Hyptia of Alexandria

Hyptia of Alexandria Hypatia of Alexandria was the first woman to make a substantial contribution to the development of mathematics. She was a Greek Neoplatonist philosopher in Roman Egypt and the head of the Platonist school at Alexandria who taught philosophy and astronomy. The life of Hypatia was one enriched with a passion for knowledge. Historians are uncertain of different aspects of Hypatia's life. Her date of birth is one that is highly debated. Some historians believe that Hypatia was born in the year 370 AD. On the other hand, others argue that she was an older woman (around 60) at the time of her death, thus making her birth in the year 355 AD. She was the daughter of Theon of Alexandria who was a teacher of mathematics with the Museum of Alexandria in Egypt. Throughout her childhood, Theon raised Hypatia in an environment of thought. Historians believe that Theon tried to raise the perfect human in her . Theon himself was a well known scholar and a professor of mathematics at the University of Alexandria. Theon and Hypatia formed a strong bond as he taught Hypatia his own knowledge and shared his passion in the search for answers to the unknown. As Hypatia grew older, she began to develop an enthusiasm for mathematics and the sciences especially in astronomy and astrology. References in letters by Synesius, one of Hypatia's students, credit Hypatia with the invention of the astrolabe, a device used in studying astronomy. Most historians believe that Hypatia surpassed her father's knowledge at a young age. However, while Hypatia was still under her father's discipline, he also developed for her a physical routine to ensure for her a healthy body as well as a highly functional mind. In her education, Theon instructed Hypatia on the different religions of the world and taught her how to influence people with the power of words. He taught her the fundamentals of teaching, so that Hypatia became a profound orator. People from other cities came to study and learn from her. Hypatia became head of the Platonist school at Alexandria in about 400 AD. There she lectured on mathematics and philosophy, in particular teaching the philosophy of Neoplatonism. Hypatia based her teachings on those of Plotinus, the founder of Neoplatonism, and Iamblichus who was a developer of Neoplatonism around 300 AD. Hypatia dressed in the clothing of a scholar or teacher, rather than in women's clothing. She moved about freely, driving her own chariot, contrary to the norm for women's public behaviour. She exerted considerable political influence in the city. Hypatia was known more for the work she did in mathematics than in astronomy, primarily for her work on the ideas of conic sections introduced by Apollonius. She edited the work On the Conics of Apollonius, which divided cones into different parts by a plane. This concept developed the ideas of hyperbolas, parabolas, and ellipses. With Hypatia's work on this important book, she made the concepts easier to understand, thus making the work survive through many centuries. Hypatia was the first woman to have such a profound impact on the survival of early thought in mathematics. According to the Suda Lexicon, a 10th-century encyclopaedia, Hypatia wrote commentaries on the Arithmetica of Diophantus of Alexandria, on the Conics of Apollonius of Perga, and on an astronomical canon .The known titles of her works, combined with the letters of Synesius who consulted her about the construction of an astrolabe and a hydroscope. The existence of any strictly philosophical works by her is unknown. Hypatia lived in Alexandria when Christianity started to dominate over the other religions. In the early 390's, riots broke out frequently between the different religions. Cyril, a leader among the Christians, and Orestes, the civil governor, opposed each other. Hypatia was a friend of Orestes and it is believed that Cyril spread virulent rumours about her. In 415 AD, on Hypatia's way home, a mob attacked her, stripped her and killed her with pieces of broken pottery. Hypatia's life ended tragically, however her life's work remained. Later, Descartes, Newton, and Leibniz expanded on her work. Hypatia made extraordinary accomplishments for a woman in her time. Philosophers considered her a woman of great knowledge and an excellent teacher. Send your valuable comments to RAJESH KUMAR THAKUR rkthakur1974@gmail.com

