A collection of articles calculus

FOREWORD

With grateful praise Allah SWT, which has shed His grace and guidance, so that articles about the benefits of the calculus of computer science can be completed on time.Artikel ini diajukan untuk memenuhi salah satu tugas mata kuliah “Dasar Kalkulus I” Management Informatika Amik Wira Nusantara, Rangkasbitung.

On this occasion we did not forget to also say many, many thanks to those who have helped smooth the manufacture of this article.

We realize there are still shortcomings here and there. Therefore, to expect our readers suggestions and criticisms for the perfection of this report. So hopefully useful. Amin.

                                                       Author

Sumber 1

From Calculus to Computers

From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, Amy Shell-Gellasch and Dick Jardine (eds), 2005, 200 pp., illustrations, $48.95 (MAA member price $39.50) paperbound.  ISBN: 0-88385-178-4. MAA notes, Catalog Code NTE-68, MAA Service Center, P.O. Box 91112, Washington, DC 20090-1112, 1-800-331-1MAA,  www.maa.org.

This collection of articles assembled by editors Amy Shell-Gellasch and Dick Jardine grew from a series of  proposed talks for MAA Math Fest during the summers of 2001 and 2002. The motivation of this volume was to address a “noticeable lack of information for educators on just how to incorporate” the mathematical history of the last 200 years in the classroom. Twenty-two articles are divided into four sections: Algebra and Calculus, Geometry, Discrete Mathematics and Computer Science, and Pedagogy. Many of these contributions are quite excellent. For example, Shai Simonson gives a very clear explanation of how Euclid’s algorithm and Fermat’s little theorem are used in modern day cryptography. A few of the articles, such as Robert Rogers on the differential, have exceeded the 200 year limit and yet are still worthwhile reading. Fewer of the articles include original source material. David J. Pengelly continues his mission of bringing this material to a greater audience with Cayley’s first paper on group theory and suggestions for its classroom use.
As the reader moves through each section he or she finds that the articles are quite varied in both focus and style as one would expect from such an assemblage. It is not difficult to question the placement of several articles into the stated sections. Perhaps, the collection could have been better organized into articles on subject matter, methods, and issues.  The most interesting articles deal with material which can be easily assimilated into particular mathematics courses. The topics considered include abstract algebra, basic calculus, elliptical curves, differential equations, modern geometry, number theory, logic and computer science. Another group of articles deal in particular with the management of including history of mathematics in particular mathematics courses. Finally, there are three or four articles that deal with clarification of historical issues particular to specific branches of mathematics. These may be of limited interest for the more general reader. While the editors readily admit that this is merely a first attempt, they have created a worthwhile collection for anyone who teaches history of mathematics or is interested in bringing an historical component into their mathematics classes.

James F. Kiernan, Adjunct Professor, Brooklyn College, CUNY

Sumber 2

The Case Against Computers in K-13 Math Education (Kindergarten through Calculus)

In Peru, as in many Third World countries, the system of public education is in crisis. Teachers' pay -- traditionally low -- is falling rapidly because of inflation. The schools are dilapidated, and there is no money for basic supplies. The government has not been responsive to the teachers' protests. Under pressure from the I.M.F. (``the International Misery Fund,'' as Egyptian President Mubarak has said), it insists that the state sector -- and that includes public schools -- must be cut back.

Yet President Fujimori has said that he wants to get computers into the schools as soon as possible. The government's priority is to ``modernize'' the economy and the educational system, and computerized learning is supposedly one way to do this. For teachers who are trying to cope with financial hardship and abysmal working conditions, what could be more demoralizing than the message that machines come before people? To the Peruvian teachers, Fujimori's advocacy of computers adds insult to injury.

On the other hand, not everyone loses. The U.S. computer industry has an interest in creating new markets in the Third World. Thanks to such strategies as interconnected products and planned obsolescence, greater and greater payments will flow to the North as countries like Peru become dependent on U.S. technology in more areas of national life.

Throughout the Third World, for about a decade pressure has been mounting to import computer learning from the wealthy countries. In February 1986, a major conference called ``Informatics and the Teaching of Mathematics in Developing Countries'' was held in Tunisia. Participants were predominantly from Northern Africa, but many came from the universities and educational establishments of other regions of the Third World. All of the mathematicians and math educators sang the praises of computers and pled for the rapid introduction of computers into their school systems. Not a single note of skepticism was raised, not a single question was explored in depth. One could have asked, for example: Are computers truly what schools in Africa need? In practice, would computers be introduced on a general level, or only in a few of the elite schools for the wealthy? Could resources be better spent in other ways -- to raise teachers' salaries, purchase classroom supplies, expand libraries? It seems bizarre that the Tunisia conference adopted as an axiom the notion that the introduction of computers should be a priority for elementary education in Africa.

It should be noted that much of the support for this conference came from the big French computer companies. Thus, even though the meeting was of little value for people seriously interested in educational issues, the funding companies must have viewed it as a great success, holding open the possibility of lucrative new markets, particularly in the former French colonies of Northern Africa.

Some Caveats

Despite my skepticism about computers in education, I do not advocate an extreme position -- I do not want to throw out the baby with the bath water. There are some appropriate uses for computers in math education. Thus, I will not argue

• that it is unwise to use calculators in class. (I have my calculus students use them.)
• that computer-based math courses are always a bad idea. (My university has an optional computer lab course, taught in conjunction with third quarter calculus, that seems to be functioning well.)
• that other technology in the classroom should be avoided. (Nice films on area, volume, etc. have been around for decades.)
• that special learning programs based on calculators or computers are never successful. (Almost any teaching method can work well under suitable conditions and with a dedicated and enthusiastic teacher.)

I will argue, however, that there has been too much hype about technology in math education, and it is time to consider the downside. In my opinion, computers should not be a major component in math education reform.

The downside can be divided into several broad areas:

• drain on resources (money, time, energy);
• bad pedagogy;
• anti-intellectual appeal;
• corruption of educators.

I shall discuss each of these objections in turn, after a few preliminary remarks.

Popular Culture

Any discussion of math education reform should take into account the cultural environment in which we live. American popular culture, which has come to dominate the entertainment industry and the mass media in many parts of the world, can easily cause distortions of the movement for reform.

Youngsters who are immersed in this popular culture are accustomed to large doses of passive, visual entertainment. They tend to develop a short attention span, and expect immediate gratification. They are usually ill equipped to study mathematics, because they lack patience, self-discipline, the ability to concentrate for long periods, and reading comprehension and writing skills.

In the U.S., attempts to popularize mathematics tend to be influenced by this ambiance. The public television program ``Square One'' is an example of a well-intentioned effort that went astray because of too much gimmickry and too little attention to content. The politics of funding adds to the pressure on reformers to water down the content. Funding agencies are impressed if grantees can demonstrate short-term success. This may explain why one often comes across pilot projects in American schools where the math content is not appropriate for the target age group -- it is too trivial. Of course, the students find it easy, and the project organizers can report great success, thereby satisfying the funding agencies. Such an approach to curricular development contributes to the ``dumbing down'' of the curriculum.

The most important example of gimmickry in math education reform is computermania.

A Drain on Resources

We already saw an extreme example of misallocated resources in Peru. But even in a ``wealthy'' country one has to be concerned about this problem. Many educators in North America share the feeling of betrayal of the teacher who said,

They can give us the axe, but they can spend thousands on computers. We have to fire our music coordinator, we have to fire our music teachers, we have shitty libraries. (Lynn, a Canadian schoolteacher, quoted in [14, p. 41])

At the University of Washington, we also have resource limitations. After considering various ways to reform the calculus course, we selected a low-tech approach using some applications-oriented lecture notes that I had written. Our calculus reform was implemented relatively quickly, painlessly, and inexpensively, largely because it was not based on computers or graphics calculators.

Cost is an issue not only for schools, but also for individual students. Ironically, it is sometimes the colleges with the highest proportion of working-class students that have become most enamored of expensive new gadgetry for teaching mathematics. At a reception for students planning to transfer to my university from community colleges, the students I talked with were complaining about having to spend more than $80 on a graphics calculator, in addition to $60 for a textbook. (By contrast, the required material for my 20-week calculus sequence costs a total of about $14.)

