34 + 35


        36 + 37 + 38 + 39 + 40 + 41 + 42  =   43 + 44 + 45 + 46 + 47 + 48


49 + 50 + 51 + 52 + 53 + 54 + 55 + 56  =   57 + 58 + 59 + 60 + 61 + 62 + 63

NUMBER PATTERN USING 1, 2, 8 AND 9

                      9 x 2 = 18

              99 x 2 = 198

            999 x 2 = 1998

          9999 x 2 = 19998

        99999 x 2 = 199998

      999999 x 2 = 1999998

    9999999 x 2 = 19999998

  99999999 x 2 = 199999998

999999999 x 2 = 1999999998

PALINDROMIC NUMBER PATTERN WITH 1

111111111
               * 111111111
--------------------------
 12345678987654321

EUCLID

FATHER OF MATHEMATICS

The Greek mathematician Euclid’s referred to as “THE FATHER OF GEOMETRY” is well known for his most famous work “ The Elements” which is a collection of geometrical theorems and “Euclidean theorem”.

EUCLID’S FAMOUS QUOTES:

“The laws of nature are but the mathematical thoughts of God”

“ There is no other Royal path which leads to geometry”.

THE ELEMENTS:

The Elements is divided into 13 books.

• The first 6 books deals with plane geometry.
• Books  7to 9 deals with number theory.
• Book 10 deals with the theory of irrational numbers .
• Books 11 to 13 deals with three-dimensional geometry .

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved.

EUCLID'S OTHER WORKS :       

• ON DIVISION deals with plane geometry.
• The book DATA discusses plane geometry and contains propositions.
• PHAENOMENA is a work by what we call today as applied mathematics, concerning the  geometry of spheres for use in astronomy. 
• THE OPTICS, corrects the belief held at the time that the sun and other heavenly bodies are actually the size they appear to be to the eye.
• CONICS was a work on conic sections.

 EUCLID'S APPROACH:

Euclid used an approach called the "synthetic approach" to present his theorems. Using this method, one progresses in a series of logical steps from the known to the unknown.

EUCLID’S CLASSICAL PROOF ON PRIME NUMBERS:

Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it, you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists.

A MATH POEM ON FRIENDSHIP

0% INVEST AND 100% INTEREST

Friendship is infinity,
containing only plus points.
So, my friend you are a Modulus.
If the world is a circle,
You are a point on the circumference,
I am a tangent,
which will touch the circle at that point.
If I am a straight lineYou are a lso the same my friend.
For we are not perpendicular,
but always coincide.

ARCHIMEDES

FATHER OF MATHEMATICS

CONTRIBUTIONS TO MATHEMATICS

Archimedes, a Greek mathematician is considered one of the three great mathematicians along with Isaac Newton and Carl Fredrick Gauss. . His greatest contributions to mathematics were in the area of Geometry. Archimedes was also an accomplished engineer and an inventor.

• Discovered the method to determine the area and volumes of circles, spheres and cones. 
• Discovered the actual value of PI.
• Archimedes‘s investigation on Method of Exhaustion led way to current form of Integral Calculus which is now updated. Though it is outdated it is believed that he invented the method of Integral Calculus 2000 years before Newton and Leibniz.

OTHER CONTRIBUTIONS

• Archimedes performed countless experiments on screws, levers, and pulleys. 
• Archimedes invented the water screw, a machine for raising water to bring it to fields.  
• His work with levers and pulleys led to the inventions of compound pulley systems and cranes.  
• His compound pulleys are highlighted in a story that reports that Archimedes moved a fully-loaded ship single-handedly while seated at a distance.  
• His crane was reportedly used in warfare during the Roman siege of his home, Syracuse. 
• Wartime inventions attributed to Archimedes include rock-throwing catapults, grappling hooks, and lenses or mirrors that could allegedly reflect thesun's rays and cause ships to catch on fire.
• Another invention was a miniature planetarium, a sphere whose motion imitated that of the earth, sun, moon, and the five planets that were then known to exist.