George Cantor

George Cantor George Cantor, the founder of transfinite set theory revolutionized mathematical thinking in this area. For centuries, the concept of infinity had been a highly controversial one both mathematically and philosophically and his ideas could not get the full impact in his life time. George Ferdinand Ludwing Philipp Cantor was born in St. Petersburg, Russia. His father George Woldemar Cantor was a stockbroker while his mother Maria Anna came from a musical family. Cantor was raised in an intensely religious atmosphere. In 1856, when young Cantor was 11, the family moved from St. Petersburg to Germany where he attended the Gymnasium school. Cantor showed all round ability there but he had special affinity with mathematics and science and his father decided that he should train as an engineer. He was therefore enrolled in Polytechnikum at Zurich but seeing his special interest in mathematics his father later decided that he should study mathematics instead of engineering and he was admitted to University of Berlin, where he proved to be a good but not excellent student. Cantor moved to the University of Berlin where he became friends with Hermann Schwarz who was a fellow student. Cantor attended lectures by Weierstrass, Kummerand Kronecker. He spent the summer term of 1866 at the University of Göttingen, returning to Berlin to complete his dissertation on number theory De aequationibus secundi gradus indeterminatis in 1867. After obtaining his doctorate in 1867, he became a school teacher for a short period and later joined University of Halle where he remained for the rest of his entire carrier. He married Vally Guttmam in 1874 and they had 6 children. While he was in Berlin, he became interested in Number Theory and came to the notice of Kronecker. His first achievement was in traditional field of trigonometric series, especially the question of uniqueness for the Fourier representation of a given function. He gradually showed that the 19th Century idea in the connection between the dimension of a set and the number of its elements were dangerously fallacious. He showed that the concepts of Cardinal and Ordinal Number could be defined mathematically in such a way that it made a good sense to talk of infinite or transfinite numbers. In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts to decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874. It is in this paper that the idea of a one-one correspondence appears for the first time, but it is only implicit in this work. Cantor went on to develop his revolutionary ideas in a series of paper published between 1879 to 1884. Cantor got the first part of his major work Beitrage zur Begrundung der transfiniten Mengenlehre (Contributions to the foundation of Transfinite set theroy) published in 1895. This book led to the circulation of Cantor’s idea to the world. Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:- I see it, but I don't believe it! In 1910 Cantor received an invitation from the University of St Andrews in Scotland to attend the 500t anniversary of the founding of the University as a distinguished foreign scholar .Cantor had hoped to meet with Russell who had just published the Principia Mathematica. However ill health and the news that his son had taken ill made Cantor return to Germany without seeing Russell. The following year Cantor was awarded the honorary degree of Doctor of Laws by the University of St Andrews but he was too ill to receive the degree in person. The second half of his life, his ideas was attacked by many mathematicians. Leopold Kronecker launched a particular vitriolic attack on Cantor that led to the decline in Cantor’s mental age and he died in a mental hospital on 6th January 1918. SEND YOUR VALUABLE COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com