Another resource issue is that techniques that work in an experimental program with an enthusiastic instructor will not necessarily work under less ideal conditions. One of the lessons of the 1960s ``new math'' fiasco is that we must look beyond the pilot programs and demonstration classes and selected testimonials, and ask ourselves what is going to happen in a typical classroom with a typical teacher.

Every week, in conjunction with a course I teach for math education majors, I spend a morning visiting just such a school. My students and I work with regular sixth grade classes at Washington Middle School (WMS), an inner-city school in Seattle. The teacher has five math classes every day, with a total of over a hundred students, many of whom have severe personal as well as learning problems.

When I started visiting her classes in 1992, she had several computers in the back of the room. But they were just gathering dust, and so have been removed. What, if not computers, does an overworked teacher like her need in order to be more effective? She herself told me that she very much regrets not having any time in the day to talk as colleagues with the other teachers -- about pupils they have in common, and about teaching methods and materials.

A fundamental problem in education is the failure to treat teachers like professionals. Teachers at schools like WMS need opportunities to upgrade their qualifications, to learn about different teaching materials, and to interact with other teachers as colleagues. This requires release time, light teaching years, or sabbaticals. If money currently used to buy computers and software were instead devoted to improving conditions for teachers, it would be money well spent. If pay and working conditions improved for teachers, then maybe more of our best students would go into teaching.

All of the fuss about computers serves to divert attention away from the central human needs of the school system -- better conditions for teachers and better teacher training.

Bad Pedagogy

In response to the grandiose claims of such computers-in-the-schools gurus as Seymour Papert, many experts in child development have pointed out that those claims rest on a flimsy pedagogical foundation, especially where young children (K - 5) are concerned.

The point is that children benefit most from material that stimulates them to exercise their imagination. For example, simple, unstructured play material like clay, sand, blocks, rag dolls, and finger-painting sets are more wholesome entertainment than TV (even ``educational TV'') and Power Ranger toys.

Harriet K. Cuffaro of the Bank Street College of Education comments on computer painting versus ordinary painting:

...in `painting' via the computer, the experience is reduced and limited by eliminating the fluid, liquid nature of paint. In this painting there are no...opportunities to become involved in the process of learning how to create shades of colors; gauging the amount of paint to be mixed; experimenting with and discovering the effects of overlaying colors; understanding the relationship of brush, paint, and paper, the effects achieved by rotating the brush and varying pressure, or how one gains control of or incorporates those unexpected, unintended drips.... There is an absence of texture, of smell, a lessening of qualitative associations with the experience of painting... computer graphics have a `stamped-out,' standardized, `painting-by-numbers' quality. Though the design or arrangement of colors, lines, and forms will vary with each child, there is a quality of sameness in appearance... individuality is flattened by the parameters of the program. [15, p. 25]

More generally, according to Douglas Sloan of Columbia Teachers College:

For the healthy development of growing children especially, the importance of an environment rich in sensory experience -- color, sound, smell, movement, texture, a direct acquantance with nature, and so forth -- cannot be too strongly emphasized.... At what points and in what ways will the computer in education only further impoverish and stunt the sensory experience so necessary to the health and full rationality of the human individual and society?... What is the effect of the flat, two-dimensional, visual, and externally supplied image, and of the lifeless though florid colors of the viewing screen, on the development of the young child's own inner capacity to bring to birth living, mobile, creative images of his[/her] own? [15, pp. 5,8]

Some have also questioned the effect of computers on teacher-student interaction. Larry Cuban, who has made a detailed study of the history of attempts since 1920 to introduce technology into American schools, writes:

In a culture in love with swift change and big profit margins, yet reluctant to contain powerful social mechanisms that strongly influence children (e.g., television), no other public institution [besides schools] offers these basic but taken-for-granted occasions for continuous, measured intellectual and emotional growth of children.... The complex relationships between teachers and students become uncertain in the face of microcomputers... introducing to each classroom enough computers to tutor and drill children can dry up that emotional life, resulting in withered and uncertain relationships. [2, pp. 89, 91]

Several educators have criticized the public's unquestioning acceptance of so-called ``computer literacy'' as a goal for education. Computer scientist Joseph Weizenbaum has said, ``There is, as far as I know, no more evidence [that] programming is good for the mind than [that] Latin is, as is sometimes claimed'' (quoted in [2, p. 94]). According to John M. Broughton of Columbia Teachers College:

Inherent in that term [`computer literacy'] is the promise of generalizability comparable with [that] of reading and writing. However, there is no evidence that programming skills transfer to other areas of psychological development, even cognitive ones. In fact, a recent comprehensive review of the literature by Pea and Kurland suggests that virtually all the claims made about the beneficial educational effects of learning to program are not only inflated, but probably incorrect. Moreover, Pea and Kurland reveal that there is not even support for the...notion that learning to program aids children's mathematical thinking. Their own research study on transfer revealed that Logo programming experiences had no effect on the planning skills that are deemed central to problem-solving skills. The tradition of grossly inflated claims identified in the artificial intelligence literature...appears to have carried over into the...area of electronic learning. [15, p. 109]

The inability to develop good translation software has been one of the most embarrassing failures of Artificial Intelligence. If the best computers in the world are unable to translate from French into English, then they certainly cannot help my calculus students do what is the main point of the course: translating word problems into mathematics. Computers in this course would only be a distraction and a waste of time and resources. If the focus in beginning calculus is put where I believe it belongs -- on analyzing real world problems and choosing the appropriate mathematical techniques -- then the course cannot be centered around computers or graphics calculators.

At my university we have an optional one-credit computer lab for students in their third quarter of calculus. At this point it makes sense to offer the technology, because

• the emphasis shifts from word problem applications to geometric applications of calculus in three dimensions, where computer graphics serve a useful purpose, and
• students at that level are sophisticated enough to benefit from computers.

At all levels of schooling we have to ask: Do the students learn to punch buttons, or do they learn mathematics? One day at Washington Middle School, we had the children play a math game that involves dividing by 7 and rounding off to the nearest integer. When the sixth graders had to find 60/7, they punched it correctly into their calculators, which displayed 8.5714... But most of them could not read or interpret the answer: they did not understand the significance of the decimal point.

Similar dangers exist among older students. For this reason, in the calculus final exams at my university we usually ask for exact (not decimal) answers. For example, sin(60°)=\sqrt{3}/2, not 0.866; the circumference of a circle is 2 pi r, not 6.283r.

Anti-Intellectual Appeal

Computers reinforce the fascination with gadgetry, as opposed to intellect, that is endemic in American popular culture. As pointed out by Brian Simpson, former education advisor to IBM in the U.K.,

Technological solutions to educational problems often have a seductive appeal, promising to make education easier and more enjoyable than ever before. In the past, extravagant claims have been made for teaching machines, educational television, language laboratories, and even such improbable, esoteric phenomena as sleep learning and learning under music-induced hypnosis. [15, p. 84]

There is a long history of people wanting and expecting technology to transform education. It has been over seventy years since the following prediction was made by a famous American:

I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks. (Thomas Edison, 1922; quoted in [2, p. 9])

Perhaps the most frequent argument for computers in the schools -- and also the most illogical -- is the inevitability argument: ``Calculators/computers are going to be everywhere, so we might as well incorporate them into math class. One can't fight against the tide.'' Using the same reasoning, one could say that, since automobiles are everywhere in our society and play a crucial role in all of our lives, therefore cars should be used as much as possible in education, and driver education should be regarded as a centrally important subject in school, much more so than such relatively useless subjects as music and art. The inevitability argument for computers in the schools is exactly the same sort of anti-intellectual non sequitur.

Computers are usually used in the classroom in a way that fosters a Golly-Gee-Whiz attitude that sees science as a magical black box, rather than as an area of critical thinking. Much computer use is ``teaching by demo'' with the student as spectator. There is then little difference between so-called electronic learning and simply watching television.

Most software is based on immediate gratification, and does not encourage sustained mental effort. While physically playing an active role, the pupil is intellectually passive, and has little opportunity to be creative. That is, the pupil is programmed to follow a path already laid out in detail by others.

Even when software designers try to get the children into a creative mode, in many cases the same could be better accomplished without the technology. Educators tend to put the cart before the horse: instead of asking whether or not technology can support the curriculum, they try to find ways to squeeze the curriculum into a mold so that computers and calculators can be used.

