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Index
- Andrew C. F. Liu, University of Alberta
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- Arithmetical Wonderland
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Contents
- Andrew C. F. Liu, University of Alberta
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7 - Numeration Systems
- Andrew C. F. Liu, University of Alberta
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Summary
“You boys have been swallowing a lot of mathematics lately, haven't you?” Alice said.
“Indeed we have,” said Tweedledum.
“And we have digested most of it,” said Tweedledee. “We only have to take a few indigestion pills now and then. On the whole, I think we have come a long way.”
“Good,” said Alice. “It does take time to ruminate, as the holy cow may tell you. Nevertheless, perhaps it is time for me to give you an examination.”
“Don't!” said the twins together. “We don't like examinations.”
“I understand how you feel,” said Alice sympathetically. “The Red Queen gave me an oral examination in arithmetic once, and I failed.”
“Do you remember any of the questions?” asked Tweedledum.
“There was only one question,” said Alice. “It was kind of long. So here it goes. A stagecoach went from London to Harwich and started out with six passengers. Do you think you can remember that?”
“Of course I can remember that,” replied Tweedledee. “There's not much to remember!”
“Very well,” replied Alice, “the coach made a stop. Two passengers got off and four passengers got on. Got that?”
“Yes,” replied Tweedledum, who was keeping count.
“Then the coach went on and made another stop. Three passengers got off. Are you following?”
“Yes,” said Tweedledee, who was faithfully keeping count.
“Then the coach went on and made another stop. Two passengers got off and two passengers got on.”
“That's the same as if the coach hadn't stopped at all!” exclaimed Tweedledee.
“Anyway,” continued Alice, “the coach went on and made another stop. Three passengers got off and five passengers got on. Are you still keeping count?”
“Yes, I am,” said Tweedledum.
“Then the coach arrived in Harwich. All the passengers got off. How many times did the coach stop?”
“Oh,” swallowed the twins. “We weren't counting that!”
“You'll never be able to pass an examination unless you can count,” said Alice.
“But we can count,” pleaded the twins together. “It's just that we counted the wrong thing!”
“That's no excuse,” replied Alice. “You should always count everything, because everything counts.”
6 - Rational and Irrational Numbers
- Andrew C. F. Liu, University of Alberta
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Summary
Tweedledum and Tweedledee invited Alice for a treat. They put eight tarts on the table. They looked so deliciously inviting that Alice picked one up and started munching right away.
“Where did you boys get these tarts?” asked Alice, suddenly suspicious.
“You don't need to know,” said Tweedledum.
“We bought them from the Duchess's Cook,” said Tweedledee.
“You haven't got the money. You boys stole them from the Queen of Hearts.”
“She has got plenty, and won't miss a few,” said Tweedledum.
“Put them back,” ordered Alice, “or you boys will be in big trouble.”
“We can't,” said Tweedledee. “It wasn't easy getting into the Queen's pantry, and if we go back, we are sure to be caught this time. However, nobody knows anything so far, except you. You are not going to tell on us, are you?”
“Oh dear,” said Alice. “I don't suppose I can now, seeing that I have finished eating one of them already. So I won't tell, but promise me that there won't be a next time.”
“We promise,” the twins said together. “Let us share the tarts among us.”
“Fairly, I suppose,” said Alice.
“Of course,” said Tweedledum.
“If I remember correctly,” said Tweedledee, “we each get two, with two left over.”
“Not this time,” said Alice. “We better not leave any remainder for the Queen to find.”
“What is a fair share then?” asked the twins.
Fractions
If we share eight tarts fairly among three people and leave no remainders, each fair share is more than two tarts but less than three. Let us focus on just one tart at a time. To share it fairly among three people, it must be cut up into three equal pieces, and each person would get one of them. The amount of each share is to be represented by a new number such that the sum of three copies of it is equal to 1. We call this number one-third and write it as ⅓. With eight tarts, each will get 8/3 tarts.
In general, if the divisor is a positive integer n, each fair share amounts to 1/n. This number is called the reciprocal of n. We could also have called it the multiplicative inverse of n, because the product of n and 1/n is equal to the multiplicative identity 1.
3 - Common Divisors and Multiples
- Andrew C. F. Liu, University of Alberta
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Summary
Alice said to the twins, “Instead of passing the day rattling each other's bones, why don't you get some gainful employment? I have just imported some apples and bananas into Wonderland. Each kind of fruit comes in boxes each of which holds exactly the same number, apples or bananas. I do not remember how many there are in each box, but it is more than 1. I do know that there are 289 apples and 221 bananas in the boxes.”
“We will give it a try,” said Tweedledum doubtfully. “I will sell apples and my twin brother will sell bananas.”