George Boole

George Boole George Boole was one of the most brilliant mathematicians England has produced. He was born on Nov 2, 1815 at Lincoln England to John Boole and Mary Ann Joyce. He was born into the lowest economic stratum of society. The lower classes into whose ranks Boole had been born simply didn’t exist in the eye of the upper class. It is said that he was born in the wrong time, in the wrong place, and definitely in the wrong class. His father John used to make shoes but he was much interested in science and in particular the application of mathematics to scientific instruments. The family was not well off, partly because John’s love of science and mathematics meant that he didn’t devote the energy to developing his business in the way he might have done. Boole was lucky enough to have a father who passed along his own love of math. Young George took to learning like a politician to a pay rise and, by the age of eight, had outgrown his father's self-taught limits. Due to poverty and having been born in lower strata Boole could not get himself enrolled in a school of good repute. At that very time knowledge of Latin was considered to be important but no Latin was taught in the school where Boole was permitted to attend. Boole decided that he would learn Latin and a friend of his father helped him a little bit but Boole took the rest journey of learning Latin all alone. By 12, he had mastered enough Latin to translate a given paragraph into English properly. Boole learned his early lesson in mathematics from his father, an amateur mathematician and optical instrument maker. His father wanted him to join the business but after finishing his schooling Boole took a commercial course. By 16, he became a school teacher. This was rather forced on him since his father's business collapsed and he found himself having to support financially his parents, brothers and sister. He spent four years teaching elementary school. Boole later decided to become a clergy man so as to support his family in a better way and thus trained himself in the language of French, German and Italian and got mastery over it. At the age of 20, Boole opened up a school to prepare his pupils in mathematics. Over the next few years, Boole prepared himself with the tough mathematical courses with the help of mathematical journals borrowed from the local Mechanic's Institute. Boole struggled with Isaac Newton's 'Principia' and the works of 18th and 19th century French mathematicians Pierre-Simon Laplace and Joseph-Louis Lagrange. He later mastered himself the Mecanuque Cleste of Laplace and Mecanique Analytique of Lagrange by his own unaided efforts. In 1837, Boole submitted some of his work to the Cambridge Mathematical Journal. The style of presentation and its originality impressed Gregory and they become a good friend for life. David Gregory advised him to take a formal course of mathematics at Cambridge but he was unable to take Duncan Gregory's advice and study courses at Cambridge as he required the income from his school to look after his parents. In the summer of 1840 he had opened a boarding school in Lincoln and again the whole family had moved with him. He began publishing regularly in the Cambridge Mathematical Journal and his interests were influenced by Duncan Gregory as he began to study algebra. By 1844 he was concentrating on the uses of combined algebra and calculus to process infinitely small and large figures, and, in that same year, received a Royal Society medal for his contributions to analysis. In 1847, Boole published a pamphlet The Mathematical Analysis of Logic which bridged the gap previously separating mathematics from formal logic. This development was crucial in advancing the potential powers of the analytic engines of Babbage. It was this paper that won him, not only the admiration of the distinguished logician Augustus de Morgan but a place on the faculty of Ireland's Queen's College. This was his first public contribution to the vast subject which his work inaugurated and in which he was to win enduring fame for the boldness and perspicacity of his vision. Boole reduced logic to an extremely easy and simple type of algebra. This book laid the foundation of new branch of mathematics called Boolean algebra. In 1849, he was appointed Professor of mathematics there. He taught there for the rest of his life, gaining a reputation as an outstanding and dedicated teacher. This post made him financially independent and he made excellent use of his comparative freedom from financial worry. Without a school to run, Boole began to delve deeper into his own work, concentrating on refining his 'Mathematical Analysis', and determined to find a way to encode logical arguments into an indicative language that could be manipulated and solved mathematically. He came up with a type of linguistic algebra, the three most basic operations of which were (and still are) AND, OR and NOT. It was these three functions that formed the basis of his premise, and were the only operations necessary to perform comparisons or basic mathematical functions. In May 1851 Boole was elected as Dean of Science, a role he carried out conscientiously. By this time he had already met Mary Everest his would be wife with whom he married on 11 September 1855. It proved a very happy marriage with five daughters. In 1854, he published his work on logic An investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. This research was based on a binary approach processing only two objects- the yes-no, true- false, on-off, zero-one approach. In 1859, he published his Treatise on the Calculus of Finite Differences. He published around 50 papers and was one of the first to investigate the basic properties of numbers, such as the distributive property, that underlie the subject of algebra. An American logician Charles Sanders Pierce spend more than 20 years modifying and expanding Boole’s treatise realising the potential for use in electronic circuitry and eventually designing a fundamental electrical logic circuit. He did introduce Boolean algebra into his university logic philosophy courses. The development of Boolean algebra was fundamental to mathematical logic, and is the basic logical tool in designing modern computer. Unfortunately, Boole's life was cut short when he died of a 'feverish cold' at the age of 49, after walking 2 miles through the rain to get to class and then lecturing in wet clothes. He died on December 8, 1864 in Ballintemple in Ireland. After his death his wife Mary Boole applied some of the ideas which she had acquired from him to rationalizing the education of young children. SEND YOUR COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com

How to read mathematics

How to Read Mathematics?

Mathematics is a language that can neither be read nor understood without initiation.

You can’t read a math book the way you read other books. It takes a special approach to read maths. Mathematics is not like a novel reading. It can never be understood if you go through it starting page 1 to the last page in one breath. In order to understand mathematics you must develop a reading protocol to get benefit. Poetry calls for a different set of strategies than fiction, and fiction requires another set of strategies than nonfiction. Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics.  Always adopt pen and paper for doing mathematics otherwise you are simply deceiving yourself. You will never be able to understand mathematics unless or until you have done it in writing.

Reading mathematics is difficult because mathematics is difficult. It is considered the fiercest subject in the world and this is due to our pre notion approach. Mathematics is indeed the success gate of science and that’s why the great mathematician C F Gauss said-- Mathematics is the queen of all subjects.