Like a quack cure in medicine, perhaps the most harmful effect of the computer craze is that it diverts people from other, more solidly grounded approaches to treating the ailments of mathematics education. Might not the Golly-Gee-Whiz-Look-What-Computers-Can-Do school of mathematical pedagogy eventually come to be regarded as a disaster of the same magnitude as the ``new math'' rage of the 1960s?

Computer Science is Not the Same As Computers

One can strongly advocate increased teaching of computer science (and related areas, such as discrete math), while opposing the use of computers. Computer scientist Michael Fellows, who is an active campaigner for computer science in grade school (see [3]), has said: ``Most schools would probably be better off if they threw their computers into the dumpster.'' Fellows uses the term ``Cargo Cult'' to refer to the fetishization of computers by the media and educational establishment.

Definition of Classical Cargo Cult: An isolated civilization comes into initial contact with European technology. Ignorant of modern science, they interpret the benefits of technology in terms of their familiar world and their familiar mode of operation. They pray and perform sacrifices, or do whatever seems to be necessary to induce the deities to bring them the Cargo.

Modern Cargo Cult: In the U.S. and many other countries, most of the general public is prescientific, in the sense of having no rational understanding of the intellectual processes that go into scientific advances or their application to the real world. On the other hand, like the classical Cargo Cultists, they realize that technology is associated with economic well-being, and that something must be done so that youngsters will later be able to reap the benefits of the ``computer age.'' The natural response, then, is to fetishize computers and fit them into the familiar world of traditional mindless school math.

The public needs to understand that math and computer science are not about computers, in much the same way that cooking is not about stoves, and chemistry is not about glassware. That is,

COMPUTER SCIENCE not= COMPUTERS

The meaning of this inequality is: What children need in order to become mathematically literate citizens in the computer age is not early exposure to manipulating a keyboard, but rather wide-ranging experience working in a creative and exciting way with algorithms, problem-solving techniques and logical modes of thought.

Money Corrupts

Much of the effort to introduce technology in the classroom is profit-driven. As Douglas Sloan says,

It does not take a flaming Bolshevik, nor even a benighted neo-Luddite, to wonder whether all those computer companies, and their related textbook publishers, that are mounting media campaigns for computer literacy and supplying hundreds of thousands of computers to schools and colleges really have the interests of children and young people as their primary concern. [15, p. 3]

In the words of Joseph Menosky, an American writer and former science editor at National Public Radio,

Certainly those who have a great deal to gain from a universal acceptance of computer literacy -- microcomputer firms selling hardware, textbook companies selling educational software, organizations selling worker and teacher retraining courses, and writers and publishers selling books and instructional guides -- have done a brilliant, if morally indefensible, job of commercial promotion. [15, p. 77]

Corporate domination of math education reform has corrosive effects. The profit motive has an excessive influence; the intrinsic value of a pedagogical idea is not considered to be as important as its saleability. Educational ideas that are not based on expensive gadgetry or new textbooks are not likely to be supported strongly. There is an excess of faddism and hype.

It is regrettable that computers have been so aggressively marketed to teachers and school systems. In speaking to parents, teachers and school boards, many company sales representatives have taken the hard-sell approach: ``If you don't buy our latest products, you will be neglecting to prepare your children for the 21st century.''

Foundations and government agencies, such as the N.S.F. in the United States, compound the problem, because they share the biases of the commercial interests. Money from corporate and foundation sources seduces educators, whose objectivity and independent judgment become compromised. Grants can corrupt.

The technology-in-education movement has some of the characteristics of a religious evangelical campaign, fueled by corporate and foundation money. One sees the same reliance on testimonials. A technology enthusiast might proclaim ``How graphics calculators have changed my life!'' -- just as a born-again Christian talks about rediscovering Jesus. Like their religious counterparts, technology boosters tend to adopt a Manichaean view, dividing educators into two camps: those who have seen the light, and the fractious infidels.

I recently had a personal encounter with the closed-mindedness of technology advocates. About two years ago, N.S.F. asked me to help evaluate calculus reform proposals. But when they learned that I am skeptical about computers and graphics calculators in calculus, they changed their minds, and decided not to send me any proposals to review. They did not want any input from a nonbeliever.

Low-Tech Is Better

To end on a positive note, the good news is that the vast majority of enrichment topics do not require high technology, but only pencil and paper and inexpensive manipulatives, such as geoboards and abacuses. (According to a Nov. 22, 1994 article in the Wall Street Journal, the abacus is making a comeback in Asian schools, replacing calculators. The sound and tactile sense give children a feel for the algorithmic dynamics -- kids say things like ``to subtract 7 you add 3 and then subtract 10'' -- whereas the use of calculators made the children feel alienated from the arithmetic.) All of the material that my math education students and I present to the Washington Middle School youngsters is low-tech.

Epilogue: First Time It's Tragedy, Second Time It's Farce

At the beginning of the article I mentioned how President Fujimori of Peru has become a fervent advocate of computers in the schools. It is interesting to note that in the late 1960s Peruvian education officials came under influences that were in some ways a foretaste of the computer rage. The ``new math'' was then the reigning paradigm in math education in the wealthy countries. The Peruvian education ministry, which at that time was heavily French-influenced, imported the new pedagogy despite the strenuous opposition of the leading Peruvian mathematicians (particularly, Professors C. Carranza and M. Helfgott). Of course, the introduction of Bourbaki-style mathematics in the schools was a disaster in Peru, as elsewhere. The Peruvian mathematicians had been right, and the French had been wrong.

Will history repeat itself? Will countries around the world again import a poorly thought out and unworkable pedagogy from the U.S. and France? This time the cost will be higher. The ``new math,'' for all its foolishness, at least was relatively inexpensive.

References

1. Consortium for Mathematics and Its Applications, For All Practical Purposes: Introduction to Contemporary Mathematics, New York: W. H. Freeman, 1988.
2. L. Cuban, Teachers and Machines: The Classroom Use of Technology Since 1920, New York: Teachers College Press, 1986.
3. M. R. Fellows, Computer science and mathematics in the elementary schools, in N. D. Fisher, H. B. Keynes, and P. D. Wagreich, eds., Mathematicians and Education Reform 1990-1991, Providence: Amer. Math. Society, 1993, pp. 143-163.
4. M. R. Fellows, A. H. Koblitz, and N. Koblitz, Cultural aspects of mathematics education reform, Notices of the Amer. Math. Society, 41 (1994), pp. 5-9.
5. M. R. Fellows and N. Koblitz, Math Enrichment Topics for Middle School Teachers, Seattle: Math Liberation Front, 1994.
6. N. Koblitz, Math 124/125 (Calculus Lecture Notes), University of Washington Mathematics Department, 1994.
7. N. Koblitz, The profit motive: the bane of mathematics education, Humanistic Mathematics Network Journal, No. 7 (1992), pp. 89-92.
8. Mathematical Sciences Education Board and National Research Council, Measuring Up: Prototypes for Mathematics Assessment, Washington: National Academy Press, 1993.
9. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, 1989.
10. National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics, 1991.
11. S. Papert, Mindstorms: Children, Computers, and Powerful Ideas, New York: Basic Books, 1980.
12. R. D. Pea and D. M. Kurland, On the cognitive effects of learning computer programming, New Ideas in Psychology, 2 (1984), pp. 137-168.
13. R. D. Pea and D. M. Kurland, Logo Programming and the Development of Planning Skills, Technical Report No. 16, New York: Bank Street College of Education, 1983.
14. K. Riel, Factors That Influence Teachers' Use and Non-Use of Computers, Masters of Arts in Education Thesis, Univ. Victoria, 1994.
15. D. Sloan, ed., The Computer in Education: A Critical Perspective, New York: Teachers College Press, 1985.
16. C. Stoll, Silicon Snake Oil: Second Thoughts on the Information Highway, New York: Doubleday, 1995.

Sumber 3

Electoral College Calculus

Computer Analysis Shows 33 Ways To End in a Tie

By Dana Milbank

Washington Post Staff Writer
Wednesday, October 27, 2004; Page A01

Could one of these electoral college nightmares be our destiny?

President Bush and Sen. John F. Kerry deadlock on Tuesday with 269 electoral votes apiece -- but a single Bush elector in West Virginia defects, swinging the election to Kerry.

