"A Passion for Mathematics" is an educational, entertaining trip through the curiosities of the math world, blending an eclectic mix of history, biography, philosophy, number theory, geometry, probability, huge numbers, and mind-bending problems into a delightfully compelling collection that is sure to please math buffs, students, and experienced mathematicians alike. In each chapter, Clifford Pickover provides factoids, anecdotes, definitions, quotations, and captivating challenges that range from fun, quirky puzzles to insanely difficult problems. Readers will encounter mad mathematicians, strange number sequences, obstinate numbers, curious constants, magic squares, fractal geese, monkeys typing Hamlet, infinity, and much, much more. "A Passion for Mathematics" will feed readers' fascination while giving them problem-solving skills a great workout!
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About the Author
CLIFFORD A. PICKOVER received his Ph.D. from Yale University and is a prolific author of books and articles, including Calculus and Pizza: A Cookbook for the Hungry Mind; Black Holes: A Traveler's Guide; and Keys to Infinity, all from Wiley. His Web site, Pickover.com, has received millions of visits.
Read an Excerpt
A Passion for Mathematics
By Clifford A. Pickover
John Wiley & SonsISBN: 0-471-69098-8
Chapter OneNumbers, History, Society, and People
In which we encounter religious mathematicians, mad mathematicians, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and "What if?" questions.
(?) Ancient counting. Let's start the book with a question. What is the earliest evidence we have of humans counting? If this question is too difficult, can you guess whether the evidence is before or after 10,000 B.C.-and what the evidence might be? (See Answer 1.1.)
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(") Mathematics and beauty. I've collected mathematical quotations since my teenage years. Here's a favorite: "Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austere, like that of sculpture" (Bertrand Russell, Mysticism and Logic, 1918).
(!) The symbols of mathematics. Mathematical notation shapes humanity's ability to efficiently contemplate mathematics. Here's a cool factoid for you: The symbols + and -, referring to addition and subtraction, first appeared in 1456 in an unpublished manuscript by the mathematician Johann Regiomontanus (a.k.a. Johann Muller). The plus symbol, as an abbreviation for the Latin et (and), was found earlier in a manuscriptdated 1417; however, the downward stroke was not quite vertical.
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(') Mathematics and reality. Do humans invent mathematics or discover mathematics? (See Answer 1.2.)
(") Math beyond humanity. "We now know that there exist true propositions which we can never formally prove. What about propositions whose proofs require arguments beyond our capabilities? What about propositions whose proofs require millions of pages? Or a million, million pages? Are there proofs that are possible, but beyond us?" (Calvin Clawson, Mathematical Mysteries).
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(!) The multiplication symbol. In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574-1660) in his book Keys to Mathematics, published in London. Incidentally, this Anglican minister is also famous for having invented the slide rule, which was used by generations of scientists and mathematicians. The slide rule's doom in the mid-1970s, due to the pervasive influx of inexpensive pocket calculators, was rapid and unexpected.
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(!) Mathematics and the universe. Here is a deep thought to start our mathematical journey. Do you think humanity's long-term fascination with mathematics has arisen because the universe is constructed from a mathematical fabric? We'll approach this question later in the chapter. For now, you may enjoy knowing that in 1623, Galileo Galilei echoed this belief in a mathematical universe by stating his credo: "Nature's great book is written in mathematical symbols." Plato's doctrine was that God is a geometer, and Sir James Jeans believed that God experimented with arithmetic. Isaac Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits.
(!) Math and madness. Many mathematicians throughout history have had a trace of madness or have been eccentric. Here's a relevant quotation on the subject by the British mathematician John Edensor Littlewood (1885-1977), who suffered from depression for most of his life: "Mathematics is a dangerous profession; an appreciable proportion of us goes mad."
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(?) Mathematics and murder. What triple murderer was also a brilliant French mathematician who did his finest work while confined to a hospital for the criminally insane? (See Answer 1.3.)
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(") Creativity and madness. "There is a theory that creativity arises when individuals are out of sync with their environment. To put it simply, people who fit in with their communities have insufficient motivation to risk their psyches in creating something truly new, while those who are out of sync are driven by the constant need to prove their worth. They have less to lose and more to gain" (Gary Taubes, "Beyond the Soapsuds Universe," 1977).
(!) Pascal's mystery. "There is a God-shaped vacuum in every heart" (Blaise Pascal, Pensees, 1670).
(!) Mathematicians and religion. Over the years, many of my readers have assumed that famous mathematicians are not religious. In actuality, a number of important mathematicians were quite religious. As an interesting exercise, I conducted an Internet survey in which I asked respondents to name important mathematicians who were also religious. Isaac Newton and Blaise Pascal were the most commonly cited religious mathematicians.
