CLICK HERE TO CHECK CLASS XII MATHS NCERT BOOK SOLUTIONS.

Happy Birthday, Srinivasa Ramanujan!

Today is the birthday of Srinivasa Ramanujan, the great Indian mathematician who studied number theory, mastered modular and partition functions, and designed summation formulas. Ramanujan was born on December 22, 1887 in Erode, a city along the banks of the Cauvery River in the southern state of Tamil Nadu. He enrolled in a local high at the age of 10, but learned more about mathematics from the college students who boarded in parents' home. According to Robert Kanigel, Ramanujan's biographer and author of The Man Who Knew Infinity, the young mathematician was deeply influenced by two borrowed books: S.L. Loney's Plane Trigonometry and George Shoobridge Carr's Synopsis of Elementary Results in Pure Mathematics. Carr's work, a list of 5000 mathematical formulas, inspired Ramanujan to develop his own proofs for these theorems. By the age of 17, Ramanujan had calculated Euler's constant to 15 decimal places and proposed a new class of numbers. Although his peers "stood in respectful awe of him", said one contemporary, "we, including his teachers, rarely understood him".
Like Albert Einstein, Srinivasa Ramanujan struggled with school and even failed his high school exams because of difficulties concentrating. In 1909, the 22-year old college dropout moved from Erode to Madras and found work as a clerk in the Accountant General's Office. Ramachandra Rao, an Indian mathematician who helped Ramanujan obtain the clerkship, encouraged the young man to publish papers and seek broader support for his work. In 1911, Ramanujan's 17-page paper about Bernoulli numbers appeared in the Journal of the Indian Mathematical Society. Two years later, the young mathematician wrote a 10-page letter with over 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory. The letter's recipient, a Cambridge mathematician named G.H. Hardy, had ignored previous communications from Ramanujan, but shared this latest letter with J.E. Littlewood, a university colleague. According to Hardy, the English mathematicians concluded that Ramanujan's results "must be true because, if they were not true, no one would have the imagination to invent them."
With Hardy's help, Ramanujan was named a research scholar at the University of Madras, a position that doubled his clerk's salary and required only the submission of quarterly reports about his work. In March 1914, Ramanujan boarded a steamship for England and, upon his arrival at Cambridge University, began a five-year collaboration with G.H. Hardy. Together, the scholars identified the properties of highly composite numbers and studied the partition function and its asymptotics. They also identified the Hardy-Ramanujan number (1729), the smallest number expressible as the sum of two positive cubes in two different ways. Individually, Ramanujan made major breakthroughs with gamma functions, modular forms, divergent series, hypergeometric series, and mock theta functions. He also developed closed-form expressions for non-simple, continued fractions (Ramanujan's continued fractions) and defined a mathematical concept known as the Ramanujan prime. "I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people," Hardy later wrote, "'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.'"
Srinivasa Ramanujan received an honorary bachelor's degree from Cambridge University in 1916, and was later appointed a Fellow of Trinity and a Fellow of the Royal Society. Despite his professional accomplishments, Ramanujan suffered from poor health and was eventually diagnosed with tuberculosis and amoebiasis, a parasitic infection of the liver. A vegetarian, he also suffered from a severe vitamin deficiency that may have been due to the shortage of fresh fruits and vegetables in wartime England. Srinivasa Ramanujan died on April 26, 1920 at the age of 33. Today, his home state of Tamil Nadu celebrates his birthday, December 22, to memorialize both the man and his achievements.
Source of this article is here.
To know more about this mathematics genius, check the following links:
1 2 3 (song about Ramanujan) 4 5 6 (Quotes)

One minute Question Paper in Ireland

Hello
Just got this in one email by my friend. very interesting. Try solving the paper before looking at the solution.
(cheating is not allowed!)



Scroll Down for Answer:

***

***

***

***

***

***

***

***




MATH SAMPLE PAPER FOR CLASS X AND XII

Dear all

Since the countdown for the board exams has begun! You ( students in class X and XII) must be preparing well for your examinations. Kindly download the Sample mathematics paper for your practice.