A computer analysis found 33 potential scenarios leading to a tie electoral vote for Sen. John F. Kerry and President Bush. If neither collects the 270 votes needed, the House will decide-with each state getting one vote. (Michael Robinson-Chavez -- The Washington Post) Free E-mail Newsletters Today's Headlines & Columnists See a Sample  |  Sign Up Now Breaking News Alerts See a Sample  |  Sign Up Now

Or Bush and Kerry are headed toward an electoral college tie, but the 2nd Congressional District of Maine breaks with the rest of the state, giving its one electoral vote -- and the presidency -- to Bush.

Or the Massachusetts senator wins an upset victory in Colorado and appears headed to the White House, but a Colorado ballot initiative that passes causes four of the state's nine electoral votes to go to Bush -- creating an electoral college tie that must be resolved in the U.S. House.

None of these scenarios is likely to occur next week, but neither is any of them far-fetched. Tuesday's election will probably be decided in 11 states where polls currently show the race too tight to predict a winner. And, assuming the other states go as predicted, a computer analysis finds no fewer than 33 combinations in which those 11 states could divide to produce a 269 to 269 electoral tie.

Normally, such outcomes are strictly theoretical. But not this time, with the election seemingly so close and unpredictable. "Flukey things probably happen in every election, but because most are not close nobody pays any attention," said Charles E. Cook Jr., an elections handicapper. "But when it's virtually a tied race, hell, what isn't important?" Cook says this election is on course to match 2000's distinction of having five states decided by less than half a percentage point.

It is still possible that the vote on Tuesday will produce a clear winner of both the electoral and popular votes. But if the winner's margin is small -- less than 1 percent of the popular vote is a rule of thumb -- the odds increase that the quirks of the electoral college could again decide the presidency and again raise doubts about a president's legitimacy.

"Let us hope for a wide victory by one of the two; the alternative is too awful to contemplate," said Walter Berns, an electoral college specialist at the American Enterprise Institute.

But many political strategists are preparing for a narrow -- and possibly split -- decision. Jim Jordan, former Kerry campaign manager now working on a Democratic voter-mobilization effort, puts the odds at 1 in 3 that Bush will share the fate Al Gore suffered in 2000: a popular-vote win but an electoral loss. "It's actually looking more and more plausible," he said, citing a number of polls showing a Bush lead nationally but a Kerry lead in many battleground states.

A repeat of 2000 -- Bush losing the popular vote but winning the electoral count -- is considered less likely because the president has been boosting his support in already Republican states and reducing his deficit in some safely Democratic states.

Even without a split between the electoral and popular votes, there is room for electoral mischief. To begin with, there are the 33 scenarios under which the battleground states could line up so that Kerry and Bush are in an electoral tie. Even if only the six most fiercely contested states are considered -- Florida, Minnesota, New Hampshire, New Mexico, Ohio and Wisconsin -- the electoral vote would be tied if Kerry wins Florida, Minnesota and New Hampshire while Bush wins New Mexico, Ohio and Wisconsin.

Under the 12th Amendment, if one candidate does not get 270 votes, the decision goes to the House, where each state gets a vote -- a formula that would guarantee a Bush victory (the Senate picks the vice president). A House-decided election could produce even more protests than the 2000 election did. That, writes Ryan Lizza of the New Republic, who spelled out 17 scenarios under which the election could end in an electoral tie, is perhaps the only way "for a second Bush term to seem more illegitimate in the eyes of Democrats than his first term."

The possibility of a tie or near-tie in the electoral college also makes it more possible for individual electors to cause havoc. In West Virginia, one of the state's five Republican electors, South Charleston Mayor Richie Robb, has said he might not vote for Bush (although he calls it "unlikely" he would support Kerry). And in Ohio, the political publication the Hotline reports, one of Kerry's 20 electors could be disqualified because he is a congressman. Such problems and "faithless electors" have surfaced before, but the elections were not close enough for it to matter.

In Maine, the state appears to be comfortably in Kerry's column. But the state splits its electoral votes based in part on the vote in each congressional district. If Bush wins in Maine's 2nd District, where Kerry has a narrow lead, the president would take one of the state's four electoral votes, a potentially decisive difference. For example, if Bush takes New Hampshire, Ohio and Wisconsin; Kerry gets Florida, Minnesota and New Mexico; and the other 44 states follow recent polls, Kerry will win the election with 270 votes -- unless Maine's 2nd District turns against him.

Conversely, Bush is favored to win Colorado's nine electoral votes. But a ballot initiative being decided Tuesday would cause the state's electoral votes to be distributed proportionally -- almost certainly meaning five electoral votes for the winner and four for the loser. Polls show the ballot initiative is likely to fail, but if it passes, the presidential election could change with it.

If Bush were to win Colorado along with the key battlegrounds of New Hampshire, New Mexico and Ohio (and other states followed polls' predictions) he would have 273 electoral votes -- but that would become a tie at 269 votes if the ballot initiative passes. Alternatively, if Kerry were to win Colorado and claim Minnesota, New Mexico and Ohio, he would have 272 votes -- until Colorado's ballot initiative returned four votes, and the presidency, to Bush.

Sumber 4

Charles Babbage

Computers have become a very important part in the life of modern society. Modern computers made ​​possible thanks to the development of electronic devices during World War II. But, the idea behind modern computer actually has thought more than a hundred years earlier by Charles Babbage. Unfortunately, the technology of that era not yet advanced enough so that he could not see the application of his ideas.

Youth

Charles Babbage was born on December 26, 1791. His father, Benjamin Babbage, wealthy merchant and banker. Babbage family living in Walworth, Surrey, on the outskirts of London. Charles was the first son of four children, but both brothers meningggal in childhood. When Charles was seriously ill in 1799, both parents are afraid of losing him too. Because of that, he was taken to Devon to get more healthy country air.
Charles started his school in Devon. He studied, among others, mathematics and accounting for simple navigation. This was the beginning of the interest that will shape his career. Calculation errors in navigation often resulting in a boating accident. Charles has devoted most of his life to develop some kind of machine that can calculate and print mathematical and astronomical tables carefully so that errors can be eliminated.
When Charles recovered from his illness, he returned to London. He went to Enfield, and the teachers immediately saw the ability in the field of mathematics. 1803, the family moved and settled in Devon. Charles went to Totnes Grammar School until 1810, and then entered Trinity College at Cambridge University.

Mathematics In Babbage's day

Babbage was so interested in studying mathematics, so he fills his spare time with reading math books, including those written in French. When asking for help to the teachers, he was surprised because they did not know the latest developments in the field of mathematics in France. At that time England and France were hostile because of the Napoleonic wars, and there are fears that the insurgency as the French Revolution will happen in the UK. Consequently, studying the work of French mathematician and scientist, like Blaise Pascal, considered unpatriotic act.
The British mathematician also ignore the developments in Germany. Gottfried Leibniz in Germany and Sir Isaac Newton in England separately and at almost the same time, have discovered calculus - a revolutionary new mathematical procedure. However, academic leaders in their respective countries claim that they are the experts who should get credit for the discovery. National competition to earn the award resulted in German and English ended up not getting anything.
Rejection of European thought is inhibiting the development of mathematics in England. This also means that those who study the progress that occurred in Europe, such as Babbage, considered a liberal group that is not patriotic, and they faced criticism from many narrow-minded colleagues. However, much of the work of Babbage later based on the progress generated by Pascal and Leibniz.
In 1812, Babbage and two friends formed the Association of Analytical (Analytical Society) in Cambridge. The second friend is a leading astronomer John Herschel (also a devoted Christian as Babbage) and mathematician George Peacock. Through this association, they sought to book the latest mathematical methods in French translated into English. But because the work went slowly, eventually they themselves who do the work. Later, the Association of Analytical very important role in renewing the teaching of mathematics at universities in England. However, this process is running very slow.