Mathematics has the distinction of having the dense writing style in the text book which is not easy to understand. It is not true. Mathematics text requires more involvement from the reader than most texts in other subject. Hence, reading mathematics takes longer time and attempting to read mathematics too fast results to frustration. According to Adler- if a text is worth reading at all it is worth three readings at least. Mathematics is not a novel where you become absorbed in the plots and characters. The scene depicted is exaggerated but mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. Both a mathematics article and a novel are telling a story and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel. Let us learn the technique to read mathematics.

1. Synthetic Reading of the Book: - In the synthetic reading, the reader proceed from the parts to the whole. The parts are the building blocks; you can’t jump on the second until you finish the first. Learn the basics and make a strong base. Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. You can get a clear picture of next chapter if you have prepared and done well in the previous chapter. Nobody understands something complex on first reading. If after devoting for an hour you understand only one or two main points of the chapter don’t be sad about this. It doesn’t mean that there is anything wrong with you. The first reading just lays the framework for you to fill in later with details.

            A) Read the chapter introduction and each section summary.                                             B) Skim the chapter; circle the new words that you don’t understand. Consult the      dictionary for those words. Once you understand the concept erase the circle.                 For better clarification, if need be, consult your teacher.                          

C) Read with concentration. While reading the textbook, highlight the important result, formula, theorem etc.                                                                                    D) Before moving to the main topic, go through the example with noticing each and every step properly. In the example you might find many steps missing. Don’t jump to the conclusion that you will do the same in the examination. Try to find the missing steps with proper reasoning. Write those steps in text book for better understanding. Remember you learn math by doing, not by reading. If you don’t understand anything, refer to another math textbook, computer software program or consult your instructor, teacher for better understanding.                                  E) Once you find yourself equipped with the content of chapter, explain it to your friend.  If there is no one else available, can you explain it aloud, without stumbling? If you can do that, you probably understand it.