Class X Class XII

All the Best.
Do well...

PLATONIC SOLIDS AND THEIR NETS

Definition REGULAR POLYHEDRA:
A 3-dimensional object bounded by regular polygons.
Can you guess how many regular polyhedra exist? SURPRISINGLY…. There are only FIVE regular solids composed of just 3 regular polygons: the triangle, square and pentagon.
The five regular polyhedrons are Tetrahedron, Cube, Octahedron, Icosahedron and Dodecahedron.
Also, if you combine 2 regular polygons triangle with square, pentagon with triangle and so on, you can make other polyhedra. There are 13 such more solids known as “Archimedian Solids”.
Click here to download the nets of Platonic Solids.
Note: Right click the above link and select the option "Save link as "

Visit the following sites to know more :
Site 1 Site 2 Site 3 Site 4 Site 5

CBSE SAMPLE PAPERS (MATHEMATICS)- CLASS X AND XII

Dear All

By this time, I hope that you must have gone through the last year Question Paper and the Sample Papers released by CBSE.
I am providing here the links for the same for your ready reference.

CLASS X (MATHEMATICS)

Sample Papers (1,2 and 3) with Solutions

HOTS Questions

HOTS by KVS MYSORE REGION

HOTS BY KVS DELHI REGION

HOTS BY KVS CHENNAI REGION

Question Bank by DOE


CLASS XII (MATHEMATICS)

Sample Papers (Released for 2009 Exams) here

Sample Papers (Released for 2010 Exams) Part 1 Part 2

HOTS by KVS banglore region

HOTS by KVS

Question Bank by DOE : For Commerce Students For Science Students

NOTE: For all other subjects visit official site of CBSE .
Last Year CBSE Question Papers, click here.
For all other subjects: Question Bank By DOE, click here.
For class IX and X , Sample papers for SA2, click here.

Sample Papers by DOE - Class X and XII with Study Tips

Dear Students

You must be now preparing for your board exams .
There are some tips and tricks that will help you a lot in preparation for the board exams. How many subjects, you touch for on daily basis and how much time you give for a single subject. What is the way of your preparation, these are some points which plays very important role in getting and scoring good marks in the board exams. Let’s talk about how to prepare:

Try to study at least four subjects out of total five per day. This will keep your memory sharp for all subjects. Then spend one and half an hour for a single subject. It means, you have to give six hours daily for your all four subjects. But don’t study continuously.

Here are a few tips that may help you reduce the exam stress:

Believe in yourself.
You are capable of passing the exam. You wouldn't have been given a place in the class or on the course if you didn't have the ability to do it.

Don’t just worry – take action!
If you don't understand some of your course material, getting stressed out won't help. Instead, take action by seeing your course tutor or asking your class mates to help you understand the problem.

Don’t put yourself under too much pressure
Aim to do your best but recognize that if you think that "anything less than A+ means I've failed" then you are creating unnecessary stress for yourself.

Take a break.
As soon as you notice you are losing concentration, take a short break – go for a walk, talk to a friend or just listen to some music. Then you will feel refreshed and able to concentrate on your revision again.

Don’t get tensed, it can result in forgetting things that you know very well. Keep studying with fresh mind every time after you take break. Studying in morning is very beneficial as compare to late night study. Studying in morning can be quite difficult but soon you will get habitual for that and you will find more interest in studying.

If you will study like this I am sure you will definitely get high marks in your all exams, CBSE board exams and all other competitive exams.

Click here to download Sample Papers (Class X) released by Directorate of Education (DOE).
For class XII Sample Papers and other related material, visit my school's blog.

In case any doubt, do feel free to write and give your comments, views and suggestions.

ALL THE BEST

Solve it - Question 9

Dear All

Due to some unavoidable circumstances, i was not able to blog for many days. Really missed it!
Anyways, I have a problem for you .Try it...

Sumit went to the shopping centre to buy supplies for his mathematics project. He spent half of what he had plus Rs. 2 in the firts shop, half of what he then had left plus Rs. 1 in the second shop; half of what he then had left plus Re. 1 in the third shop and, in the fourth shop half of all he had left. Three Rs. were left over. How much money did he start with?