New Directions

Charles Babbage received a degree in mathematics in 1814. That same year, he married Georgina Whitmore. They had eight children, but only five lived beyond childhood. Georgina died in 1827. Shortly after marriage, Babbage decided to become a pastor. He applied to several churches. Unfortunately, church leaders too believe allegations that Babbage was a liberal who is not patriotic, so her application was rejected.
Loss of the church be a boon for mathematics. Charles and his wife moved to London in 1815. Here, he demonstrated practical abilities in mathematics and provide a series of lectures on the benefits of experimentation, in addition to the mathematical theory. Thanks to these activities, in 1816, he was elected as a member of the Royal Society - the most prestigious association for British scientists. Babbage earn a master's degree in 1817.
Over the next few years, Babbage made ​​important contributions in the field of pure mathematics, like algebra and function theory. But the main desire is to practice math. With the support of mathematicians, navigators, and scientists, he began work on the analytical engine.

Previous Calculation Tools

Calculating machine which was first known to the world is the "abacus", which used the Chinese nation since around the year 600 BC. This device consists of beads that hung on strings in the frame, and the beads that moved along the strings during the calculation. Each bead memunyai specific numerical value.
In 1614, Scottish mathematician, John Napier, published his first paper on his findings, called logarithms. By using a series of rods, now known as "Napier's bones", Babbage simplify multiplication and division by turning it into the process of addition and subtraction are more modest. The slide rule Edmund Gunter's findings in 1620 also uses this principle.
The next advances in calculators appeared in 1642, when Blaise Pascal invented the first count, which can add and subtract. This machine consists of a set of wheels, each with numbers 0 through 9. The wheels are connected to the gear, so if one wheel spins full, will move the wheels at the next one-tenth of one round. But this machine is expensive and difficult to operate. In 1671, Gottfried Leibniz improve the ability of Pascal's machine by adding trains that can be moved. This machine can now multiply and divide.
Beginning in the 1820s, Babbage began working to create a calculating machine with a capacity of twenty decimal. He began by making a small six-wheeled totalizer which can count carefully. This little machine diperagakannya before the Royal Society and received full support from member institutions. Thanks to the support of this, the government agreed to provide financial assistance for continued development of the "difference engine" is.

Babbage Difference Machine

Babbage designed a difference engine to calculate and print mathematical tables automatically. Thus, human error which may be made can be nullified. He made a table of logarithms in 1827 by wearing a smaller version of the engine.
Although Babbage was a professor of mathematics at Cambridge University from the years 1826 - 1835, he was rarely asked to give lectures. This enabled him to devote most of his time for research. However, the manufacture of larger machines are expensive, while government funding is insufficient and bureaucracy greatly inhibited. The project could only proceed after Babbage inherited from his father who died in 1827.

Babbage's Analytical Machine

Babbage's difference engines continue to improve until the 1830s. Then he got the idea to create "analytical engine". This machine consists of four parts warehouse that became a memory, the factory where did the math, a system of gears and lever for the transfer of data between factories and warehouses, and one unit of input / output (this arrangement in accordance with the arrangement of the modern computer, even though its components vary) .
Warehouse analytical machine using wheels with ten different positions for storing numbers, as did the difference engine. Warehouse that can store up to 1,000 numbers with 50 digits of each number.
The idea of ​​this feedback mechanism Babbage obtained from an unusual source, namely the French silk weaving industry. In 1801, Joseph Marie Jacquard created a weaving machine using perforated cards for the "program" the desired pattern into the loom. Thus, the same pattern can be printed in large quantities. Babbage realized that this system can be used to enter data and store instructions into the machine.
Unfortunately, Babbage did not succeed in forming a working model for the analytical engine. He constantly facing financial difficulties because of the cost to design and create a new machine. But the biggest problem is the inability at that time engineering techniques to produce components sufficiently accurate and flexible. The failure of this technology makes Babbage was very disappointed.
"Babbage seek something that is impossible with the means which he possesses. However, the concept and principles behind the analytical engine was absolutely correct." This was revealed when Babbage's notebooks were discovered in 1937 and studied re-design. With the technology in the 1940s, modern computers become a reality.
Babbage was not only designed the forerunner of computer hardware (engine) of the present, but also has mengonsepsikan essential elements of the software (programs) the computers we know today. Babbage's conception of how to structure an analytical engine program is very similar to the technique used to program a modern computer.

Other donations

Babbage was concerned because advances in mathematics and sciences of Europe unacceptable in Britain. In the paper entitled "Reflections on the Decline of Science in England" in 1830, he imposed some of the blame for these issues arose at the Royal Society. Society has become very large, with about 630 members. However, only about a hundred people who actually practice as a scientist. Previous scientific debates very preferably also have been lost. That's why Babbage founded and became a member of the British Association for the Advancement of Science in 1831. This society is still functioning as an arena for scholarly discussion until now.
Babbage participate mesndirikan Royal Astronomical Society in 1820. He also co-founded the Statistical Society in 1834. He set the tables of a reliable estimate of the first calculation, ie the tables "risk" is used by insurance companies. He also helped define the modern postal system in England.
Babbage's findings pretty much, including the speedometer, the cow-catcher (cowcatcher) used in front of the locomotive, and ophtalmoskop (a tool doctors use to examine the inside of the eye). He also manipulate hundreds of tools and equipment for factory machinery. The results of the other engineering applied in mining, architecture, and construction of bridges.
Apart from industrial equipment to engineer, Babbage also advocated a new approach in the industry and government, known as "operational research" (operations research). The Heritage Dictionary defines operational research as a "mathematical or scientific analysis of the systematic efficiency and performance of manpower, machinery, equipment, and policies in government, military, or trade." 1832, Babbage published his approach was in the book "On the Economy of Machinery and Manufactures".
Rekacipta Babbage and operations research techniques play an important role in the development of British industrial technology, when the country emerged as a world industry leader. However, Babbage always campaigned for reforms in government policies to further encourage the development of scientific research. But generally, his appeal was ignored.

Christian character

In his biography written by his friend, HW Buxton, Babbage described as being "warm and generous; her friends are loyal and dependable". Babbage described as someone with integrity. "If he believes a principle, he would defend it despite the challenges." Although frustrated for not able to convince others of the need to maintain the progress of British science and industry, Babbage never criticize them who do not support it. Buxton said, "bad-mouth other people had nothing in his character."

Harmony Science And Christianity

Many of Babbage's work in the fields of mathematics and science has been published. In 1837, he also wrote one of the discussion of Bridgewater. This is a series of papers entitled "On the Power, Wisdom, and Goodness of God, as manifested in the Creation", published by the Royal Society and funded by the nobles Bridgewater. As written in the biography of Babbage Anthony Hyman, "Babbage believe that the scientific method which functioned until the maximum limit, entirely in harmony with revealed religion, and he wrote 'Ninth Bridgewater Treatise' to prove it."

Peace in the Christian Certainty

Babbage faith is more than simply recognizing the harmony of science and Christianity. As Buxton, Babbage "believe that the study of nature with scientific accuracy is the preparation to be done, in order to understand and interpret the nature of the testimony of wisdom and goodness of the divine Creator".
Charles Babbage died on October 18, 1871 in London, at the age of 79 years. Hyman states that at the time breathed his last, he conceived a great sense of peace because of his beliefs, especially about the certainty of the Christian life after death. Babbage is remembered not only as the father of modern computer science, but also as a Christian who submit completely to his God.

Taken and edited as needed from:
The original title of the article: Charles Babbage (1791-1871)
The original title of the book: 21 Great Scientists Who Believed the Bible
Title of the book: Trusting Divine Scientists
Writer: Ann Lamont
Translators: Lillian D. Tedjasudhana
Publisher: YKBK (Yayasan Bina Love Communications) / OMF, Jakarta 1997
Pages: 134-146

Sumber 4 Translite

Charles Babbage

Komputer telah menjadi bagian yang sangat penting dalam kehidupan masyarakat modern sekarang. Komputer modern dimungkinkan berkat perkembangan alat elektronik selama Perang Dunia II. Tapi, gagasan di balik komputer modern sebenarnya telah dipikirkan lebih dari seratus tahun sebelumnya oleh Charles Babbage. Sayang, teknologi pada zaman itu belum cukup maju sehingga dia tidak bisa menyaksikan penerapan gagasannya.

Masa Muda

Charles Babbage lahir tanggal 26 Desember 1791. Ayahnya, Benjamin Babbage, saudagar dan bankir kaya. Keluarga Babbage tinggal di Walworth, Surrey, di pinggiran kota London. Charles adalah putra pertama dari empat bersaudara, namun kedua saudara laki-lakinya meningggal pada masa kanak-kanak. Ketika Charles sakit parah tahun 1799, kedua orang tuanya takut kehilangan dia juga. Karena itu, dia dibawa ke Devon untuk mendapatkan udara pedesaan yang lebih sehat.