In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics struggle to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, as well as impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek "ideal," immutable, nonmaterial truths and then often venture to apply these truths in the real world. Are mathematics and religion the most powerful evidence of the inventive genius of the human race? In "Reason and Faith, Eternally Bound" (December 20, 2003, New York Times, B7), Edward Rothstein notes that faith was the inspiration for Newton and Kepler, as well as for numerous scientific and mathematical triumphs. "The conviction that there is an order to things, that the mind can comprehend that order and that this order is not infinitely malleable, those scientific beliefs may include elements of faith."
In his Critique of Pure Reason, Immanuel Kant describes how "the light dove, cleaving the air in her free flight and feeling its resistance against her wings, might imagine that its flight would be freer still in empty space." But if we were to remove the air, the bird would plummet. Is faith-or a cosmic sense of mystery-like the air that allows some seekers to soar? Whatever mathematical or scientific advances humans make, we will always continue to swim in a sea of mystery.
(?) Leaving mathematics and approaching God. What famous French mathematician and teenage prodigy finally decided that religion was more to his liking and joined his sister in her convent, where he gave up mathematics and social life? (See Answer 1.4.)
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(!) Ramanujan's gods. As mentioned in this book's introduction, the mathematician Srinivasa Ramanujan (1887-1920) was an ardent follower of several Hindu deities. After receiving visions from these gods in the form of blood droplets, Ramanujan saw scrolls that contained very complicated mathematics. When he woke from his dreams, he set down on paper only a fraction of what the gods showed him.
Throughout history, creative geniuses have been open to dreams as a source of inspiration. Paul McCartney said that the melody for the famous Beatles' song "Yesterday," one of the most popular songs ever written, came to him in a dream. Apparently, the tune seemed so beautiful and haunting that for a while he was not certain it was original. The Danish physicist Niels Bohr conceived the model of an atom from a dream. Elias Howe received in a dream the image of the kind of needle design required for a lock-stitch sewing machine. Rene Descartes was able to advance his geometrical methods after flashes of insight that came in dreams. The dreams of Dmitry Mendeleyev, Friedrich August Kekule, and Otto Loewi inspired scientific breakthroughs. It is not an exaggeration to suggest that many scientific and mathematical advances arose from the stuff of dreams.
(!) Blaise Pascal (1623-1662), a Frenchman, was a geometer, a probabilist, a physicist, a philosopher, and a combinatorist. He was also deeply spiritual and a leader of the Jansenist sect, a Calvinistic quasi-Protestant group within the Catholic Church. He believed that it made sense to become a Christian. If the person dies, and there is no God, the person loses nothing. If there is a God, then the person has gained heaven, while skeptics lose everything in hell.
Legend has it that Pascal in his early childhood sought to prove the existence of God. Because Pascal could not simply command God to show Himself, he tried to prove the existence of a devil so that he could then infer the existence of God. He drew a pentagram on the ground, but the exercise scared him, and he ran away. Pascal said that this experience made him certain of God's existence.
One evening in 1654, he had a two-hour mystical vision that he called a "night of fire," in which he experienced fire and "the God of Abraham, Isaac, and Jacob ... and of Jesus Christ." Pascal recorded his vision in his work "Memorial." A scrap of paper containing the "Memorial" was found in the lining of his coat after his death, for he carried this reminder about with him always. The three lines of "Memorial" are
Complete submission to Jesus Christ and to my director. Eternally in joy for a day's exercise on the earth. May I not forget your words. Amen.
(") Transcendence. "Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature" (Freeman Dyson, in the introduction to Nature's Imagination).
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(?) The value of eccentricity. "That so few now dare to be eccentric, marks the chief danger of our time" (John Stuart Mill, nineteenthcentury English philosopher).
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(?) Counting and the mind. I quickly toss a number of marbles onto a pillow. You may stare at them for an instant to determine how many marbles are on the pillow. Obviously, if I were to toss just two marbles, you could easily determine that two marbles sit on the pillow. What is the largest number of marbles you can quantify, at a glance, without having to individually count them? (See Answer 1.5.)
* * *
(?) Circles. Why are there 360 degrees in a circle? (See Answer 1.6.)
* * *
(") The mystery of Ramanujan. After years of working through Ramanujan's notebooks, the mathematician Bruce Berndt said, "I still don't understand it all. I may be able to prove it, but I don't know where it comes from and where it fits into the rest of mathematics. The enigma of Ramanujan's creative process is still covered by a curtain that has barely been drawn" (Robert Kanigel, The Man Who Knew Infinity, 1991).