ANSWER:
Have fun and enjoy mathematics.

Orientation Program - class IX about CCE

Dear All

On 7 November 2009, an orientation program for class IX about Continuous and Comprehensive Evaluation (CCE) was organised in the school.
Parents of the students studying in class IX were invited for the same. Parents were addressed by Principal Sir, Vice-Principal Ma'am and me. The basic idea was to create awareness among parents about the new evaluation system, which we will be followed in our school for class IX and then will be continued till class X.

I was a part of the school's team which attended a teacher-training workshop organized by CBSE for the same. It was really a wonderful experience to share the acquired knowledge with all.

For more details you can visit school's blog.

Believe it or not - 12

Dear All

The study room of the famous mathematician Sir Isaac Newton (1642-1727) in the Royal Society which had not been used after his death, was made available to Ramanujan (1887-1920), the greatest Indian Mathematician. He was the first Indian to be elected to be coveted fellowship of the Royal Society of England.

Believe it or not - 11

Dear All

Russian King Peter the Great announced that no member of the Royal Family should marry until they pass in arithmetic, geometry and navigation.

Any comments...

Annual Day Celebration on 15 October 2009

Dear Students,

You must be knowing by now that CRPF School, Rohini is going to celebrate its Annual day on Thursday ie. 15 October 2009.
The function will start at 5:30 pm. A large number of children are participating in various activities - Orchestra, Choir, Ballet, Dance etc.
All those alumni students who wants to propose their help and support for the smooth functioning of the program, can contact me at amitbajajcrpf@gmail.com for further details. It is to mention that students who really love their institution and
are willing to help are supposed to contact.Of course you will be getting an opportunity to meet with your friends and teachers also.
No students without the prior permission from Principal/Teacher will be allowed to attend the program.
You can also visit www.crpfpsrohini.blogspot.com for details of the program (will be updated soon).

With lots of love
Amit Sir

PROJECT WORK TOPICS FOR CLASS IX AND X


-->
Dear All
We are well aware that CBSE is acknowledging the concept of internal assessment in Mathematics for the past three years wherein a student studying in class IX and X is supposed to submit a project work in Mathematics. This project can be in the form of extended learning in the subject or related to real life situations as well. Click here to download the evaluation performa for the same.
I have observed that students are usually confused while selecting the topic for their project work. That is why; I want to share a few possible topics to help them pick up a suitable topic which suits their aptitude and attitude (as separate posts). Students will have the liberty to take the decision of taking up individual or a group projects (though they need the consent of their concerned math's teacher!). I would also discuss (in brief) about each topic and share some related resources which may prove to be helpful.
few suggested topics are:
1. day on given date
2. tessellations
3. fibonnaci numbers
4. paper sizes
5. tower of hanoi
6. divisibility rules for prime numbers
7. unit digit of x^y
8. joseph shooting problem
9. magic squares
10. polyominoes
11. platonic and archemedian solids
12. optical illusion in maths
13. some uncommon results in maths
14. great mathematicians and their contributions
15. vedic maths and its relevance
16. various proofs of pythagoras theorem
17. golden ratio in nature / human body
18. generation of solids- 2d to 3d model
19. working model of ellipse
20. square root spiral
21. soma cube
22. common error in maths
23. fractals (sprinski triangle, kosh snowflake etc)
You are welcome to clarify any of your doubts. Also, do suggest other topics you know about. I would want you to share some more trivia related to already suggest topics. I will be waiting for your response…

All the best and happy reading!

Believe it or not - 10

Dear All

There are certainly at least two peoples in the world with exactly the same number of hairs on their heads!
(There are more people than the hairs on any one head).

Believe it or not - 9

Dear all

Descartes (1596-1650) wrote his coordinate geometry without the use of negative numbers.

Believe it or not - 8

Dear All

Believe it or not, three legged stool is more stable than any other stool. The principle underlying is this fact is we can draw only one cirlcle through three given points which are not collinear. That is why old people walk with the assistance of a stick (third leg) , or even think of tripod.