Charles memulai sekolahnya di Devon. Ia mempelajari antara lain matematika untuk navigasi sederhana dan akuntansi. Inilah awal dari minat yang akan membentuk kariernya. Kesalahan perhitungan dalam navigasi sering kali mengakibatkan kecelakaan kapal. Charles telah mengabdikan sebagian besar hidupnya untuk mengembangkan beberapa jenis mesin yang dapat menghitung dan mencetak tabel matematik dan astronomik secara cermat sehingga kesalahan dapat ditiadakan.

Ketika Charles sembuh dari sakitnya, ia kembali ke London. Dia bersekolah di Enfield, dan guru-gurunya segera melihat kemampuannya dalam bidang matematika. Tahun 1803, keluarganya pindah dan menetap di Devon. Charles bersekolah di Totnes Grammar School sampai tahun 1810, dan kemudian masuk Trinity College di Universitas Cambridge.

Matematika Pada Zaman Babbage

Babbage begitu berminat mempelajari matematika, sehingga dia mengisi waktu senggangnya dengan membaca buku-buku matematika, termasuk yang ditulis dalam bahasa Perancis. Ketika minta tolong kepada guru-gurunya, dia heran karena mereka sama sekali tidak mengetahui perkembangan terakhir dalam bidang matematika di Perancis. Waktu itu Inggris dan Perancis sedang bermusuhan karena perang Napoleon, dan ada kekhawatiran kalau pemberontakan seperti Revolusi Perancis akan terjadi di Inggris. Akibatnya, mempelajari karya ahli matematika dan ilmuwan Perancis, seperti Blaise Pascal, dianggap tindakan yang tidak patriotik.

Para ahli matematika Inggris juga mengabaikan perkembangan di Jerman. Gottfried Leibniz di Jerman dan Sir Isaac Newton di Inggris secara terpisah dan pada waktu hampir bersamaan, telah menemukan kalkulus -- suatu prosedur matematika baru yang revolusioner. Namun, para pemimpin akademik di masing-masing negara mengklaim bahwa ahli merekalah yang patut memperoleh penghargaan atas penemuan itu. Persaingan nasional untuk memperoleh penghargaan itu mengakibatkan Jerman maupun Inggris akhirnya tidak memperoleh apa-apa.

Penolakan pemikiran dari Eropa ini menghambat perkembangan matematika di Inggris. Ini juga berarti bahwa mereka yang mempelajari kemajuan yang terjadi di Eropa, seperti Babbage, dianggap kelompok liberal yang tidak patriotik, dan mereka menghadapi kecaman dari banyak teman sejawat yang picik. Namun demikian, banyak dari karya Babbage kelak didasarkan atas kemajuan yang dihasilkan oleh Pascal dan Leibniz.

Tahun 1812, Babbage dan dua temannya membentuk Perhimpunan Analitis (Analytical Society) di Cambridge. Kedua temannya itu adalah astronom terkemuka John Herschel (juga Kristen yang penuh pengabdian seperti Babbage) dan ahli matematika George Peacock. Melalui perhimpunan itu, mereka berupaya agar buku-buku metode matematika terbaru dalam bahasa Perancis diterjemahkan ke dalam bahasa Inggris. Tapi karena pekerjaan itu berlangsung lamban, akhirnya mereka sendirilah yang mengerjakan tugas tersebut. Kelak, Perhimpunan Analitis sangat berperan penting dalam memperbarui pengajaran matematika di perguruan-perguruan tinggi di Inggris. Namun, proses ini berjalan sangat lamban.

Arah Baru

Charles Babbage mendapat gelar dalam bidang matematika tahun 1814. Tahun itu juga, dia menikah dengan Georgina Whitmore. Mereka memiliki delapan anak, tapi hanya lima yang hidup melewati usia kanak-kanak. Georgina meninggal tahun 1827. Tidak lama setelah menikah, Babbage memutuskan untuk menjadi pendeta. Dia melamar ke beberapa gereja. Sayang, para pemimpin gereja terlalu memercayai tuduhan bahwa Babbage adalah seorang liberal yang tidak patriotik, sehingga lamarannya ditolak.

Kerugian gereja menjadi keuntungan bagi matematika. Charles dan istrinya pindah ke London tahun 1815. Di sini, dia menunjukkan kemampuan praktisnya dalam matematika dan memberikan serangkaian ceramah mengenai manfaat eksperimen, di samping teori matematika. Berkat kegiatan ini, tahun 1816, dia terpilih sebagai anggota Royal Society -- perkumpulan paling bergengsi untuk para ilmuwan Inggris. Babbage memperoleh gelar master tahun 1817.

Selama beberapa tahun berikutnya, Babbage memberikan sumbangan penting dalam bidang matematika murni, seperti aljabar dan teori fungsi. Tapi keinginannya yang utama adalah mempraktikkan matematika. Dengan dukungan para ahli matematika, navigator, dan ilmuwan, dia mulai mengerjakan mesin analitis.

Alat-Alat Kalkulasi Sebelumnya

Mesin hitung yang pertama kali dikenal dunia adalah "abakus", yang dipakai bangsa Tionghoa sejak sekitar tahun 600 SM. Alat ini terdiri dari manik-manik yang digantung pada dawai dalam bingkai, dan manik-manik itulah yang digerakkan di sepanjang dawai selama perhitungan. Setiap manik memunyai nilai angka tertentu.

Tahun 1614, ahli matematika Skotlandia, John Napier, menerbitkan karya pertamanya mengenai temuannya yang disebut logaritma. Dengan menggunakan sederet batang, yang sekarang dikenal sebagai "tulang-tulang Napier", Babbage menyederhanakan perkalian dan pembagian dengan mengubahnya menjadi proses penambahan dan pengurangan yang lebih sederhana. Mistar hitung temuan Edmund Gunter tahun 1620 juga memakai asas ini.

Kemajuan berikutnya dalam alat hitung muncul tahun 1642, ketika Blaise Pascal menemukan mesin hitung yang pertama, yang mampu menambah dan mengurangi. Mesin ini terdiri dari seperangkat roda, masing-masing dengan angka 0 sampai 9. Roda-roda dihubungkan dengan gir, sehingga apabila satu roda berputar penuh, akan menggerakkan roda di sebelahnya sepersepuluh dari satu putaran. Tapi mesin ini mahal dan sukar dioperasikan. Pada tahun 1671, Gottfried Leibniz meningkatkan kemampuan mesin Pascal dengan menambah kereta yang dapat digerakan. Mesin ini sekarang dapat mengalikan dan membagi.

Awal tahun 1820-an, Babbage mulai bekerja untuk membuat mesin hitung dengan kapasitas dua puluh desimal. Dia mulai dengan membuat mesin hitung kecil beroda enam yang bisa menghitung secara cermat. Mesin kecil ini diperagakannya di hadapan Royal Society dan mendapat dukungan penuh dari anggota lembaga tersebut. Berkat dukungan itulah, pemerintah setuju memberi bantuan keuangan demi kelanjutan perkembangan "mesin perbedaan" ini.

Mesin Perbedaan Babbage

Babbage merancang mesin perbedaannya untuk menghitung dan mencetak tabel matematika secara otomatis. Dengan demikian, kesalahan yang mungkin dibuat manusia bisa ditiadakan. Dia membuat tabel logaritma tahun 1827 dengan memakai versi yang lebih kecil dari mesinnya.

Meskipun Babbage adalah profesor matematika di Universitas Cambridge dari tahun 1826 -- 1835, dia jarang diminta memberi kuliah. Ini memungkinkannya mengabdikan sebagian besar waktunya untuk penelitian. Namun, pembuatan mesin yang lebih besar membutuhkan biaya mahal, sementara dana dari pemerintah tidak cukup dan birokrasi sangat menghambat. Proyek itu baru bisa dilanjutkan setelah Babbage menerima warisan dari ayahnya yang meninggal pada tahun 1827.