 A FAMOUS STORY 

There are many stories about how Archimedes made his discoveries. A famous one tells how he uncovered an attempt to cheat King Hieron.

The king ordered a golden crown and gave the crown's maker the exact amount of gold needed. The maker delivered a crown of the required weight, but Hieron suspected that some silver had been used instead of gold. He asked Archimedes to think about the matter. One day Archimedes was considering it while he was getting into a bathtub. He noticed that the amount of water overflowing the tub was proportional (related consistently) to the amount of his body that was being immersed (covered by water). This gave him an idea for solving the problem of the crown. He was so thrilled that he ran naked through the streets shouting, "Eureka!" (Greek for "I have discovered it!").

There are several ways Archimedes may have determined the amount of silver in the crown. One likely method relies on an idea that is now called Archimedes's principle. It states that a body immersed in a fluid is buoyed up (pushed up) by a force that is equal to the weight of fluid that is displaced (pushed out of place) by the body. Using this method, he would have first taken two equal weights of gold and silver and compared their weights when immersed in water. Next he would have compared the weight of the crown and an equal weight of pure silver in water in the same way. The difference between these two comparisons would indicate that the crown was not pure gold. 

MATH LIMERICK

A Dozen, a Gross, and a Score,

plus three times the square root of four,

divided by seven,

plus five times eleven,

equals nine squared and not a bit more.

-- Jon Saxton

CAREERS IN MATHEMATICS

Actuary
Actuaries are hired by insurance companies (life, health, casualty, etc.), pension plans, businesses, consulting firms (business and actuarial), and government agencies. To become an actuary (an Associate or a Fellow), one must pass a series of examinations administered by the Society of Actuaries. The initial exams are primarily mathematics, including probability and statistics, and can be taken while still an undergraduate student.


Computational Scientist
A computational scientist is an applied mathematician who interprets problems arising from the physical sciences and engineering in mathematical form and develops mathematical solutions to these problems. Very large and sophisticated computers are used intensively. Potential employers include government laboratories, the chemical industry, and the biotech industry. 

Operations Research Analyst
Also called management science analysts, operations research analysts help organizations coordinate activities and operate in the most efficient manner, by applying scientific methods and mathematical principles to organizational problems. Computers are used extensively in their work.

Systems Engineer or Systems Analyst
A systems engineer or analyst usually has substantial course work in engineering or another technical field. This enables him/her to apply mathematical techniques to solve the problems unique to the industry of their employer.

Scientific Communication
The scientific publishing industry has a need for scientifically trained individuals for sales and editing.

Software Engineer or Software Consultant
A software engineer generally designs and writes software that performs nonnumerical functions, such as graphics. A background in math and computer science is needed. Employers include consulting firms and large corporations which do their own software development. There is also room in this field for the entrepreneur or consultant.

 Statistician
Statistics is both a very applied field and also a theoretical one. Many, but not all, statisticians are active in both applications and the development of new theory, but the greatest potential in terms of jobs is in applied statistics. Statisticians generally work with people in other fields, therefore communication skills are very important. Statistical
applications nearly always include the analysis of data and hence some knowledge and experience in computing is very important. There are opportunities for statisticians in the government, in industry, business, medicine, and in academia

FLORENCE NIGHTINGALE'S CONTRIBUTION TO MATHEMATICS

The rare photograph of Florence Nightingale was taken by Lizzie Caswall Smith in 1910 .The black and white image of the silver-haired nursing pioneer shows her in the imposing bedroom of her home just off London's Park Lane, before her death in 1910 at the age of 90.


Florence Nightingale is most remembered as a pioneer of nursing and a reformer of hospital sanitation methods. For most of her ninety years, Nightingale pushed for reform of the British military health-care system and with that the profession of nursing started to gain the respect it deserved.

During the American Civil War, Nightingale was a consultant on army health to the United States government. She also responded to a British war office request for advice on army medical care in Canada. Her mathematical activities included ascertaining "the average speed of transport by sledge" and calculating "the time required to transport the sick over the immense distances of Canada."

