Saturday, 25 May 2013
Study Tips for Math  Algebra
Math Studying & Test Taking Tips
 

Friday, 24 May 2013
HPU observes Mathematics Awareness Month
Howard Payne University’s Department of Mathematics recognized April as Mathematics Awareness Month. The national theme for this year, according to the American Mathematical Society, was Mathematics of Sustainability, “balancing needs and seeking solutions for a complex changing world.”
The month of April was an exciting time for the students and faculty in the math department as they hosted HPU’s second annual Math Day and participated in many other events.During the second week of April, two faculty members and four students from the math department traveled to Texas Tech University to attend the Texas Section Meeting of the Mathematical Association of America. While there, the students in attendance represented HPU in the Annual Calculus Bowl. The HPU team consisted of senior mathematics majors Laura Peisker of Cedar Park and Kathryn Barrackman of Houston, as well as juniors James Smith of Comanche and Tim Marlin of Uvalde. Improvement was shown from last year’s team, the first in HPU’s history to compete, as the 2013 team advanced from one of the two preliminary rounds into finals.
“This activity is both fun and competitive for students as it heightens interest in calculus, which is a core discipline for mathematics majors,” said Dr. Kenneth Word, department chair and professor of mathematics.
Toward the end of the month, the School of Science and Mathematics hosted the HPU General Education Math Day. The keynote speaker was Michael Sullivan, professor of mathematics at Joliet Community College in Illinois, the oldest community college in the U.S. Sullivan is the author of “Fundamentals of Statistics,” a Pearson Education publication, which is the textbook used for Introduction to Probability and Statistics at HPU. Sullivan demonstrated a statistics software program called StatCrunch used for data analysis and descriptive statistics that accompanies the MyStatLab software used with his textbook.
The School of Science and Mathematics also hosted the Engineering Math Day, featuring guest speaker Dr. Ron Larson, professor of mathematics at Penn State University in Erie. Dr. Larson is known as the lead author of a comprehensive program for mathematics which spans middle school, high school and college courses. Dr. Larson is even more wellknown on HPU’s campus as the author of the precalculus text and coauthor of the calculus text, both Cengage publications, currently used by the math department. Dr. Larson spoke about six mathematical models that are the basis for most mathematical, physics and engineering problems. He particularly discussed the linear model used to determine Hooke’s Law and the linear regression of a natural logarithmic model used to determine Kepler’s Law concerning the orbits of the planets around the sun.
Also speaking at Engineering Math Day was Thomas Hummel, a graduate of Texas A&M University in College Station and Central Texas College near Fort Hood. An IBM circuit design engineer and retired Army veteran, Hummel is also a veteran of the Iraqi campaign. He spoke of the importance of mathematics in building one’s resume for career prospects in science or engineering.
According to Dr. Word, Mathematics Awareness Month has become a tradition for the HPU mathematics department over the last two years.
“This tradition not only benefits the department but the whole campus as well,” he said.
Monday, 20 May 2013
How to Study Math
Steps
 Learn all of the basic addition, subtraction, multiplication, and division facts by memory. Failure to learn the basic facts properly will make advanced math impossible.
 Learn Mathematical Definitions (vocabulary). Have your teacher restate (and/or explain) the words you don’t understand. Even if your current teacher doesn’t use the terminology often, you can be certain that other teachers will.
 Four Squared, Four Cubed, Factor of Four, Four Factorial all have different meanings. Knowing the definitions will make solving them easier.
 Try to anticipate the assignment of math problems by your teacher. This may seem like extra work, but it will be an advantage.
 Work on all of the problems (odd and even) from your textbook before they are assigned.
 Some teachers always assign the even questions, so the students can’t cheat. Some teachers always assign odd questions, so the students can correct their own work. Some teachers assign one set of problems for homework, and use the others for tests!
Ask the teacher for help for any problems troubling you, even problems that were not assigned. Remember, you are trying to learn. Problems that are unassigned often end up on tests. And the extra difficult problems give you a chance to earn extra credit.
When the teacher discusses the subject (probably before assigning the problems), ask questions that may occur to you (because of the work you have already done.) This is the most useful aspect of doing the work early: All of the other students are thinking, "HUH?" while you are thinking about a specific question you need answered.