Believe it or not - 7

Dear All

A plane can be filled with squares, equilateral triangles and regular hexagons. No other regular polygons can fill the plane without gaps.
visit this link to know more.

Solve it - Question 8

Dear All

One of my student (now not studying in CRPF School)shared this math problem with me. Let us all try together to solve it.
check website www.shubhamsuman.co.nr (Must Visit)

The question is as follows:

Let two cyclists A and B start from a point 100kms apart from each other with the speed of 10kms towards each other. As they start a bee takes off from the nose of a cyclist a and heads towards B's nose.It turns back and heads towards A's nose.It continues this back and forth movement till it gets squashed between cyclist.in doing this how much total distance does the bee covers? Given speed of bee is 20 km/hr.

Well, he also shared this piece of information. Hope all the readers will like it...

HANGING NECKLACE
Borrow a necklace from a shop and hold it up in the air by its two ends at the same level so that it hangs downwards.The necklace assumes the shape of a curve.This curve is named as catenoid.
the problem posed by jacob bernoulli was to find the equation for the curve.
the curve is not a parabola.
it was found that
y=a(e^(x/a) + e^(-x/a))/2 + b
a and b are constants
a =depends on physicsal characteristics e.g. density how far ends were held .
b =the placement on x axis which can be 0 also.
ITS APPLICATION
Among all possible shapes that hanging string can assume ,catenary is one for which potential energy of string is minimised .
centre of gravity at this shape of string is at the lowest.
Dip the two identical circular hoops of wire into a soap solution in contact with one another along their entire circumference.Let the hoops be pulled out of soap solution and drawn apart carefully.Soap solution produce s a soap film connecting the two hoops, and the shape assumed by the film is a catenoid.The soap film the shape which minimises the total potential energy.

Believe it or not - 6

Dear All

What is the largest number of circles that can touch another circle of equal size? The answer is 6. In case of spheres this number is 12 (total 13). Newton discovered this property in 1694. Believe it or not this property was proved by R. Hoppe only in 1874 after 180 years of its discovery.

MOBIUS STRIP

Dear All

If we take a rectangular strip of paper, then make a half twist and join the ends we come up with a Mobius strip. If we were to draw a line through the center of the strip without lifting the pencil off the paper, we would come back to the starting point but on the "opposite" side of the paper. Logically, this is only possible if the surface has only one side and only one boundary, meaning that while the Mobius strip appears to have two sides, it actually has one.FEW EXPERIMENTS WITH MOBIUS STRIP:

Experiment 1: Draw a line through the center of the strip. We would have to go round the loop twice to get back to the starting point. This is a key feature of the Mobius strip because it's what describes it as a non orientable surface.
Experiment 2: Cut through the center line. In general, if we cut a rectangular strip of paper lengthwise through the middle from end to end, we would expect to get two strips. This is not the case with the Mobius strip. Instead of getting two strips, we get one long strip with two full twists in it. If we cut the strip again through the center line, we come up with two strips wound around each other.
Experiment 3: Cut through the line about one third from the edge. (note: we have to go twice round the loop), we get two separate strips, one of which is thinner, but of the same length as the original strip. The other will be a long strip whose length is twice that of the original strip.
Experiment 4: Cut once round the loop through the line about one third from the edge, then cut through the center line of the resultant thicker strip. We get three strips wound around each other, one in the middle and one on either side.

A strip with an odd number of twists will behave the same way as a Mobius strip, that is, it will have one edge and one side. On the other hand, a strip with an even number of twists will have two boundaries and sides.

APPLICATIONS:

Mobius strips have been used as conveyor belts because their one sided nature allows equal wearing of "opposite sides" of the belt. This makes the belts last longer. The strip has also been used in recording tapes to double the playing time without having to manually take out the tape and change the side playing. It is also used in numerous electronic appliances especially those which have resistors and superconductors.