Unknown to many, Florence Nightingale is credited with developing a form of the pie chart now known as the polar area diagram, or occasionally the Nightingale rose diagram, equivalent to a modern circular histogram to dramatize the needless deaths caused by unsanitary conditions and the need for reform during the Crimean War .

The legend reads:

The Areas of the blue, red, & black wedges are each measured from the
centre as the common vertex.
The blue wedges measured from the centre of the
circle represent area for area the deaths from Preventable or Mitigable
Zymotic diseases, the red wedges measured from the centre the deaths from
wounds, & the black wedges measured from the centre the deaths from all
other causes.
The black line across the red triangle in Nov. 1854 marks the
boundary of the deaths from all other causes during the month.
In October 1854, & April 1855, the black area coincides with the red, in January
& February 1855,(*) the blue coincides with the black.
The entire areas may be compared by following the blue, the red, & the black lines
enclosing them.

With her analysis, Florence Nightingale revolutionized the idea that social phenomena could be objectively measured and subjected to mathematical analysis.

A Strange Prime Number - 73,939,133

The prime number 73,939,133 has a very strange property.

If you keep removing a digit from the right hand end of the number, each of the remaining numbers is also prime.

It's the largest number known with this property.

Take a look: 73,939,133 and 73,939,13 and 73,939,1 and 73,939 and 7,393 and 739 and 73 and 7 are all prime!

MAGIC 1089 - A COOL MATH TRICK

• Write down a three-digit number whose digits are decreasing.
• Then reverse the digits to create a new number.
• Subtract the reversed number from the original number.
• With the resulting number, add it to the reverse of itself.

The number you will get is 1089!

EXAMPLE :

• Let us take 532 (three digit number in decreasing order).
• Then the reverse is 235.
• Subtract 532-235 to get 297.
• Now add 297 and its reverse 792,
• The resulting answer is 1089!

Division by addition

Multiplication TABLE 9 using FINGERS

MATHS AND MUSIC

WHY IS A CIRCLE DEFINED AS 360 DEGREES?

A circle has 360 degrees, but it also has 400 gradients and approximately 6.2831853 radians. It all depends on what *units* you measure your angles with.

You think 360 is a terrible number, and you think that you want a circle to have 100 "somethings" in it. Well, you divide up the circle into 100 equal angles, all coming out from the center, and then you call one of these angles a "deeg." Then you've just defined a new way to measure a circle. 100 deegs are in a circle.

This invented unit, the deeg, is much like the degree, except the degree is smaller (why?). They are both angles. Just as 1 inch = 2.54 centimeters, although the centimeter is smaller, the inch and centimeter are both units of length. So the ancient Babylonians (not the Greeks), decided that a circle should contain 360 degrees. In one degree there are 60 minutes (though they have the same name, one minute-angle is not the same as one minute-time). Furthermore, in one minute there are 60 seconds (again, one second-angle is not one second-time, though they are the same word).

The British military chose a different way to divide the circle, specifically, 400 gradients in one circle. So one gradient is a tad bit smaller than a degree. And what's a radian? It's what mathematicians use because there's a way to divide the circle into a number of parts that happen to make certain computations easy. The way they decided this was that they took a circle, say with radius 1 cm. Then they took a piece of string, and made marks on it, evenly spaced 1 cm apart. Then they took the string and wrapped it around the circle. They then asked how many little 1 cm pieces of string fit around the circle, and they got the answer of about 6.2831853 pieces. They decided that the angle that a 1 cm piece of string covers as it is wrapped about the edge of a circle of radius 1 cm should be called one radian. Weird but true.

Now, one might wonder why the Babylonians chose the number 360. The reason is that their number system was based on the number 60. To compare, we base our number system on 10. For us, 10 is a nice, round number and we find it very convenient to count in multiples of 10, like millimeter, centimeter, meter, kilometer, etc. But the Babylonians liked 60.