 Some college mathematical professors teach their classes entirely by answering questions from the students. The students are expected to come to class having completed some of the work.
 Another benefit of completing work early (including extra work) is that if you need to turn something in late, your teacher will know that you are not trying to take advantage of his/her goodwill and will “give you a break”.
If there is a sample problem in your book, or an example problem, work through it yourself, and use the sample in your book to guide you.
Identify your errors. Discovering your own errors is the best way not to repeat them.
Don't go on in your book until you have learned the material completely. Math builds upon itself.
 A math book is like a novel, it doesn’t make sense unless you start at the beginning.
Neatly complete your work. Endeavor to make all of your numbers look the same way every time. The more complex the math, the more neatness counts.
Form a study group. When one person in the group has a problem, others can help.
 Do not let anyone in the group COPY OR CHEAT! Copying will be discovered eventually, one way or another. Besides, why would you wish someone else to get credit for your work?
 use it as much as possible in life
Tips
 The best thing you can do when studying unfamiliar math is to leave a paper trail. Take the extra time to make your handwriting look nice, and when you’re having trouble, try not to skip steps. If you really want your math to look good, use rulers for your straight lines and fraction bars. It’s much easier to study or retrace your own work when it is written out clearly.
 Don’t worry if you feel like you don’t understand something immediately. This might be a good thing, and can stretch your mind. You might have to live with a problem, or a step in a proof, or even a definition, for a few days or a week before your mind can absorb it and grasp it. Stop working on it, do something else, and go back to it later. Stay with it, but in intervals. Look at other descriptions or similar problems in other books for ideas.
 If you are fuzzy about something that the book expects you to know before you read it, and the book does not review it, or review it in enough depth so that you are not fuzzy, you should look that up, and learn it first.
 If you’re learning an odd theorem or property, make up and write down some examples that satisfy the preconditions, and see if you can find the results yourself. This will allow you to get the properties “under your fingers.” This is especially useful in abstract algebra and number theory.
 Use all of your senses and sensations to learn Math and fix it in your mind. Write things out(uses visual brain and kinesthetic brain). Say definitions and theorems out loud(uses your auditory brain).
 If you need to study for long periods, take breaks at regular intervals for a set amount of time.
Warnings
 Don’t be too hard on yourself. Realize that many have struggled with the same areas you are learning. Some people just take longer to understand math. Eventually, with enough, perseverance, you can succeed in math.
 Don’t be afraid to ask for help from someone who has been doing math for a long time or is further along.
 please if your a child first learn the basics than you can highleveled math.
 Do not push self too far on maths it if you do your mind will be confused.
Things You’ll Need
 notebooks
 pencils
 mathbooks
 blue and black pen
Matematik bukan subjek hafalan
SEBAGAI ahli akademik, saya hairan dengan sikap segelintir pelajar khususnya di sekolah menengah yang gemar membaca buku Matematik seolaholah membaca buku Bahasa Melayu atau novel. Matematik bukan mata pelajaran untuk dibaca apatah lagi dihafal. Banyak kesan negatif yang timbul hasil hafalan terhadap konsep tertentu dalam Matematik.
Saya percaya apabila terlalu banyak formula atau rumus tertentu dalam Matematik dihafal tanpa pemahaman, pelajar sukar menjawab ujian atau peperiksaan. Pelajar akan ‘mengeluarkan’ segala yang dihafal dan apabila terlalu banyak formula atau rumus dihafal tanpa pemahaman, pelajar keliru untuk mengaplikasikan formula tertentu dalam sesuatu soalan Matematik.
Jika dapat menjawab soalan Matematik sekalipun, mereka terus beranggapan belajar Matematik hanya untuk lulus peperiksaan.
Pelajar yang ‘membaca’ Matematik tidak dapat meningkatkan kemahiran berfikir. Ini dibuktikan dalam bidang geometri umpamanya dengan tahap pemikiran pelajar boleh diukur menggunakan pemikiran van Hiele yang terdiri daripada lima tahap.
Kajian menunjukkan budaya menghafal dalam Matematik khususnya topik yang berkaitan dengan geometri hanya membantu pelajar mencapai tahap satu atau dua dalam model berkenaan.