INTERNATIONAL MATHEMATICS OLYMPIAD (IMO)

Dear All

National Science foundation organises IMO every year. The benefits of participating in IMO are as follows:

  • Every participant of Level I is awarded a Certificate of Participation. Toppers are awarded merit certificate.
  • Students get a chance to be assessed at international level and are awarded performance-based rankings.
  • Students gain confidence to contest at international level in a world where from admission to elite courses to selection for choice jobs is becoming more and more competitive.
  • The spirit of competition and sense of recognition inspire students to excel in the chosen field of study and set higher goals.
  • Top three students from each class are awarded Gold, Silver and Bronze medals.
  • The top 500 winners (class-wise) are endowed with cash prizes and scholarships, courtesy of the official sponsor.
  • Every participant who scores 50% or more is awarded a Certificate of Participation. Toppers are awarded merit certificate.

CLASSES: II TO XII

EXAM FEE: RS 100

EXAM DATE: 10 DECEMBER

SYLLABUS: AS PER CBSE

EXAM PATTERN: 50 MULTIPLE CHOICE QUESTIONS (MCQ) TO BE ANSWERED IN ONE HOUR


HOW TO PREPARE: IMO WORKBOOKS FOR II TO X CLASS ARE AVAILABLE

SAMPLE PAPER: CLICK THIS LINK.

VISIT www.sofworld.org FOR FURTHER DETAILS

LAST DATE OF REGISTRATION: CONTACT YOUR SUBJECT TEACHER

Believe it or not - 5

Dear All

There are only five Heronian triangles (triangles having integer sides) such that the perimeter is equal to the area. They are : (5, 12, 13) , (6, 8, 10) , (6, 25, 29) , (7, 15, 20) and
(9, 10, 17). The first two triangles are right triangles also.

Solve it - Question 7

Dear All

Solve the following problem. Is the remainder theorem applicable here?



Solution:

(Click image to enlarge)

Solve it - Question 6

Dear All

A rectangle with perimeter 44 units is partitioned into 5 congruent rectangles, as indicated in the diagram. Find the perimeter of each of the congruent rectangle.

I hope that the hint figure given below will help you to answer this!

Believe it or not - 4

Dear All

Do you think that anything and everything in Mathematics has been discovered till date, or is there a scope for NEWER Maths?
Read below and then think again!!!

The number of periodicals that publish mathematical research papers and articles are given below:

Prior to 1700 AD

17 periodicals

18th Century

210 periodicals

19th Century

950 periodicals

20th Century

2000 periodicals


According to one estimate every year nearly 2,00,000 (two lakhs, no typing mistake) new theorems and methods appear in the journals and the books.

So... What are you thinking? To derive a new maths formula or result!
Good luck!

Solve it - Question 5

Dear All

Try this question:

The time on electronic digital watch is 11 : 11. How many minutes before this would the watch have shown a time with all digits identical?

Here is the Solution:

The required digits on the clock before 11:11 would be 5:55 (Note that times like 6.66, 7.77 are not possible).
This gives a time difference of 316 minutes (11 : 11 – 5.55).

Believe it or not - 3

Dear All

Most of us know that the sum of two sides of any triangle is greater than the third side. But very few of us know that the sum of three sides of a triangle is greater than the sum of the bisectors of the angles of the triangle.

Well... whats about the relationship between four sides of any quadrilateral? any guess....
yes, you are right....
The sum of any three sides of a quadrilateral is always greater than the fourth side.

Believe it or not - 2

Dear All

A side of a regular pentagon, of a regular hexagon, and of a regular decagon inscribed in the same circle constitute the sides of the right triangle.

Believe it or not - 1

Dear All

Believe it or not there is no equilateral triangle , the coordinates of whose vertices are all integers.
Well... it can be proved very easily. Try it!

Solve it - Question 4

Dear All
Try this question:

Each of the integers 226 and 318 have digits whose product is 24. How many three digit positive integers have digits whose product is 24?

Quite simple...
Check your answer here.

Solve it - Question 2

Dear All

The given figure is formed by two squares. The side of each square is a whole number. If area of the figure is 58 cm^2, find its perimeter.
Click here to check your solution.