Why this was nice for them, nobody knows, but 60 is a nice number too, because 60 = 2 x 2 x 3 x 5 and 360 = 2 x 2 x 2 x 3 x 3 x 5. What's so neat about that, you ask? Well, you will find that 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, and 20. There are few other numbers as small as 360 that have so many different factors. This makes the degree a very nice unit to divide the circle into an equal number of parts. 120 degrees is 1/3 of a circle, 90 degrees is 1/4, and so on.

INTERESTING PI FACTS

• Pi is an irrational number. It means that it cannot be written as the ratio of two integer numbers. 22/7 is a popular one used for Pi but it is only an approximation, which equals to 3.142857143...
• Another characteristic of pi as an irrational number is the fact that it takes an infinite number of digits to give its exact value, i.e. you can never get to the end of it.
• One of the most accurate fractions for Pi is 104348 / 33215. it is accurate to 0.00000001056%
• First 100 digits
  3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...
• 1.2411 trillion digits (1,241,100,000,000) digits of pi have been dicovered
• You can determine your hat size by measuring the circumference of your head, then divide by Pi and round off to the nearest one_eighth inch.
• The height of an elephant (from foot to shoulder) = 2 * Pi * the diameter of its foot.
• The Babylonians, in 2000 B.C. were the first people known to find a value for Pi.
• Pi day is celebrated on March 14 at the Exploratorium in San Francisco (March 14 is 3/14) at 1:59 PST which is 3.14159.
• Pi Approximation Day is on the 22 / 7 - that is, July 22. For the past few years, people at Chalmers University have celebrated it.
  

Real Life Applications of Imaginary Numbers

• Complex numbers enter into studies of physical phenonomena in ways that most people can't imagine.

• For example: A differential equation, with coefficients like the a, b, and c in the quadratic formula, that models how electrical circuits or forced spring/damper systems behave. The movement of the shock absorber of a car as it goes over a bump is an example of the latter. The behavior of the differential equations depends upon whether the roots of a certain quadratic are complex or real. If they are complex, then certain behaviors can be expected. These are often just the solutions that one wants.

• Closely related to the electrical engineering example is the use of complex numbers in signal processing, which has applications to telecommunications (cellular phone), radar (which assists the navigation of airplanes), and even biology (in the analysis of firing events from neurons in the brain).

• In modeling the flow of a fluid around various obstacles, like around a pipe, complex analysis is very valuable for transforming the problem into a much simpler problem. When everything from large structures of riveted beams to economic systems are analyzed for resilience, some very large matrices are used in the modeling.

• In everyday use, industrial and university computers spend some fraction of their time solving polynomial equations. The roots of such equations are of interest, whether they are real or complex

THE INTEGRAL SYMBOL

• The ∫ symbol is used to denote integral in mathematics.

•  The notation was introduced by the German mathematician Gottfried Wilhelm von Leibniz towards the end of the 17th century.  

• The symbol chosen to be a stylized script "S" (long s ) because the integral is a limit of sums.

Euler's depiction on SWISS 10 FRANC NOTE

EULER'S POWERS OF MEMORY AND CONCENTRATION

Euler's powers of memory and concentration were legendary :

Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy.

Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as " Cyclops".

Euler later suffered a cataract in his good left eye, rendering him almost totally blind a few weeks after its discovery in 1766.

Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory.

For example: Euler could repeat the Aeneid of virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

JOHANN CARL FRIEDRICH GAUSS

 PRINCE OF MATHEMATICS

Johann Carl Friedrich Gauss , a German mathematician who had a remarkable influence in many fields , including number theory, statistics, analysis, differential geometry, electrostatics, astronomy and optics is ranked as one of history's most influential mathematicians.

CHILD PRODIGY:

At the age of three he amazed his father by correcting an arithmetical error.

In primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of integers. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher. 

Gauss's presumed method, which supposes the list of numbers was from 1 to 100, was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums:

1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050 .