Jadi adakah matlamat kurikulum Matematik sekolah menengah iaitu untuk melahirkan individu yang berupaya untuk berfikir secara Matematik tercapai? Matematik ialah subjek yang indah untuk diterokai. Namun amalan pembelajaran Matematik menjadikan ia ditakuti sebilangan pelajar.
Justeru, amalan pembelajaran pada hari ini perlu dipelbagaikan agar pelajar dapat merasai keindahan Matematik. Banyak strategi pembelajaran boleh diaplikasikan. Tambahan pula, banyak teknologi pendidikan dan alat bantu mengajar boleh digunakan untuk membantu pembelajaran Matematik.
Mungkin guru berpendapat kekangan masa ialah faktor utama mengapa strategi pembelajaran tidak dipelbagaikan dan alat bantu mengajar yang dibekalkan kementerian tidak dimanfaatkan. Justeru, rancangan kementerian untuk memansuhkan peperiksaan utama terutama peperiksaan Penilaian Menengah Rendah (PMR) yang sedikit masa dulu diperkatakan, dilaksanakan bagi memastikan masa pembelajaran pelajar khususnya Matematik lebih berkualiti dan sentimen belajar Matematik hanya untuk lulus dalam peperiksaan dapat dihapuskan.
ABDUL HALIM ABDULLAH,
Fakulti Pendidikan, Universiti Kebangsaan Malaysia.
Saya percaya apabila terlalu banyak formula atau rumus tertentu dalam Matematik dihafal tanpa pemahaman, pelajar sukar menjawab ujian atau peperiksaan. Pelajar akan ‘mengeluarkan’ segala yang dihafal dan apabila terlalu banyak formula atau rumus dihafal tanpa pemahaman, pelajar keliru untuk mengaplikasikan formula tertentu dalam sesuatu soalan Matematik.
Pelajar yang ‘membaca’ Matematik tidak dapat meningkatkan kemahiran berfikir. Ini dibuktikan dalam bidang geometri umpamanya dengan tahap pemikiran pelajar boleh diukur menggunakan pemikiran van Hiele yang terdiri daripada lima tahap.
Kajian menunjukkan budaya menghafal dalam Matematik khususnya topik yang berkaitan dengan geometri hanya membantu pelajar mencapai tahap satu atau dua dalam model berkenaan.
Jadi adakah matlamat kurikulum Matematik sekolah menengah iaitu untuk melahirkan individu yang berupaya untuk berfikir secara Matematik tercapai? Matematik ialah subjek yang indah untuk diterokai. Namun amalan pembelajaran Matematik menjadikan ia ditakuti sebilangan pelajar.
Justeru, amalan pembelajaran pada hari ini perlu dipelbagaikan agar pelajar dapat merasai keindahan Matematik. Banyak strategi pembelajaran boleh diaplikasikan. Tambahan pula, banyak teknologi pendidikan dan alat bantu mengajar boleh digunakan untuk membantu pembelajaran Matematik.
Mungkin guru berpendapat kekangan masa ialah faktor utama mengapa strategi pembelajaran tidak dipelbagaikan dan alat bantu mengajar yang dibekalkan kementerian tidak dimanfaatkan. Justeru, rancangan kementerian untuk memansuhkan peperiksaan utama terutama peperiksaan Penilaian Menengah Rendah (PMR) yang sedikit masa dulu diperkatakan, dilaksanakan bagi memastikan masa pembelajaran pelajar khususnya Matematik lebih berkualiti dan sentimen belajar Matematik hanya untuk lulus dalam peperiksaan dapat dihapuskan.
ABDUL HALIM ABDULLAH,
Fakulti Pendidikan, Universiti Kebangsaan Malaysia.
Ilmu matematik ajar manusia hidup lebih bersistem, efisien
"APA saiz kasut awak?"
“Berapa ketinggian awak?”
“Berapa berat badan awak?”
Demikian antara pertanyaan profesor matematik terhadap setiap orang ditemuinya dalam novel Yoko Ogawa, The Housekeeper and The Professor (2009). Bagi profesor yang mengalami penyakit memori singkat itu, hanya angka dan ungkapan matematik saja yang mudah difahami dan diingatinya. Selain itu, memang tidak diingati lebih 80 minit termasuk rupa, nama, sejarah, peristiwa, benda, foto apatah lagi konsep bukan matematik.