“We need to know how many fruit there are in each box,” said Tweedledee.
“Well, see if you can figure that out before I come back,” said Alice. “Perhaps until you do, you should just sell whole boxes of fruit.”
“That is a good idea,” said the twins together.
The Duchess's Cook came in shortly after Alice took off, went to Tweedledum and said, “I see that you have more boxes of apples than Tweedledee has boxes of bananas. I want to buy as many boxes of apples as he has boxes of bananas.”
Tweedledum made the sale, and recorded that now he had 289 − 221 = 68 apples left.
The next three to come were the Dormouse, the March Hare and the Mad Hatter. Each of them bought as many boxes of bananas from Tweedledee as Tweedledum had boxes of apples. Tweedledee recorded that now he had 221 − 68 − 68 − 68 = 17 bananas left.
The next four customers were the Two, the Five, the Seven and the Knave of Hearts. Each bought as many boxes of apples from Tweedledum as Tweedledee had boxes of bananas. Now Tweedledum had no apples left.
Tweedledee said, “I still have 17 bananas left.”
“They are still inside unopened boxes because neither of us had opened one at any time. This means that the number of bananas in each box is a divisor of 17.”
“The only divisors of 17 are 1 and 17,” said Tweedledee. “Since Alice told us the number of bananas in each box is not 1, it must be 17.”
“Let me check,” said Tweedledum. “We have 221 ÷ 17 = 13 and we have 289 ÷ 17 = 17.
So 17 indeed divides both 221 and 289.”
About the Author
- Andrew C. F. Liu, University of Alberta
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Frontmatter
- Andrew C. F. Liu, University of Alberta
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Introduction
- Andrew C. F. Liu, University of Alberta
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Summary
We are deeply saddened by the passing of Martin Gardner on May 22, 2010. It is the end of an era in the world of popular and recreational mathematics. Thus this book is dedicated to his memory.
Martin's legacy in popular and recreational mathematics is legendary, though he modestly insisted that he was not a mathematician but a mathematics journalist. His background was in literature and philosophy, and his most famous book is The Annotated Alice. In honor of Martin, we invited the denizens of Lewis Carroll's Alice in Wonderland and its companion volume Through the Looking Glass to grace the pages of this volume and guide us through our mathematical adventures.
We strongly believe that the primary reason many students are having difficulty with mathematics is that they are bothered by the language of mathematics. By having Alice and the twins, Tweedledum and Tweedledee, serve as the protagonists in this volume, it is hoped that the students may find the conversational style less daunting to read than formal mathematics.
Alice and the twins are featured prominently in Chapter Zero, which is intended as assigned reading. They also introduce us to all subsequent chapters. As the students become more comfortable with the subject, the conversations recede to the background so as not to be overly distracting.
We have taken many excerpts from both books, but also bent them to our purpose. We hope Lewis Carroll would pardon the liberty we had taken, especially in mixing the characters between the two books. We wish to arouse sufficient interest in the students so that they would actually read and enjoy these two books, other works of Lewis Carroll, and classical literature in general.
At the University of Alberta, this book is used as a text for a contents course, offered by the Faculty of Science, for students in the elementary education program. These students also take Curriculum and Instruction courses from the Faculty of Education. Although the book has these students in mind all the time, it does not offer much pedagogical advice. We leave this to the experts.
Even as a text for a contents course, the book is still somewhat idiosyncratic and unorthodox. Many instructors may see it more as a classroom resource. Also, because its entry point is set at the ground level, the book is very suitable for self-study, life-long learning as well as liberal arts education.
Appendix - A Legacy of Martin Gardner
- Andrew C. F. Liu, University of Alberta
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Summary
Martin Gardner is famous, among other things, for his “Mathematical Games” columns in “Scientific American”, which ran for a quarter of a century. The columns can be read independently. They cover a great variety of topics. While he professed not to have been a mathematician, he did have a fine sense for what is beautiful in the subject. His writing is most inspiring, and opens up many avenues for readers, both old and young.
Later, the columns were anthologized and became chapters, in fifteen volumes. Unfortunately, since these books were put out by several publishers, they varied greatly in format. The Mathematical Association of America has packaged all of them into a compact disc, but maintaining the original format.
A new edition, published jointly by the Association and Cambridge University Press, will have uniform format and updated material. Three volumes were edited by Martin himself before his passing in 2010, but other volumes will continue to appear. Here are the titles.
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi.
Origami, Eleusis, and the Soma Cube.
Sphere Packing, Lewis Carroll, and Reversi.
Knots and Borromean Rings, Rep-Tiles, and Eight Queens.
Klein Bottles, Op-Art, and Sliding-Block Puzzles.
Sprouts, Hypercubes, and Superellipses.
Nothing and Everything, Polyominoes, and Game Theory.