Mesin Analitik Babbage

Babbage terus meningkatkan kemampuan mesin perbedaannya hingga tahun 1830-an. Kemudian dia mendapat gagasan untuk menciptakan "mesin analitis". Mesin ini terdiri dari empat bagian gudang yang menjadi memori, pabrik tempat melakukan perhitungan matematika, suatu sistem roda gigi dan pengumpil untuk pemindahan data antara pabrik dan gudang, serta satu unit masukan/keluaran (susunan ini sesuai dengan susunan komputer modern, meskipun komponennya berbeda).

Gudang mesin analitis memakai roda dengan sepuluh posisi yang berbeda untuk menyimpan angka, sebagaimana dilakukan mesin perbedaan. Gudang itu bisa menyimpan sampai 1.000 angka dengan 50 digit setiap angka.

Ide tentang mekanisme masukan ini diperoleh Babbage dari sumber yang tidak biasa, yakni industri penenunan sutra Perancis. Tahun 1801, Joseph Marie Jacquard menciptakan mesin tenun yang memakai kartu berlubang-lubang untuk "memprogram" pola yang diinginkan ke dalam mesin tenun. Dengan demikian, pola yang sama bisa dicetak dalam jumlah banyak. Babbage menyadari bahwa sistem ini dapat dipakai untuk memasukkan data dan menyimpan instruksi ke dalam mesin.

Sayangnya, Babbage tidak berhasil membentuk model kerja untuk mesin analitisnya. Dia terus-menerus menghadapi kesulitan keuangan karena besarnya biaya untuk merancang dan membuat mesin baru. Tapi masalah terbesar adalah ketidakmampuan teknik rekayasa pada masa itu untuk menghasilkan komponen-komponen yang cukup akurat dan fleksibel. Kegagalan teknologi ini membuat Babbage sangat kecewa.

"Babbage mengupayakan sesuatu yang mustahil dengan sarana yang dia miliki. Namun, konsep dan asas di balik mesin analitis memang mutlak benar." Hal ini terungkap ketika buku catatan Babbage ditemukan tahun 1937 dan rancangannya dipelajari kembali. Dengan teknologi tahun 1940-an, komputer modern menjadi kenyataan.

Babbage tidak hanya merancang cikal bakal peranti keras komputer (mesinnya) masa kini, tapi juga telah mengonsepsikan unsur-unsur penting dari peranti lunak (program) komputer yang kita kenal sekarang. Konsepsi Babbage mengenai cara menyusun program mesin analitis sangat mirip dengan teknik yang dipakai untuk memprogram komputer modern.

Sumbangan Lain

Babbage prihatin karena kemajuan matematika dan ilmu dari Eropa sukar diterima di Inggris. Dalam tulisan berjudul "Reflections on the Decline of Science in England" tahun 1830, dia membebankan sebagian kesalahan atas timbulnya masalah ini pada Royal Society. Perhimpunan ini telah menjadi sangat besar, dengan sekitar 630 anggota. Namun, hanya sekitar seratus orang yang benar-benar berpraktik sebagai ilmuwan. Perdebatan ilmiah yang sebelumnya sangat diutamakan juga telah hilang. Karena itulah Babbage mendirikan dan menjadi anggota British Association for the Advancement of Science tahun 1831. Perkumpulan ini masih berfungsi sebagai arena diskusi ilmiah hingga sekarang.

Babbage turut serta mesndirikan Royal Astronomical Society tahun 1820. Dia juga ikut mendirikan Statistical Society tahun 1834. Dia menyusun tabel-tabel perkiraan kalkulasi pertama yang andal, yakni tabel-tabel "risiko" yang dipakai oleh perusahaan asuransi. Dia juga membantu menentukan sistem pos yang modern di Inggris.

Temuan Babbage cukup banyak, antara lain spidometer, penangkap sapi (cowcatcher) yang dipakai di depan lokomotif, dan ophtalmoskop (alat yang dipakai dokter untuk memeriksa bagian dalam mata). Dia juga merekayasa ratusan alat dan perlengkapan mesin untuk pabrik. Hasil rekayasanya yang lain diterapkan dalam pertambangan, arsitektur, dan konstruksi jembatan.

Selain merekayasa peralatan industri, Babbage juga menganjurkan pendekatan baru dalam industri dan pemerintahan yang dikenal sebagai "penelitian operasional" (operations research). The Heritage Dictionary mendefinisikan penelitian operasional sebagai "analisis matematis atau ilmiah terhadap efisiensi sistematik dan kinerja tenaga manusia, mesin-mesin, perlengkapan, dan kebijakan dalam pemerintahan, militer, atau perdagangan." Tahun 1832, Babbage menerbitkan pendekatannya itu dalam buku "On the Economy of Machinery and Manufactures".

Rekacipta Babbage dan teknik penelitian operasionalnya berperan penting dalam perkembangan teknologi industri Inggris, sewaktu negara itu muncul sebagai pemimpin industri dunia. Namun, Babbage senantiasa mengampanyekan reformasi dalam kebijakan pemerintah untuk lebih mendorong perkembangan penelitian ilmiah. Tapi umumnya, seruannya tidak dihiraukan.

Watak Kristiani

Dalam biografinya yang ditulis oleh temannya, H.W. Buxton, Babbage dilukiskan sebagai orang yang "hangat dan dermawan; dia teman yang setia dan bisa diandalkan". Babbage digambarkan sebagai orang yang memiliki integritas. "Bila dia meyakini suatu prinsip, dia akan mempertahankannya meskipun menghadapi tantangan." Meskipun frustrasi karena tak berhasil meyakinkan orang lain mengenai perlunya mempertahankan kemajuan ilmu dan industri Inggris, Babbage tak pernah mengecam mereka yang tidak mendukungnya. Buxton berkata, "Menjelek-jelekkan orang lain sama sekali tidak ada dalam wataknya."

Keserasian Ilmu Dan Kekristenan

Banyak karya Babbage dalam bidang matematika dan ilmu sudah diterbitkan. Tahun 1837, dia juga menulis satu dari Pembahasan Bridgewater. Ini adalah serangkaian tulisan yang berjudul "On the Power, Wisdom, and Goodness of God, as Manifested in The Creation", yang diterbitkan oleh Royal Society dan dibiayai oleh bangsawan Bridgewater. Sebagaimana ditulis Anthony Hyman dalam biografi Babbage, "Babbage percaya bahwa metode ilmiah yang difungsikan sampai batas maksimalnya, seluruhnya serasi dengan agama yang diwahyukan, dan dia menulis 'Ninth Bridgewater Treatise' untuk membuktikannya."

Damai dalam Kepastian Kristen

Iman Babbage lebih dari sekadar mengakui keserasian ilmu dan kekristenan. Sebagaimana dikatakan Buxton, Babbage "percaya bahwa pengajian alam dengan ketelitian ilmiah adalah persiapan yang harus dilakukan, agar bisa memahami dan menafsirkan kesaksian alam mengenai kearifan dan kebaikan Penciptanya yang ilahi".

Charles Babbage meninggal tanggal 18 Oktober 1871 di London, dalam usia 79 tahun. Hyman menyatakan bahwa pada waktu menghembuskan napasnya yang terakhir, Babbage merasakan damai sejahtera yang besar karena keyakinannya, terutama mengenai kepastian orang Kristen akan kehidupan sesudah kematian. Babbage tidak hanya dikenang sebagai bapak ilmu komputer modern, tapi juga sebagai orang Kristen yang berserah sepenuhnya kepada Tuhan-nya.

Sumber 5

DIFFERENTIAL CALCULUS

Differential calculus is a calculus branch of mathematics that studies how the value of a function is changed by changing the input value. The main topics in differential calculus is an instance of learning. The derivative of a function at a certain point explain the properties of the function close to the input value. For real-valued functions with a single real variable, the derivative at a point equal to the slope of the tangent line graph of the function at that point. In general, the derivative of a function at a point in determining the best approach to a linear function at that point.
The search process is called pendiferensialan derivative (differentiation). Fundamental theorem of calculus states that pendiferensialan invertibility is the process of integration.
Derivatives have applications in all areas of quantitative. In physics, a derivative of the displacement object with respect to time is velocity, and the derivative of the velocity versus time is accelerating. Newton's second law of motion states that the derivative of the momentum of an object is equal to the force applied to the object.
Reaction rate of chemical reactions is also derived. In operations research, derivatives determine the most efficient way of moving materials and design. By applying game theory, derivatives can provide the best strategy for companies that are competing.
Derivatives are often used to search for ekstremum point of a function. Equations involving derivatives are called differential equations and is very important in describing natural phenomena. Derivatives and perampatannya (generalization) often appear in different areas of mathematics, such as complex analysis, functional analysis, differential geometry, and even abstract algebra.