Solve it - Question 1

Dear All

Find the sum of digits of number 1000^20 – 20 expressed in decimal notation.



Click here for answer.

A view of Mathsland


DEAR ALL


To most people, mathematics is that subject they have always had difficulty understanding. It is a form of communication, a kind of strange language in which complete sentences must have something called an equals sign or some other equally strange symbol. It appears to be a form of the English language but interlaced with rows of austere symbols and incomprehensible formulae (some Martian language!).

For different reasons, the majority of the world's `educated' population, by the time they graduate from high school, have already made up their minds that mathematics is difficult and that nothing new ever happens in mathematics. My suspicion is that this uninformed majority spreading these unfounded rumors have no personal experience with mathematics. They probably heard this story from a friend who in turn had heard rumors from elder brothers and sisters that mathematics is a difficult subject. Believing this lie and obviously lacking self-confidence and motivation, most decide to give up before they even give it a try.

The world of mathematics is an ever-changing one. It is a world highly endowed with provocative ideas, very rich in poetry and full of vivid images.

To read the complete article, click here.



'True' life story - Nobert Weiner



Dear All

Weiner, a great mathematician of this century, was very absent minded. The following story is told about him: When he and his family were moving house, his wife was so certain that he would forget that they had moved and where they had moved to that she wrote down the new address on a piece of paper, and gave it to him. Naturally, in the course of the day, an insight occurred to him. He reached in his pocket, found a piece of paper on which he furiously scribbled some notes, thought it over, decided there was a fallacy in his idea, and threw the piece of paper away.
At the end of the day he went home --- to his old address. When he got there he realized that they had moved, that he had no idea where they had moved to, and that the piece of paper with the address was long gone. Fortunately inspiration struck. There was a young girl on the street and he conceived the idea of asking her where he had moved to, saying, "Excuse me, perhaps you know me. I'm Norbert Weiner and we've just moved. Would you know where we've moved to?" To which the young girl replied, "Yes daddy, mommy thought you would forget."

Why is there no Nobel Prize in Mathematics?


Dear All


Six Nobel Prizes are awarded each year, one in each of the following categories: literature, physics, chemistry, peace, economics and medicine. Notably absent from this list is an award for Mathematics. The reason for this conspicuous omission has been subject of extensive speculations, some of which are discussed below.

One of the most common reasons as to why Nobel decided against a Nobel prize in math is that [a woman he proposed to/his wife/his mistress] [rejected him beacuse of/cheated him with] a famous mathematician. Gosta Mittag-Leffler is often claimed to be the guilty party.

There is no historical evidence to support the story.

For one, Mr. Nobel was never married.

Click here to read the complete article.

The prof's age

Dear All
Its just for fun...

Professsor: Given the age of the light as 3 x 10^8 m/s and the density of water as 1km/m^3, what's my age?

Student: I'd say about -- 46.

Professor: Excellent! How did you know that?

Student: You see, sir, my brother's 23 and he's only half-mad.

Train alert!


A group of people were asked the following question. ``Say there are 2 people tied to a railway track, and a train is fast approaching. You have time to save just one of them. Which one would you save?''

The politician replied ``After arranging for a television crew to be present, and preferably making sure there was a press conference afterwards, I would rescue the one with the louder screams.''

The accountant replied ``I would rescue the one who offered to pay me most.''

The journalist replied ``I would ask the train driver to delay the collision till I could get a camera.''

The lawyer replied ``I would jump into the train and offer my services to the driver, who will almost certainly be sued by relatives of the two. ''

The physicist replied ``I would derail the train, as your question did not place any limitations on the safety of those in the train.''

The statistician replied ``I would toss a coin to pick one of them. Then I would toss the coin another hundred times to make sure that that was not a statistical fluke. Then I'd rescue the one selected.''

The Mathematician replied ``I would make two rescues, each saving half a person. But since persons come in whole numbers, in the mathematics of persons 1/2 is always rounded to 1, so each rescue saves a person. There are two rescues, so 2 people are saved.''