Maka sepanjang perbualan dalam novel itu, profesor itu hanya mengungkapkan semua hal membabitkan angka dan formula matematik. Menariknya, pembantu rumah dan anak lelakinya juga terpaksa ‘belajar’ matematik untuk membolehkan mereka berinteraksi dengan profesor itu.
Novel itu begitu bermakna kepada saya. Setiap kali saya dijadualkan bertemu tokoh matematik, saya akan membaca kembali novel itu untuk membolehkan saya mengungkapkan pemikiran saya secara matematik walaupun pada hakikatnya pengetahuan matematik saya sangat terbatas.
Sepanjang minggu ini, kampus UTM Johor Bahru dikunjungi sekumpulan ahli matematik diketuai Prof John Ockendon, Pengasas Oxford Centre for Collaborative Applied Mathmetics (OCCAM), Universiti Oxford. Selari dengan program kerjasama akademik antara Oxford dan UTM, mereka mengendalikan bengkel Mathematics in Industry Study Group (MISG) yang ditaja King Abdullah University of Science and Technology (KAUST), Arab Saudi.
MISG, jenama dikaitkan dengan Prof John Ockendon. Menerusi MISG, pihak industri tampil mengemukakan pelbagai isu dan masalah dari segi proses, kejuruteraan, pengurusan, perancangan dan sebagainya. Di satu pihak lain, ahli matematik khususnya di kalangan akademik akan berusaha memahami persoalan sebenar dan mengungkapkannya dalam formula matematik. Banyak kaedah boleh digunakan termasuk permodelan, simulasi dan sistem pakar yang akhirnya membolehkan pihak industri merancang dan menggunakan sumber secara lebih efektif serta efisien.
Sepanjang minggu ini lebih enam isu industri dibincangkan oleh lebih 100 ahli matematik dari pelbagai bidang dan institusi. Program yang turut ditaja MIMOS itu juga dijangka akan menjadi acara tahunan bagi membolehkan ahli matematik lebih menyerlah dalam konteks pembangunan industri.
Kita juga berharap kehadiran ramai tokoh matematik ternama ke Malaysia akan membuka lebih banyak ruang membudayakan matematik dalam konteks lebih luas dan produktif. Dalam konteks yang lebih tepat, matematik perlu dihayati untuk menjadikan kehidupan kita lebih bermakna, kreatif dan santai.
Ketika berbual dengan Prof Ockendon semasa melawat pejabatnya di Oxford tahun lalu, beliau memperihalkan peranan matematik dalam mengubah minda dan menyediakan landasan pemikiran lebih kukuh.
Ketika menikmati hidangan makan malam pada awal minggu ini di Johor Bharu, saya cuba berbual dengan penuh angka dan rumus matematik. Secara berjenaka saya menyebut dalam satu ucapan bahawa saya boleh meneka latar belakang akademik seseorang penutur ketika mereka memberi ucapan, khususnya ahli matematik.
Beliau bertanya apa kelainan ahli matematik? Jawab saya, gaya dan cara mereka berucap seperti persamaan terbitan (differential equation). Jawapan sedemikian memang mengundang perbahasan dan lebih banyak tanda tanya. Itu yang menggegarkan perbualan kami sepanjang dua jam mengadap hidangan makan malam!
Hal ini tidak jauh bezanya dengan perbualan profesor dan pembantu rumah dalam The Housekeeper and the Professor. Segala hal dituturkan berkait dengan angka, formula dan teori matematik, walaupun dengan seorang pembantu rumah dan anak lelakinya.
Keindahan matematik hanya boleh difahami mereka yang mencintai ilmu, menghayati resam kesarjanaan yang tinggi dan utuh. Itu sebabnya hampir semua tokoh ilmu sebelum bermula Revolusi Industri di Eropah adalah pakar dalam sesuatu bidang yang juga mempunyai pengetahuan matematik yang mendalam.
Dalam tradisi keilmuan Barat, Leonardo da Vinci, misalnya pakar matematik, sekali gus ahli falsafah dan pelukis ternama. Begitu juga dalam tradisi Islam menerusi Ibn Sina dan Alkhawarizmi. Hakikatnya matematik adalah asas ilmu untuk membolehkan kita berfikir secara logik, teratur dan bersistem.
