Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started. In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a, b, c) with a 2+b 2 = c 2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra. Geometric problems relating to similar figures, area and volume were also studied and values obtained for p.
The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea’s paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration. The theory of conic sections show a high point in pure mathematical study by Apollonius.
The Essay on Chinese Remainder Theorem Mod Number Mathematics
CHINESE REMAINDER THEOREM WHAT IS IT Mathematics plays an important role in our world, as we know it. It is used everyday to solve all kinds of complex problems enabling us to create a better place for everyone to live in. One particular aspect of Mathematics known as the Chinese Remainder Theorem has a very interesting background. The Chinese Remainder Theorem was believed to have been created ...
Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry. The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics.
From about the 11 th Century Abelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe. Major progress in mathematics in Europe began again at the beginning of the 16 th Century with Paci oli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionized the applications of mathematics to the study of the universe. The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Steven in Belgium and Vi ” eye in France. The 17 th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator y science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17 th Century. Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton’s theory of gravitation and his theory of light take us into the 18 th Century. However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18 th Century rather than that of Newton.
The Term Paper on Roosevelt’s New Deal Works Progress Administration
On July 2, 1932, at the Democratic National Convention, the crowd listened intently to the phrase," I pledge you, I pledge myself to a new deal for the American people." The New Deal name was soon applied to the program of reform and recovery instituted by Franklin Delano Roosevelt. During the early part of the Great Depression, the economy had ground to a halt as a result of the stock market ...
Leibniz’s influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application. The most important mathematician of the 18 th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat. Toward the end of the 18 th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace’s great work on celestial mechanics as well as major progress in synthetic geometry by Mong e and Carnot. The 19 th Century saw rapid progress.
Fourier’s work on heat was of fundamental importance. In geometry Pl”u cker produced fundamental work on analytic geometry and Steiner in synthetic geometry. Non-euclidean geometry developed by Lobachevsky and Bol yai led to characterization of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionize the topic. He also contributed in a major way to astronomy and magnetism.
The 19 th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois’ introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20 th Century. Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.
Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grass mann. The end of the 19 th Century saw Cantor invent set theory almost single handed ly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers Analysis was driven by the requirements of mathematical physics and astronomy. Lie’s work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionize the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.
Adult Learning Skills Stress Life Work
Do you feel stress in your life? Does this affect the way you live and work? Many things currently going on in one's life, such as work, health, family and finances, can cause stress. It is how we individually identify the root cause and begin working on managing them effectively. As adult learners, there are various aspects of our work life that cause each of us some form of stress. We discovered ...
The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm’s work led to Hilbert and the development of functional analysis. Notation and communication There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation. For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals.
What is M LXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals. What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz’s notation lead more easily to extending the ideas of the calculus, while Newton’s notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton’s notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz. Let us think for a moment how dependent we all are on mathematical notation and convention.
Ask any mathematician to solve ax = b and you will be given the answer x = b / a. I would be very surprised if you were given the answer a = b / x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities. It was not always like this: Harriet used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Vi ” eye who used vowels for unknowns and consonants for known’s.
The Term Paper on Sociocultural Factors That Influence The Direction Of Scientific Discovery
CAN CHANGES BE MADE IN THE GOALS AND PRINCIPLES OF SCIENCE BE EXPLAINED BY THE SOCIO-CULTURAL FACTORS AT WORK DURING THAT TIME? Philosophical analysis and scientific practice: The arguments about these rival ontological and epistemological views cannot be safely left or judged without first looking more closely at the complex relationship between the general analytical interests of philosophers ...
Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign ‘ = ‘ was introduced by Record in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written! Brilliant discoveries? It is quite hard to understand the brilliance of major mathematical discoveries. On the one hand they often appear as isolated flashes of brilliance although in fact they are the culmination of work by many, often less able, mathematicians over a long period. For example the controversy over whether Newton or Leibniz discovered the calculus first can easily be answered.
Neither did since Newton certainly learnt the calculus from his teacher Barrow. Of course I’m not suggesting that Barrow should receive the credit for discovering the calculus, I’m m early pointing out that the calculus comes out of a long period of progress staring with Greek mathematics. Now we are in danger of reducing major mathematical discoveries as no more than the luck of who was working on a topic at ‘the right time’. This too would be completely unfair (although it does go some why to explain why two or more people often discovered something independently around the same time).
There is still the flash of genius in the discoveries, often coming from a deeper understanding or seeing the importance of certain ideas more clearly. How we view history We view the history of mathematics from our own position of understanding and sophistication.
There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past. There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses.
The Term Paper on The Discovery Telomerase
The linear chromosomes of eukaryotes are more complex than the circular chromosomes of the bacteria. Due to the biochemical properties of DNA polymerases, the replication of the eukaryotic chromosomes poses a special problem: the maintenance of the length of the linear chromosomes. However, it was discovered that a unique enzyme complex appears to play a crucial in maintaining the length of ...
This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics. Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children.
Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting. A challenge If you think that mathematical discovery is easy then here is a challenge to make you think. Napier, Briggs and others introduced the world to logarithms nearly 400 years ago. These were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs. Then the world changed.
The pocket calculator appeared. The logarithm remains an important mathematical function but its use in calculating has gone for ever. Here is the challenge. What will replace the calculator? You might say that this is an unfair question.
However let me remind you that Napier invented the basic concepts of a mechanical computer at the same time as logs. The basic ideas that will lead to the replacement of the pocket calculator are almost certainly around us. We can think of faster calculators, smaller calculators, better calculators but I’m asking for something as different from the calculator as the calculator itself is from log tables. I have an answer to my own question but it would spoil the point of my challenge to say what it is. Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, and everything else to do with MATH! Einstein and his Theory What do you think when some one says Einstein, is it Relativity, or E = MC 2? What do you think E = MC 2 means, well it means Energy = Mass x Speed of Light Squared. He was way ahead of his own time, he was a genius!.
The Essay on Apple Inc in 2012
On October 5, 2011, Steve Jobs tragically died of cancer. The recently retired CEO of Apple Inc. was a legend: he had changed Apple from a company on the verge of bankruptcy to one of the largest and most profitable companies in the world. Moreover, he had revolutionized several industries in the process. Few companies’ successes were so closely identified with its CEO. Jobs’s passing promised to ...