(?) Calculating [pi]. Which nineteenth-century British boarding school supervisor spent a significant portion of his life calculating [pi] to 707 places and died a happy man, despite a sad error that was later found in his calculations? (See Answer 1.8.)
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(!) The special number 7. In ancient days, the number 7 was thought of as just another way to signify "many." Even in recent times, there have been tribes that used no numbers higher than 7.
In the 1880s, the German ethnologist Karl von Steinen described how certain South American Indian tribes had very few words for numbers. As a test, he repeatedly asked them to count ten grains of corn. They counted "slowly but correctly to six, but when it came to the seventh grain and the eighth, they grew tense and uneasy, at first yawning and complaining of a headache, then finally avoided the question altogether or simply walked off." Perhaps seven means "many" in such common phrases as "seven seas" and "seven deadly sins." (These interesting facts come from Adrian Room, The Guinness Book of Numbers, 1989.)
(?) The world's most forgettable license plate? Today, mathematics affects society in the funniest of ways. I once read an article about someone who claimed to have devised the most forgettable license plate, but the article did not divulge the secret sequence. What is the most forgettable license plate? Is it a random sequence of eight letters and numbers-for example, 6AZL4QO9 (the maximum allowed in New York)? Or perhaps a set of visually confusing numbers or letters-for example, MWNNMWWM? Or maybe a binary number like 01001100. What do you think? What would a mathematician think? (See Answer 1.7.)
* * *
(!) Carl Friedrich Gauss (1777-1855), a German, was a mathematician, an astronomer, and a physicist with a wide range of contributions. Like Ramanujan, after Gauss proved a theorem, he sometimes said that the insight did not come from "painful effort but, so to speak, by the grace of God." He also once wrote, "There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example, touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science."
(") Genius and eccentricity. "The amount of eccentricity in a society has been proportional to the amount of genius, material vigor and moral courage which it contains" (John Stuart Mill, On Liberty, 1869).
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(") Mathematics and God. "The Christians know that the mathematical principles, according to which the corporeal world was to be created, are co-eternal with God. Geometry has supplied God with the models for the creation of the world. Within the image of God it has passed into man, and was certainly not received within through the eyes" (Johannes Kepler, The Harmony of the World, 1619).
(!) James Hopwood Jeans (1877-1946) was an applied mathematician, a physicist, and an astronomer. He sometimes likened God to a mathematician and wrote in The Mysterious Universe (1930), "From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician." He has also written, "Physics tries to discover the pattern of events which controls the phenomena we observe. But we can never know what this pattern means or how it originates; and even if some superior intelligence were to tell us, we should find the explanation unintelligible" (Physics and Philosophy, 1942).
* * *
(!) Isaac Newton (1642-1727), an Englishman, was a mathematician, a physicist, an astronomer, a coinventor of calculus, and famous for his law of gravitation. He was also the author of many books on biblical subjects, especially prophecy.
Perhaps less well known is the fact that Newton was a creationist who wanted to be known as much for his theological writings as for his scientific and mathematical texts. Newton believed in a Christian unity, as opposed to a trinity. He developed calculus as a means of describing motion, and perhaps for understanding the nature of God through a clearer understanding of nature and reality. He respected the Bible and accepted its account of Creation.
(!) Leonhard Euler (1707-1783) was a prolific Swiss mathematician and the son of a vicar. Legends tell of Leonhard Euler's distress at being unable to mathematically prove the existence of God. Many mathematicians of his time considered mathematics a tool to decipher God's design and codes. Although he was a devout Christian all his life, he could not find the enthusiasm for the study of theology, compared to that of mathematics. He was completely blind for the last seventeen years of his life, during which time he produced roughly half of his total output.
Euler is responsible for our common, modern-day use of many famous mathematical notations-for example, f(x) for a function, e for the base of natural logs, i for the square root of -1, [pi] for pi, [summation] for summation. He tested Pierre de Fermat's conjecture that numbers of the form [2.sup.n] + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8, and 16, and showed that the next case [2.sup.32] + 1 = 4,294,967,297 = 641 x 6,700,417, and so is not prime.
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Table of ContentsAcknowledgments.
1. Numbers, History, Society, and People.
2. Cool Numbers.
3. Algebra, Percentages, Weird Puzzles, and Marvelous Mathematical Manipulations.
4. Geometry, Games, and Beyond.
5. Probability: Take Your Chances.
6. Big Numbers and Infinity.
7. Mathematics and Beauty.
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