Selepas mengikuti pengajian wajib selama 11 tahun di sekolah, seluruh warga Malaysia seharusnya sudah mampu mengira dengan betul, memahami pelbagai teori asas matematik, biarpun tidak sampai ke tahap si pembantu rumah dalam karya Ogawa itu dengan teori angka dan teori terakhir Ferment. Sekurangkurangnya kita wajib memahami bahawa 28 ialah nombor sempurna!
Penulis ialah Naib Canselor Universiti Teknologi Malaysia (UTM)
“Berapa ketinggian awak?”
Demikian antara pertanyaan profesor matematik terhadap setiap orang ditemuinya dalam novel Yoko Ogawa, The Housekeeper and The Professor (2009). Bagi profesor yang mengalami penyakit memori singkat itu, hanya angka dan ungkapan matematik saja yang mudah difahami dan diingatinya. Selain itu, memang tidak diingati lebih 80 minit termasuk rupa, nama, sejarah, peristiwa, benda, foto apatah lagi konsep bukan matematik.
Maka sepanjang perbualan dalam novel itu, profesor itu hanya mengungkapkan semua hal membabitkan angka dan formula matematik. Menariknya, pembantu rumah dan anak lelakinya juga terpaksa ‘belajar’ matematik untuk membolehkan mereka berinteraksi dengan profesor itu.
Novel itu begitu bermakna kepada saya. Setiap kali saya dijadualkan bertemu tokoh matematik, saya akan membaca kembali novel itu untuk membolehkan saya mengungkapkan pemikiran saya secara matematik walaupun pada hakikatnya pengetahuan matematik saya sangat terbatas.
Sepanjang minggu ini, kampus UTM Johor Bahru dikunjungi sekumpulan ahli matematik diketuai Prof John Ockendon, Pengasas Oxford Centre for Collaborative Applied Mathmetics (OCCAM), Universiti Oxford. Selari dengan program kerjasama akademik antara Oxford dan UTM, mereka mengendalikan bengkel Mathematics in Industry Study Group (MISG) yang ditaja King Abdullah University of Science and Technology (KAUST), Arab Saudi.
MISG, jenama dikaitkan dengan Prof John Ockendon. Menerusi MISG, pihak industri tampil mengemukakan pelbagai isu dan masalah dari segi proses, kejuruteraan, pengurusan, perancangan dan sebagainya. Di satu pihak lain, ahli matematik khususnya di kalangan akademik akan berusaha memahami persoalan sebenar dan mengungkapkannya dalam formula matematik. Banyak kaedah boleh digunakan termasuk permodelan, simulasi dan sistem pakar yang akhirnya membolehkan pihak industri merancang dan menggunakan sumber secara lebih efektif serta efisien.
Sepanjang minggu ini lebih enam isu industri dibincangkan oleh lebih 100 ahli matematik dari pelbagai bidang dan institusi. Program yang turut ditaja MIMOS itu juga dijangka akan menjadi acara tahunan bagi membolehkan ahli matematik lebih menyerlah dalam konteks pembangunan industri.
Kita juga berharap kehadiran ramai tokoh matematik ternama ke Malaysia akan membuka lebih banyak ruang membudayakan matematik dalam konteks lebih luas dan produktif. Dalam konteks yang lebih tepat, matematik perlu dihayati untuk menjadikan kehidupan kita lebih bermakna, kreatif dan santai.
Ketika berbual dengan Prof Ockendon semasa melawat pejabatnya di Oxford tahun lalu, beliau memperihalkan peranan matematik dalam mengubah minda dan menyediakan landasan pemikiran lebih kukuh.
Ketika menikmati hidangan makan malam pada awal minggu ini di Johor Bharu, saya cuba berbual dengan penuh angka dan rumus matematik. Secara berjenaka saya menyebut dalam satu ucapan bahawa saya boleh meneka latar belakang akademik seseorang penutur ketika mereka memberi ucapan, khususnya ahli matematik.