Derivative

Let x and y are real numbers where y is a function of x, ie y = f (x). One of the simplest type of function is a linear function. This is a graph of the function of a straight line. In this case, y = f (x) = mx + c, where m and c are real numbers which depend on the line where the graph is determined. m is referred to as the slope of the formula:

where the symbol Δ (delta) means "change in value". This formula is true because
y + Δy = f (x + Δx) = m (x + Δx) + c = mx + c + y + m Δx = mΔx.
Followed Δy = m Δx.
However, the things above are only applicable to linear functions. Nonlinear function has no definite value of the slope. Derivative of f at point x is the best approach to the idea of ​​the slope of f at point x, is usually marked with f '(x) or dy / dx. Together with the value of f at x, the derivative of f determines the linear approach most closely, or so-called linearization, of f near the point x. These properties are usually taken as the definition of a derivative.
A term closely related to each other is a differential function derivative.

When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph f 'at the point x. Because the source and target of f-dimensional one, the derivative of f are real numbers. If x and y are vectors, then the linear approach that comes closest to the graph f f depends on how the changes in several directions simultaneously. By taking a linear approach most closely in one direction determines a partial derivative, usually characterized by ∂ y / ∂ x. Linearization of f to all directions simultaneously is referred to as total derivatives. This total derivative is a linear transformation, and it determines the closest hiperbidang graph of f. Hiperbidang is referred to as hiperbidang oskulasi; this same concept by taking tangents to all directions simultaneously.

The application of derivative

optimization

If f is a function that can be derived on R (or an open interval) and x is a local maximum or local minimum of f, the derivative of f at point x is zero; the points where f '(x) = 0 is called the critical point or pegun point (and the value of f at x is called the critical value). (The definition of the critical point is sometimes extended to include the points at which the derivative of a function does not exist.) Rather, the critical point x of f can be analyzed using the second derivative of f at x:

• if the second derivative is positive, x is a local minimum;
• if the second derivative is negative, x is a local maximum;
• if the second derivative is zero, x may be a local maximum, local minimum, or not both. (For example, f (x) = x ³ has a critical point at x = 0, but the point is not the point of maximum or minimum point; otherwise f (x) = ± x4 has a critical point at x = 0 and that point is the minimum point and the maximum.)
It is named as the second derivative test. An alternative approach, the first derivative test involves the value of f 'on both sides of the critical point.
Reduce the function and look for critical points is usually one simple way to find local minima and local maxima, which can be used for optimization. In accordance with ekstremum value theorem, a function which is continuous on a closed interval must have a minimum value and maximum at least once. If the function can be lowered, minima and maxima can only occur at a critical point or end point.
It also has its own application in the process of sketching the graph: if we know the local minima and maxima of functions that can be derived, an approximate graph can we get from the observation that he would rise and decline of the critical points.
In higher dimensions, the critical point of the scalar function is the point where the gradient function is zero. The second derivative test can still be used to analyze the critical points by using the eigenvalues ​​of the Hessian matrix of second partial derivatives of the function at the critical point. If all eigenvalues ​​are positive, then the point is a local minimum; if everything is negative, then it is a local maximum point. If there are some positive and some negative, then the critical point is a saddle point, and if none of the above conditions are met (eg there are some eigenvalues ​​are zero) then the test inkonklusif.

calculus of variations

One example optimalisai problem is finding the shortest curve in advance of two points on a surface with the assumption that the curve should be on the surface. If the surface is flat field, then the shortest curve is a straight line. But if the surface is not the field, then we can not know for certain that the shortest curve. This curve is called a geodesic, and one of the simplest problems in the calculus of variations is to find other geodesik.Contoh is to find the smallest surface area bounded by closed curves in three dimensional space. This surface is referred to as a minimum surface, and these can be searched by using the calculus of variations.

Physics

Calculus is very important in physics. Many physical processes that can be described by the derivative, referred to as differential equations. Physics specifically studied changes in the quantity of time, and the concept of "derivative time"-time rate of change to the change-is extremely important as the precise definition on some important concepts. For example, the time derivative of the position of objects is very important in physics Newtonan:
• speed is the derivative of the position of the object with respect to time.
• acceleration is the derivative of velocity versus time, or the second derivative of the position of objects with respect to time.
For example, if the position of an object in a line are:

then the speed of the object is:

and acceleration of the object it is:

Differential Equation

Differential equations is the relationship between a group of functions with their derivatives. Ordinary differential equation is a differential equation linking a variable to a function with derivatives of the variable itself. Partial differential equations are differential equations which connect the functions that have more than one variable to the partial derivative. Differential equations arise naturally in the physical sciences, mathematical models, and in mathematics itself. For example, Newton's second law describes the relationship between the acceleration of the position can be initiated by ordinary differential equations:

Heat equation in one space variable that describes how the heat diffuses through a straight rod that is partial differential equations.

Here u (x, t) is the temperature of the stick at the position x and time t and α is a constant that depends on how fast the heat diffuses.

Average value theorem

Average value theorem gives the relationship between the value of the derivative to the value of the original function. If f (x) is real-valued function and a and b are numbers with a <b, then the Average value theorem says that the slope between two points (a, f (a)) and (b, f (b)) is equal with the slope of the tangent line at point f c between a and b. In other words:

In practice, this theorem Average value of controlling a function of its derivatives. For example, suppose that f has a derivative equal to zero at every point, then the function must be horizontal. Average value theorem proves that this should be true, that the slope between two points on the graph f must equal the slope of a tangent line at f All of the slope is zero, so the line between any one point with another point in the function has zero slope. But it also says that the function does not go up or down.

Polinomial Taylor dan deret Taylor

Derivative gives the best linear approach, but this approach can be very different from the original function. One way to improve this approach is to use the quadratic approach. Linearization of real-valued function f (x) at a point x0 is the linearization polynomial a + b (x - x0), and it is possible to get a better approach by using a quadratic polynomial a + b (x - x0) + c (x - x0) ². Still better when using cubic polynomial a + b (x - x0) + c (x - x0) ² + d (x - x0) ³, and this idea can be expanded up to a high degree polynomial. For each of these polynomials, there is a choice to be the most appropriate coefficient values ​​for a, b, c, and d which makes this approach as closely as possible.
For a, the best option is always worth the value f (x0), and for b is always worth f '(x0). For c, d, and other high-degree coefficients, these coefficients are determined with a high degree derivative of f. c must be f''(x0) / 2, and d be f'''(x0) / 3!. By using this coefficient, we get the Taylor polynomial of f. Taylor polynomial of degree d is a polynomial of degree d which gives the best approach to f, and the coefficient can be determined by perampatan of the above formula. Taylor's theorem gives the constraints that will detail how well the approach. If f is a polynomial of degree smaller or equal to d, then the Taylor polynomial of degree d equal to f.
Limitation of Taylor polynomials is infinite series known as Taylor series. Taylor series is usually the approach quite close to the original function. The same functions with Taylor series is referred to as analytic functions. It is not possible for functions that are not continuous or have a sharp angle to be analytic functions. But there are also non-analytic smooth function.

Teorema fungsi implisit

Several natural geometric shapes, like circles, can not be drawn as a graph of the function. If F (x, y) = x ² + y ², then the circle is the set of pairs (x, y) where F (x, y) = 0. The set is referred to as the set of zeros (zero set) (not the empty set) of F. This is not the same as the graph F, which form a cone. Implicit function theorem to change the relation as F (x, y) = 0 be a function. This theorem states that if F is continuously differentiable, then around most of the points, the set of zeros of F looks like a graph of the function are coupled together. Point where this is not correctly specified in the conditions of derivative F. The circle can be combined together with the graphs of two functions. In every neighborhood of the circle except the point (-1, 0) and (1, 0), one of the two functions has a graph similar to the circle. (The two functions are also met in the (-1, 0) and (1, 0), but this is not confirmed by the implicit function theorem).
Implicit function theorem closely related to the inverse function theorem that determines when a function graph looks similar to the irreversible functions are coupled together.