1. Learn Mathematical Symbols and vocabulary: - Before you attempt to understand the mathematical terms, you should learn the mathematical symbols commonly used in your textbook. In Algebra, you may see symbols like Σ (sigma) for addition, Π for multiplication. In trigonometry you may see the symbols like α (Alpha), β (Beta), γ (Gamma), δ (Delta), θ (Theta), φ (Phi) etc. Make a list of symbol and learn it. If possible, copy the list of math symbol from your math book or math dictionary and paste it on the wall so that you can revise the list every now and then. Moreover, it is also advisable to consult math dictionary for new words. Mathematics obtains much of its power by constructing a very precise vocabulary. A strong vocabulary can help you to understand the content of the book more easily and effectively. The most fundamental issue involved in reading mathematics for meaning is to get some sense of the mathematical words, phrases and applications. Mathematical text is a combination of jargon. Therefore we should not attempt to read mathematics as we would other type of reading. Reading mathematically is more than reading the printed words on the page. It requires linking the words with the mathematical ideas that are involved. So , making sense of mathematical prose is a complex process involving understanding mathematical terms and this can be done only when you learn the math vocabulary.
2. Learn Mathematical statement, formula: - In order to understand the mathematical concepts, you must learn the formula, statement of theorems, axioms etc. so that you may have the better understanding of the chapter. Try to learn the way how the formulas are generalized. It is better to learn the way formula is constructed rather than remembering the bags of formula. Make a poster of formula and paste it on the wall so that you can have a look of it every morning. Spend at least 5-10 minutes daily in revising these. You can develop your own way of remembering the formula. It is advisable to consult your teacher and ask him/ her to teach you the innovative ways of making and learning formula. You can also enquire about Pascal Triangle that is very helpful in understanding the binomial expansion of any positive power. Likewise you can also learn AFTER SCHOOL TO COLLEGE to remember the mathematical formula for sin(90+ө) , sin (180 ± ө), sin (270 ±ө) and sin (360±ө) etc. I still remember my first day in Trigonometry class when teacher wrote PANDIT BADRI PRASAD, HARI HARI BOLE to define the different Trigonometric ratios such as sine, cos, tan etc. I shall therefore put much emphasis on learning different techniques to learn formula rather than just learning the hundreds of formula.
3. Don’t be a passive Reader: - Mathematics is all about putting an extra effort. Go the extra mile and do some research works to find out what the particular page you are struck into want to tell you. Always keep a pen and paper with you while you read math text book. Write steps and solve the step by your own rather than seeing what is written in the book. Many results must be given in your book of which the details are suppressed. It is expected from the readers to fill the gap. If you are doing the work without pen and paper, you may probably not be able to understand certain thing. Always remember—to a great extent, people think mathematically through writing. It is hard to do in your head. Read the chapter introduction and each section summary. Skim the reading material that will allow you to see that if problems presented in one chapter is being explained in next chapter or not. As you skim the chapter, circle the new words that you don’t understand and ask these words to your parents first and then to teacher in the next day if you don’t understand these new words after reading the assignment .Mathematics says a lot with a little. Math uses special words to mean specific things. Sometimes, words are used differently in math than in regular language. For example PRIME, SET, VOLUME, COMPLEX, REAL, RATIONAL etc have different meaning in maths than they usually do. Understanding maths term will help you understand topic. Remember a math text book is very difficult. It might take you half an hour to read and understand just one page so don’t get impatient in such situation. You must be an active partner. At every stage you must decide whether or not the idea being presented is clear. Ask yourself these questions: ---                                                                                                                          Why is this idea true?                                                                                                          Is there any better option to challenge the idea?                                                              Could I convince someone else that it is true?                                                                       Why did the author use a different argument?                                                                         Why this particular formula or theorem was used here?
4. Study the examples and figures: - Most textbooks include examples with detailed solutions. The author of textbook design the exercise on the basis of examples discussed. Paying attention to the examples discussed provide an excellent opportunity for you to assess your readiness to begin the assigned exercises at the end of the section. If you go through the introduction of the chapter and read the example and correlate the matter, you may find the exercise handling is a cake walk for you.
5. Take a break when you are struck on a problem: - Many a time when you are trying to solve a math problem, you may force yourself to keep working at it until you find the solution. This may not be as useful as you think. If you get struck on one example, put it aside for a while.                                                                                                                                Take a break of 5-10 minutes.                                                                                                 Drink a glass of water or gat a snack.                                                                                          Do something completely different that will get your mind on something else. When you feel refreshed, go to the problem and read it with a fresh eye. This time you will have the solution. If problem still persists, ask your parents for immediate help.
6. Don’t make too much selection: - I have seen students making selection in reading. In the age of technology, students are aware of the pattern of the questions, blue prints of the syllabus and they make choices. A mediocre student selects the list of chapters that are easy to understand and estimate the necessary marks to pass by reading those few selected chapters. There is no harm in making selection of chapters but never thing that your ultimate goal is to pass in the examination with minimum passing marks. Go through all the chapters thoroughly and do all types of questions. Don’t make selection of important question at the very beginning. Remember                                                                                                                  Better know everything of something                                                                                    rather than something of everything                                                                                                       AND                                                                              Nothing like if you know everything of everything.                                              Selection of questions and chapters at the beginning will limited your knowledge and though you may feel good that this is the smartest way to pass the examination but it will do much harm to you in later phase of your life.
7. Put a special mark in your book: - Your teacher tells you to do some questions 5-10 times because that question is important. Your teacher with his experience knows those particular questions are important for the examination and lets you know in advance. My advice is in order to simplify the reading and also to select the important questions for your revision work before examination you make a regular habit of marking such questions with red ink to identify it and ease your learning during examination. You may mark with the initial letter of your name. Moreover, you can give special mark to those questions which you feel are difficult or require some special attention. You may write the special character at the first page of your book such as—R = Revision, I = Important, R*** = Revise it three  times etc. don’t include in this list those question which are based on the direct application of formula and which you think you can do.