Beliau bertanya apa kelainan ahli matematik? Jawab saya, gaya dan cara mereka berucap seperti persamaan terbitan (differential equation). Jawapan sedemikian memang mengundang perbahasan dan lebih banyak tanda tanya. Itu yang menggegarkan perbualan kami sepanjang dua jam mengadap hidangan makan malam!
Hal ini tidak jauh bezanya dengan perbualan profesor dan pembantu rumah dalam The Housekeeper and the Professor. Segala hal dituturkan berkait dengan angka, formula dan teori matematik, walaupun dengan seorang pembantu rumah dan anak lelakinya.
Keindahan matematik hanya boleh difahami mereka yang mencintai ilmu, menghayati resam kesarjanaan yang tinggi dan utuh. Itu sebabnya hampir semua tokoh ilmu sebelum bermula Revolusi Industri di Eropah adalah pakar dalam sesuatu bidang yang juga mempunyai pengetahuan matematik yang mendalam.
Dalam tradisi keilmuan Barat, Leonardo da Vinci, misalnya pakar matematik, sekali gus ahli falsafah dan pelukis ternama. Begitu juga dalam tradisi Islam menerusi Ibn Sina dan Alkhawarizmi. Hakikatnya matematik adalah asas ilmu untuk membolehkan kita berfikir secara logik, teratur dan bersistem.
Selepas mengikuti pengajian wajib selama 11 tahun di sekolah, seluruh warga Malaysia seharusnya sudah mampu mengira dengan betul, memahami pelbagai teori asas matematik, biarpun tidak sampai ke tahap si pembantu rumah dalam karya Ogawa itu dengan teori angka dan teori terakhir Ferment. Sekurangkurangnya kita wajib memahami bahawa 28 ialah nombor sempurna!
Penulis ialah Naib Canselor Universiti Teknologi Malaysia (UTM)
Sunday, 19 May 2013
Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80
Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80.
The cause was esophageal cancer, which was diagnosed in October, his son Andrew said.
Since the time of Euclid and Pythagoras, proofs of mathematical theorems had consisted of long strings of equations or geometric notations that any mathematician could read and quibble with, all marching logically, step by step, toward a conclusion. But the proof that Dr. Appel and a colleague, Wolfgang Haken, established in 1976 was of a different order.
Their conclusion, that four colors would suffice for any map, depended on 1,200 hours of computer time — the equivalent of 50 days — and 10 billion logical decisions all made automatically and out of sight by the innards of an I.B.M. computer at the University of Illinois in Urbana.
Hailed in some circles, including this newspaper, as “a major intellectual feat,” the proof shepherded computers toward a greater role in higher math. But it made many mathematicians uneasy; they worried about computer bugs and wondered how they could check or understand a “proof” they could not see. And it ignited a longrunning debate about what constitutes a mathematical proof.
“Like a landmark Supreme Court case, the proof’s legacy is still felt and hotly debated,” said Edward Frenkel, a mathematician at the University of California, Berkeley.
Kevin Short, a mathematician at the University of New Hampshire, where Dr. Appel spent his later years, called the feat “a watershed for modern mathematics.”
“It has spawned whole fields of study,” he said.
Kenneth Ira Appel (pronounced ahPEL) was born on Oct. 8, 1932, in Brooklyn and grew up in Queens, where he graduated from Queens College with a degree in mathematics in 1953. His father, Irwin, was an electrical engineer, and his mother, the former Lillian Sender, had been an office worker.
After a short stint as an actuary and two years in the Army, Kenneth Appel enrolled at the University of Michigan, where he earned a Ph.D. in math in 1959. During the summers, he programmed computers for Douglas Aircraft.
Dr. Appel soon went to work for the Institute for Defense Analyses in Princeton, N.J., doing research in cryptography and number theory for the federal government. He joined the University of Illinois as a professor in 1961. Long interested in Democratic politics, he also served a term on the Urbana City Council.
Some of the thorniest problems in math are simple to state but hideously complex under the surface. Such is the case with the fourcolor theorem, first enunciated by an English mapmaker, Francis Guthrie, in 1852. He asserted that to create a map in which no adjacent countries are the same color, only four colors are needed. Although everyone believed it was true, proof had eluded a century of mathematicians until Dr. Appel attended a lecture in 1972 by Dr. Haken.