Measuring the height of trees

Dear All

Some Native Americans had a very interesting way of doing this. To see how high a tree was, they would find a spot where, looking under their legs (as shown), they could just see the top of the tree. The distance from such a spot to the base of the tree was approximately the height of the tree.

Why does this work? The reason is quite simple. For a normal, healthy adult, the angle formed by looking under one's legs is approximately 45o. Hence, the distance to the tree must be around the same as the height of the tree.

Poor Man's space travel

I met a youngster rummaging through a dust bin. He seemed to be interested only in large sheets of paper.
"What are you doing?"
"I am trying to get to the moon."
"Are you going to make a paper spaceship?"
"No, it's much simpler than that."
He put one piece of newspaper down and stood on it.
"I am now nearer to the moon."
He doubled the paper and stood on that. Then doubled again and stood on top of that -- there were now 4 thicknesses of paper, say a total of 4/10 of a millimetre and he carried on doubling.
After a few more doublings I began to get the idea. It is roughly 400000km to the moon. How many times must he double?
Surprisingly, the answer is only 43.
The pattern is 1, 2, 4, 8, 16,..., each term doubling the previous one. Such a sequence is called a Geometric Progression and the nth term is given by 2^(n-1).
Geometric Progressions (GP's) often have terms which get very big like this one. For some GP's however, the terms get smaller, look at the series1, 1/2, 1/4, 1/8, 1/16,... for example.
The population explosion is often analysed by considering population increase, e.g. 3% as being added at the end of a year. The GP is made by multiplying by 1.03 to get successive terms.
``Population increases geometrically: Food increases arithmetically. Population will therefore always outstrip food supply until famine, disease or war ensue.''

The most touching story in Mathematics


``And if you divide any number by itself, you get 1.''

The teacher in a small high school in southern India turned round to see a tiny hand trying to reach the ceiling. Oh by the gods, him again! That Aiyangar boy with his horribly difficult and quite irrelevant questions. Like last week, when he wanted to know how long it would take for a steam train to reach Alpha Centauri. As if he would be able to afford the fare if he knew. Well, he couldn't let him exercise his hands too much.

``Yes Ramanujan?''

The small boy with shining eyes stood up. He spoke slowly, with the calm confidence of one who did not need to be told he was the best in the class.

``Is zero divided by zero also equal to one?''

Unfortunately for all those other teachers who've been asked this question at least twenty times in their lives, the response to the question is unknown. But the life of the boy, Srinivasa Ramanujan Aiyangar, certainly isn't.

To read this complete article, click here.

Fermat's Last Theorem

Dear All


In the margin of his copy of a book by Diophantus, Pierre de Fermat wrote that it is possible to have a square be the sum of two squares, but that a cube can not be the sum of two cubes, nor a fourth power be a sum of two fourth powers, and so on. Further, he wrote that he had found a truly marvelous proof which the margin was too small to contain.

Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.

That is to say, there are no integers x, y, z such that x3 + y3 = z3 or integers x, y, z such that x7 + y7 = z7.

Although this is easily stated, it has proved to be one of the most puzzling problems in the whole history of mathematics. Long after all the other statements made by Fermat had been either proved or disproved, this remained; hence it is called Fermat's Last Theorem (actually, Conjecture would be more accurate than Theorem).

This conjecture was worked on by many famous mathematicians. Fermat himself proved this theorem for n = 4, and Leonhard Euler did n = 3. Special cases were dispatched one after another. New theories were developed to attack the problem, but all attempts at a general proof failed. They failed, that is, until this decade, when, building on work of many famous mathematicians, Prof. Andrew Wiles of Princeton University finally proved it in 1996. His method could not have been known to Fermat. Fermat's "truly marvelous proof" is now believed to have been faulty.

The actual proof is very indirect, and involves two branches of mathematics which at face value appear to have nothing to do either with each other or with Fermat's theorem. The two subjects are elliptic curves and modular forms and involve work done previously by Taniyama and Shimura. The greatest difficulty was in proving that the Taniyama-Shimura conjecture was true. This is the contribution made by Andrew Wiles, and the final stage in establishing Fermat's Last theorem.

To know more, click here.