8. Be a regular reader: - If you love watching cartoons/ serials on the Television and you miss a episode or two, you will still be in a position to judge what had happened in the last episode. But the same is not true for mathematics. Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. A mathematics article and a serial/ cartoon shown on TV are telling a story but a math article does the job with a tiny fraction of the words and symbols of those used in a serial/ cartoons. The TV serial uses language to evoke emotions and present themes which defy precise definition but the beauty in a mathematical article is in the elegant efficient way it concisely describe precise ideas of great complexity.                                                                         So if you are jumping one chapter or a single step in mathematics than there is greater chances that you fail to understand the whole concept and deflect the way. Always remember, mathematics can’t be mastered in a single day. You have to read mathematics regularly for at least 1 hour a day if you can’t put maximum in a particular day. Keeping yourself away for a week or two from math book will bring you to the mark zero and you have to pad up again to regain the knowledge because it is sure that you have forgotten almost 90% of what you have learnt a week or 15 days ago.  As you watch the television serial whether it is an entertainment channel or informative one, you must have seen that before the serial begins it recast the glimpses of the previous day. In the same manner, you must have to revise the odd words, formulas, important facts of the previous chapter before switching to the next.
9. Get your problem solved instantly: - Mathematics can be learnt if you resolve the problem instantly. Keep the phone number of your class teacher and your tutor and call him or her when you feel helpless in understanding a particular step or problem. It is better to consult the doctor when you feel uneasy rather than wait for the problem to grow. You can’t wait for a week to consult your class teacher to resolve the problem you face a week ago. Suppose you are reading a chapter from Algebra and while solving a particular sum you faced a problem. My simple advice is you first consult your parents. If they are good at mathematics then you will get the answer instantly. If you can’t get the answer, look at the example given in that chapter. If you have kept some reference book than you can consult the book too. If your labour fails in vain then don’t get hesitate in calling your teacher or tutor after all they are to help you. But don’t call them late night.
10. Keep the Reference Book/ Journal with you: - Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. When you read a math book, you may or may not encounter several problems. Don’t assume that understanding each phrase, word, symbol etc. will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and colour but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You may not get the detail in your text book itself so you have to consult some other book or journals. A class text book has a limited number of problems and you may not get varieties of problem in it but a good reference book will help you to solve varieties of problem boosting your confidence. A good reference book will help you to understand the definition, properties of numbers, geometrical shapes etc and also the algorithm to solve problem of particular type.
11. Pay attention to your anxious feelings:- Some people feel like they are simply not able to learn math. They may have been unsuccessful in learning math earlier or may have been told that they could not do math. This is called math anxiety. Math anxiety has nothing to do with abilities. If you feel that you can’t do math simply doesn’t mean that you are unable to do math. The feeling can get in the way. If you see a problem that is difficult for you, you may unknowingly tell yourself that you can’t do it. A key to getting over math anxiety is to figure out what is going on and manage them until the problem over power you. Make a journal and write your good/ bad feeling in it. If you feel happy after solving a particular problem, write it down. In the similar way if you feel distressed or nervous when you fail to solve a problem, also write it. As you know thinking in math is related to doing. This exercise of writing your feeling allows your mind to critically analyse the problem. You analyse to anticipate problem areas when you write down your feeling and this way you are helping yourself in another way.
12. Don’t read too fast: - In the very beginning, I have told you that mathematics book should not be read as novel. You may read 10-20 pages in half hour if you are a avid novel reader but the same may not be true for mathematics. The same amount of hour in a math book may finishes with 1-2 pages depending on the chapter you are reading and how experienced you are at reading mathematics. There is no substitute of work and time. You can speed up your math reading skill by practicing. In novel reading, you may skip the unwanted paragraph but still you can understand what novel is trying to say. This may not be true in case you are reading math. A single paragraph may have hundreds of hidden facts that will be essential to understand the next paragraph. So, please be patience and hold your breath. Mathematics is called the queen of all subjects and you can’t make please queen in your zig-zag style of reading.
13. Practice, practice and practice: -       Mathematics is not a subject you learn in a single attempt. Keep pen and paper and do as many problems as is required to ensure that you understand the concept. The amount of practice may vary from person to person but you can’t skip practicing. You will want to practice a concept until it makes sense and until you are fluent at finding solutions to various problems within the concept readily. When you complete a set of questions in a row, you are probably to the point of understanding. Re- visit the same problem after a month to check whether you are still capable of doing the same problem with the same amount of easiness or not. Think of math the way one thinks about a musical instrument. Most of us don’t just sit down and play an instrument. We first take lessons, practice it several times before moving to the next lesson. A good musician takes out time to review and never stops practicing. Mathematics is like the same. You need to practice more and more. Do extra exercise. Go beyond what is asked for. If you are asked to do 20 odd questions, do it but never put yourself in a particular boundary. Buy another book and practice more and more until you reach to the point of fluency with the concept. Doing the extra practice questions will help you to grasp the concept more readily. Be sure to re- visit the exercise a few months later to ensure that you still have a grasp of it.
14. Discuss what you learnt with your friends:-  The best way to get good at math is to discuss what you have learnt to your friend. It is said- Two heads are better than one. When you discuss the problem in a group, you are clarifying the concept for you by looking at it in a different way. When you learn something new from your parents, tutor or book you read that is completely different from what you have learnt in class, never forget to share it with your friend. Remember what William Glasser says-                               10% of what we READ, 20% of what we HEAR, 30% of what we SEE , 50% of what we SEE and HEAR,70% of what is DISCUSSED with OTHERS ,80% of what is EXPERIENCED PERSONALLY , 95% of what we TEACH TO SOMEONE ELSE.                       
15. Read backwards and forwards: - Mathematical knowledge can’t be gained by straight reading. You have to move in all the directions. You may not fully learn something in chapter 1 until you are halfway through chapter 2 or chapter 3. Hence it is a good idea to look back once in a while over previous sections so that you can have the control of what you have learnt previously and your knowledge don’t get outdated.  