FAMOUS QUOTE:

“Mathematics is the queen of the sciences and number theory is the queen of mathematics”

“Ask her to wait a moment,I am almost done” (he told this while working when he was informed that his wife is dying).

CONTRIBUTIONS:

• In Disquisitiones Arithmeticae, one of the most brilliant achievements in mathematics, Gauss systematized the study of number theory. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
• Gauss proved the Fundamental Theorem of Algebra, which states that every polynomial has a root of the form a+bi.
• He also discovered the Cauchy Integral theorem for analytic functions
• Gauss's work in mathematical physics contributed to potential theory and the development of the Principle of Conservation of Energy.
• Theoria motus corporum celestium (theory of motion of the celestial bodies) is his most significant work on applied mathematics.
• Gauss discovered Ceres, the largest of the asteroids orbiting around the Sun.
• His Theory of Celestial Movement remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant.
• Introduced the Method of Least Squares, a procedure used in all sciences to this day to minimize the impact of measurement error.

142857 - CYCLIC NUMBER

142857 is a cyclic number, the numbers of which always appear in the same order but rotated around when multiplied by any number from 1 to 6.

142857 * 2 = 285714

142857 * 3 = 428571

142857 * 4 = 571428

142857 * 5 = 714285

142857 * 6 = 857142

INTERESTING FACTS

• The word "MATHEMATICS" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning".

• The word "FRACTION" derives from the Latin " fractio - to break".

• GEOMETRY(Ancient Greek: γεωμετρία; geo = earth, metria = measure) is a part of mathematics.

• "ALGEBRA" comes Arabic word (al-jabr, literally, restoration)

• There are just four numbers (after 1) which are the sums of the cubes of their digits:

153 = 1^3 + 5^3 + 3^3

370 = 3^3 + 7^3 + 0^3

371 = 3^3 + 7^3 + 1^3

407 = 4^3 + 0^3 + 7^3

• 111,111,111 multiplied by 111,111,111 equals 12,345,678,987,654,321.

• Among all shapes with the same area circle has the shortest perimeter.

• For every object there is a distance at which it looks its best.

• The Five Most Important Numbers in Mathematics in One Equation :

e^i*pi + 1 = 0

QUOTES ON MATHEMATICS

• Pure mathematics is, in its way, the poetry of logical ideas. ~Albert Einstein

• Mathematics are well and good but nature keeps dragging us around by the nose. ~Albert Einstein

• The essence of mathematics is not to make simple things complicated, but to make complicated things simple. ~S. Gudder

• Go down deep enough into anything and you will find mathematics. ~Dean Schlicter

• The laws of nature are but the mathematical thoughts of God. ~Euclid

• The highest form of pure thought is in mathematics - Plato

• Mathematics is the queenof sciences and Arithmetic is the queen of mathematics - Gauss

• Perfect numbers like perfect men are very rare - René Descartes

• Mathematics is the language with which God has written the universe.- Galileo

• The essence of mathematics lies in its freedom - Cantor

• The moving power of mathematical invention is not reasoning but imagination – De Morgan

SIMPLE MATH PUZZLES

PUZZLE 1

Divide 110 into two parts so that one will be 150 percent of the other. What are the 2 numbers?

PUZZLE 2

The following number is the only one of its kind. Can you figure out what is so special about it? 8,549,176,320.

PUZZLE 3

Jennifer took a test that had 20 questions. The total grade was computed by awarding 10 points for each correct answer and deducting 5 points for each incorrect answer. Jennifer answered all 20 questions and received a score of 125. How many wrong answers did she have?

PUZZLE 4

When asked about his birthday, a man said:"The day before yesterday I was only 25 and next year I will turn 28."This is true only one day in a year - when was he born?

PUZZLE 5

The following equation is wrong: 101 - 102 = 1. Insert one mathematical operator to make it correct.

PUZZLE 6

NOTHING BUT ZEROS!
Use FIVE 0's and any math operators to get 120.

PUZZLE 7

What is the next number in the series? 01, 1011, 111021, 31101211, ?