Because of the bewildering variety of map configurations, Dr. Haken was contemplating using computers to solve the problem, but as he related in his lecture that evening, experts had convinced him that it was not possible.
Dr. Appel, familiar with computers from his defense and government work, was more optimistic.
“I don’t know of anything involving computers that can’t be done; some things just take longer than others,” he said to Dr. Haken afterward, according to an account in the journal Social Studies of Science by Donald MacKenzie of the University of Edinburgh. “Why don’t we take a shot at it?”
The two started off by showing that the universe of all possible maps must contain what mathematicians call an “unavoidable set” of 1,936 different configurations. One configuration might be a country surrounded by four neighbors, for example.
Their task, then, was to prove that each of these configurations could be rendered on a map using only four colors in such a way that no two adjacent land areas were of the same color. That was where the heaviest computation would come in. To help, they recruited a computer science graduate student, John Koch, and Dr. Appel persuaded the university to let them use its I.B.M. 370168 computer, newly acquired for administrative services.
Those were the days when computers filled an entire room, although their memory capacities were minuscule compared with a modern smartphone. Dr. Short recounted an occasion, as described by Dr. Appel, when the computer gave an unexpected answer.
“Oh, that wire must have fallen out again,” Dr. Appel said.
Dr. Appel began to think of the computer as a partner, though with a different kind of brain, with almost “an artificial intelligence,” he told Dr. MacKenzie.
Shakuntala Devi, 83, ‘Human Computer’
Shakuntala Devi, an Indian mathematical wizard known as “the human computer” for her ability to make incredibly swift calculations, died on Sunday in Bangalore, India. She was 83.
The cause was respiratory and cardiac problems, said D. C. Shivadev, a trustee of the Shakuntala Devi Educational Foundation Public Trust.
Ms. Devi demonstrated her mathematical gifts around the world, at colleges, in theaters and on radio and television. In 1977, at Southern Methodist University in Dallas, she extracted the 23rd root of a 201digit number in 50 seconds, beating a Univac computer, which took 62 seconds.
In 1980, she correctly multiplied two 13digit numbers in only 28 seconds at the Imperial College in London. The feat, which earned her a place in the 1982 edition of the Guinness Book of World Records, was even more remarkable because it included the time to recite the 26digit solution.
(The numbers, selected at random by a computer, were 7,686,369,774,870 and 2,465,099,745,779. The answer was 18,947,668,177,995,426,462,773,730.)
Shakuntala Devi was born in Bangalore on Nov. 4, 1929. Her father was a trapeze artist and lion tamer in a circus. Survivors include a daughter and two grandchildren.
She was about 3 and playing cards with her father when he discovered that she was a mathematical prodigy with an uncanny ability to memorize numbers. By the time she was 5, she had become an expert at solving math problems.
Ms. Devi won fame demonstrating her math skills at the circus, and later in road shows arranged by her father.
“I had become the sole breadwinner of my family, and the responsibility was a huge one for a young child,” she once said. “At the age of 6, I gave my first major show at the University of Mysore, and this was the beginning of my marathon of public performances.”
She toured Europe in 1950. When she appeared on the BBC, her answer to a difficult calculation was different from the interviewer’s. It turned out that she was right. Similarly, at the University of Rome, one of her answers to a problem was found to be wrong, until the experts reexamined their own calculations.
When Ms. Devi performed in New York in 1976, an article in The New York Times marveled at her abilities: “She could give you the cube root of 188,132,517 — or almost any other number — in the time it took to ask the question. If you gave her any date in the last century, she would tell you what day of the week it fell on.”
In a 1990 journal article about Ms. Devi, Arthur R. Jensen, a researcher on human intelligence at the University of California, Berkeley, noted that unlike the Dustin Hoffman character in the movie “Rain Man,” an autistic savant who was also a mathematical prodigy, “Devi comes across as alert, extroverted, affable and articulate.”
He posited that for Ms. Devi, “the manipulation of numbers is apparently like a native language, whereas for most of us arithmetic calculation is at best like the foreign language we learned in school.” But he added that she built on her inherent skills through intense practice as a child.
Ms. Devi was also a successful astrologer, cookbook author and novelist.
This is a more complete version of the story than the one that appeared in print.
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