I hope my advice will help you to read mathematics in a planned manner and you know when any work is done with a full proof planning an outstanding result is sure to come.    

Send your comments to

Rajesh kumar Thakur

rkthakur1974@gmail.com

Mathematics and Hindu Religion lecture series

Fermat

Pierre De Fermat

Born: - August 17, 1601                     Died: - January 12, 1665

Fermat was one of the leading mathematicians of early 17^th century. Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery. He was an amateur mathematician.

 He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius Plane loci to one of the mathematicians there. From Bordeaux Fermat went to Orléans where he studied law at the University and spent his working life as a magistrate in the small provincial town of Castres.

After he moved to Toulouse , he gained a new mathematical friend Carcavi.  In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall. Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. Fermat sent a letter to Mersenne containing two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines.

His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published in Cursus mathematicus a work by Herigone. With Pascal, Fermat stands as one of the founder of mathematical theory of probability. Pierre de Fermat independently founded the new branch of mathematics called Analytical Geometry. This work led to violent controversies over question of priority with Rene Descartes. Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").^[

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.

He is probably best known for his work on number theory. He also left one of the famous unsolved problems in maths called- Fermat’s last theorem. This theorem states that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2. in the margin of Bachet's translation of Diophantus's Arithmetica , Fermat worte “ I have discovered a truly remarkable proof which this margin is too small to contain.” These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries. The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles.

                                          

 

The second stamp was released after it was proved by Andrew Wiles.

In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. 

He died at Castres, Tarn on January 12, 1655. The oldest and most prestigious high school in Toulouse is named after him: 

Send your comments to

rkthakur1974@gmail.com

Galois

Evariste Galois

Evariste Galois was a French mathematician who made significant contribution to the theory of functions, the theory of equations and number theory. His work became the basis for Galois Theory and Group theory. He was the first to use the word Group as a technical term in mathematics to represent a group of permutations. 

 The mathematical genius Evariste Galois was born on 25^th October 1811 in France. He was the second son of Nicholas Gabriel Galois and Adelaide Marie. His father was the director of a boarding school and later mayor of Bourg – la- Reive. Evariste had a happy childhood. He seemed to have a phenomenal memory.  While still in teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long standing problem. Up to the age of 12, he was educated by his mother who instilled in him knowledge of classics and a sceptical attitude towards religion.

He entered the College Royale de Louis le Grand in Paris in 1823 where his precocious mathematical genius first emerged. He studied the Legendre’s text book Elements de Geometrie and Lagrange’s work on Differential Equation and Analytic Function. By the age of 16, he had published many papers. He appeared for the examination of Ecole Polytechnic but failed. The next year Galois had the good fortune to be studying maths under a distinguished teacher Louis Paul Emile Richard who recognised his exceptional gifts. Richard was enthusiastic about Galois mathematical work and this led to his publication of first paper on continued fraction. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences but the Academy refused to accept them for publication.

On 28 July 1829 Galois's father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority and Louis Philippe became the king. His anti monarchist view led him to imprisonment leading his expulsion from the college.

Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government. They were later acquitted of all charges.