SOLUTIONS

PUZZLE 1 : 44 and 66


PUZZLE 2 : It's the only number that contains all of the digits in alphabetical order.


PUZZLE 3 : 5. She had five wrong answers. If Jennifer had answered all 20 questions correctly, she would have scored 200. Since she only scored 125, she must have lost 75 points. Since each incorrect answer results in a total loss of 15 points (10 for not getting it correct and 5 for answering incorrectly) she must have missed 5 questions. 5 x 15 = 75, 200 - 75 = 125.


PUZZLE 4 : He was born on December 31st and spoke about it on January 1st.


PUZZLE 5 : 101-10^2 = 1


PUZZLE 6 : ( 0! + 0! + 0! + 0! + 0! )! = 120


PUZZLE 7 : 132110111221...... first its 01 (one 0, one 1) etc.... so its one 3 two 1 one 0 one 1 one 2 and two 1!!!!

FOR MORE PUZZLES CLICK HERE

SEQUENCE OF NUMBERS WITHOUT 8

12345679 x 09 = 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

NUMERIC PALINDROMES WITH 1'S

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

SEQUENTIAL 8'S WITH 9

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

SEQUENTIAL 1'S WITH 9

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
12345678 x 9 + 9 = 111111111

123456789 x 9 + 10 = 1111111111

SEQUENTIAL INPUT OF NUMBERS WITH 8

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 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

ORIGIN OF ARABIC NUMERALS

This chart shows the origin of Arabic numerals, which were defined according to the number of angles .

REGULAR POLYGONS LISTED BY NO. OF SIDES

enagon or monogon {1}


Digon {2}


Triangle {3}


Quadrilaterals {4}


Pentagon {5}


Hexagon {6}


Heptagon {7}


Octagon {8}


Nonagon {9}


Decagon {10}


Hendecagon {11}


Dodecagon {12}


Tridecagon {13}


Tetradecagon {14}


Pentadecagon{15}


Hexadecagon{16}


Heptadecagon{17}


Octadecagon{18}


Enneadecagon{19}


Icosagon{20}

1729 - TAXICAB NUMBER

1729 is smallest taxicab number , i.e., the smallest number representable in two ways as a sum of cubes. It is given by

1729 = 12³ + 1³

1729 = 10³ + 9³

The number derives its name from the following story:

G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

1729 is the second taxicab number (the first is 2= 1^3 + 1^3). The number was also found in one of Ramanujan's notebooks dated years before the incident.

"Every positive integer is one of Ramanujan's personal friends."—J.E. Littlewood, on hearing of the taxicab incident

More details are available on the attached link.
Source(s):
http://en.wikipedia.org/wiki/1729_(number)

ONE EQUALS ZERO

Consider the following integral:
INTEGRAL (1/x) dx
Perform integration by parts:
let u = 1/x , dv = dx du = -1/x2 dx , v = x
Then obtain:
INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x2) dx
= 1 + INTEGRAL (1/x) dx
which implies that 0 = 1.

USES OF LAPLACE TRANSFORMS

 PIERRE SIMON LAPLACE

• The Laplace transform is a simple way of converting functions in one domain to functions of another domain.

Here's an example :
Suppose we have a function of time, such as cos(w*t). With the Laplace transform, we can convert this to a function of frequency, which is

cos(w*t) ----L{}-----> w / (s^2 + w^2)

• This is useful for a very simple reason: it makes solving differential equations much easier.

• The development of the logarithm was considered the most important development in studying astronomy. In much the same way, the Laplace transform makes it much easier to solve differential equations. 

• Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial. Since polynomials are much easier to solve, we would rather deal with them. This occurs all the time.

• In brief, the Laplace transform is really just a shortcut for complex calculations. It may seem troublesome, but it bypasses some of the most difficult mathematics.  

• Laplace transform is a technique mainly utilized in engineering purposes for system modeling in which a large differential equation must be solved.

• The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.

• Laplace Transform is used in electrical circuits for the analysis of linear time-invariant systems