During his imprisonment, he continued developing his mathematical ideas. On the advice of Simon Poisson he submitted his work on the Theory of Equation for publication but Poisson decleared his work incomprehensible. Galois reacted violently to the rejection and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. 

During 1830s Galois published several papers in Bulletin des Sciences mathematiques of Baren de Ferussace, a review that normally only published work by established scientist. These articles contained most of his work on Theory of Equation which is now known as Galois Theory.

In 1831, he was arrested again for keeping illegal weapon. He was released on April 29, 1832. On May 25 he wrote to Chevalier, a close friend of his from the Ecole Normale, expressing his complete disenchantment with life, and hinting that a broken love affair was the reason. The woman in question was Stephanie Dumotel, the daughter of the resident physician at the hostel where Galois stayed during last months of his life.

On May 30, Galois fought a duel with pistol and was shot in the abdomen. Galois remained unattended for hours until a passerby took him to the hospital. He refused the services of a priest and died of peritonitis the following day, at the age of 20. A night before the duel was scheduled he recorded his mathematical idea in a later to his former school master Auguste Chevalier where he had out lined his work on elliptical integrals and permutations of groups.

Hermann Weyl, a mathematician, said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

Galois' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not solvable by radicals.

 Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

Send your comments to

rkthakur1974@gmail.com

Euclid

  Euclid

Euclid of Alexandria is better known as the Father of Geometry. He is one of the most prominent mathematicians of antiquity best known for his treatise The Elements. Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown. But whatever speculated data are available shows that he was born about 325 BC in Alexandria and died about 265 BC.

An Arabian author, al-Qifti (d. 1248), recorded that Euclid's father was Naucrates and his grandfather was Zenarchus, that he was a Greek, born in Tyre and lived in Damascus. But there is no real proof that this is the same Euclid. During the reign of Ptolemy I he taught at Alexandria. Ptolemy had created the great library at Alexandria, which was known as the Museum, because it was considered a house of the muses for the arts and sciences. Many scholars worked and taught there, and that is where Euclid wrote The Elements.

There is a very interesting story about Euclid---

A student who had begun to learn geometry with Euclid asked him “What shall I get by learning all these things in Geometry?” Euclid called his slave and told him to give three pence since he must make gain out of what he learns.

Euclid said to his students that --- There is no royal road to Geometry.

             A page from Euclid’s Elements

Euclid's most famous for his treatise on mathematics called The Elements. The book is a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid. The Elements is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry.

Euclid organized the known geometrical ideas, starting with simple definitions, axioms; formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. One of the proofs was for the theorem of Pythagoras, proving that the equation is always true for every right triangle. The Elements was the most widely used textbook of all time. It has appeared in more than 1,000 editions since it was first printed in 1482 and is thought to have sold more copies than any book other than the Bible. Original of Euclid's Elements have not been preserved, but Arabic mathematicians obviously had a full copy as an Arabic version of The Elements appeared at the end of the 8th century BC.

 Euclid believed that we can't be sure of any axioms without proof, so he devised logical steps to prove them. There are 5 axioms and 5 postulates found in the book of Euclid. He called Axiom "Common Notions," because they were common to all sciences.

Axioms

1.    Things which are equal to the same thing are also equal to one another.

2.    If equals are added to equals, the sums are equal.

3.    If equals are subtracted from equals, the remainders are equal.

4.    Things which coincide with one another are equal to one another.

5.    The whole is greater than the part.

Postulates

1.      You can draw a straight line between any two points.

2.      You can extend the line indefinitely.

3.      You can draw a circle using any line segment as the radius and one end point as the center.

4.      All right angles are equal.

5.      Given a line and a point, you can draw only one line through the point that is parallel to the first line.

The fifth postulates later became the cause of discovery of new branch of geometry known as Non Euclidian Geometry.

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time.

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. Mathematicians since Euclid have attempted without success to find a pattern to the sequence of prime numbers.

Although best known for its geometric results, the Elements also include number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Euclid also wrote Data, which contains 94 propositions, Phaenomena, concerning spherical astronomy, Caloptrics, about mirrors, Optics, the theory of perspective, and a work of music theory. In his works about optics, Euclid made light rays part of geometry, working with them as if they were straight lines. Many of the works ascribed to Euclid are no longer in existence or are incomplete.

Send your comments to

rkthakur1974@gmail.com