Random Walks, Hyperspheres, and Palindromes.
Words, Numbers, and Combinatorics.
Wheels, Life, and Knotted Molecules.
Knotted Doughnuts, Napier Bones, and Gray Codes.
Tangrams, Tilings, and Time Travel.
Penrose Tiles, Trapdoor Ciphers, and the Oulipo.
Fractal Music, Hypercards, and Chaitin's Omega.
Hydras, Eggs and Other Mathematical Mystifications.
Solution to Odd-numbered Exercises
- Andrew C. F. Liu, University of Alberta
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4 - Linear Diophantine Equations
- Andrew C. F. Liu, University of Alberta
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Summary
“I had a very strange dream last night,” Tweedledum said to Tweedledee. “I dreamt that we were not twins but quintuplets.”
“What were the names of the others?” asked Tweedledee.
“One of them was called Tweedledoo. I don't remember the other two, but they were also Tweedle-something. We had done something that was only possible in a dream. We made the Queen of Hearts happy. She rewarded us with enough tarts to fill the entire pantry. It was still daytime then, and we were to share the tarts equally in the evening.”
“I wish I have dreams like that, even if it is only a dream. What happened next?”
“The pantry was guarded by the Duchess's Cook. Sometime during the day, she suggested to me that I should make sure that I got my fair share. She took me inside, and helped me divide the tarts into five equal piles. There was one left over. I gave that to her while I ate my pile. Then we put the rest of the tarts back together. During the course of the day, I noticed that you and the other three went into the pantry one at a time with the Duchess's Cook, and each time, she came out eating a tart.”
“So the Duchess's crook made the same crooked deal with all of us.”
“By the time we divided the tarts in the evening, there were a lot less than before. Nobody spoke up. So I assume that each of us was as guilty as the others. The tarts now came out in five equal shares.”
“How many tarts were there altogether?”
“I was too stuffed to count. Let us ask Alice and see if she can figure it out.”
“This problem belongs to the type called Diophantine problems,” said Alice, “named after the Greek mathematician Diophantus. Such problems lead to what are called Diophantine equations. They look just like ordinary algebraic equations, except that only integral solutions are accepted.”
“This is not unreasonable,” said Tweedledum. “The answers to most problems are supposed to be positive integers.”
“So how many tarts were there in my twin brother's dream?”
Bézoutian Algorithm
An important result in the last chapter is the following.
5 - Prime Factorizations
- Andrew C. F. Liu, University of Alberta
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Summary
Tweedledum told Alice, “The Queen of Hearts had put the Ace to Ten of Hearts in jail, ten days ago.”
“What had they done?” exclaimed Alice.
“Probably nothing,” said Tweedledee. “With Her Majesty, anything you do may be an offence. You are lucky if you still keep your head.”
“The Knave of Hearts had made an appeal to the King of Hearts, pleading the innocence of the prisoners,” said Tweedledum.
“What could His Majesty do?”
“Nothing that Her Majesty would not allow,” said Tweedledee.
“Ah!” said Tweedledum. “His Majesty had decided to be the Devil just this once, and unlocked all ten prison cells.”
“Hooray for His Majesty!” said Alice.
“His Majesty may live to regret this,” said Tweedledee, “or may not live to regret this.”
“His Majesty realized that too. So the next day, he went back to the jail. Starting with the second one, he relocked every second cell.”
“Were the Ace to Ten of Hearts placed in order in the cells?” asked Alice.
“Yes,” said Tweedledum.
“So now the Two, Four, Six, Eight and Ten of Hearts were locked up while the Ace, Three, Five, Seven and Nine were free,” said Tweedledee.
“Yes. On the third day, His Majesty had another change of mind. He started with the third cell, and altered the lock of every third cell.”
“What exactly did he do?” asked Alice.
“His Majesty would lock a cell if it was open, but unlock it otherwise. So now the Three and Nine of Hearts were locked up, but the cell of the Six of Hearts was unlocked again.”
“Knowing His Majesty,” said Tweedledee, “I bet he would continue in this manner for the next seven days.”
“Yes,” said Tweedledum. “Thus on the fourth day, he altered the lock of the fourth and the eighth cells, and today, he just altered the lock of the tenth cell. The latest news is that the Queen of Hearts has granted the prisoners a partial pardon, so that those whose cells are unlocked now are set free.”
“Who are the lucky prisoners?” asked Alice.
1 - Divisibility
- Andrew C. F. Liu, University of Alberta
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Summary
Alice asked Tweedledum and Tweedledee what 6 divided by 3 was. Both twins said 2, but Alice said, “Wrong!”
They stared at Alice as though she had been munching mushrooms again. After a while, Tweedledum asked timidly, “What is the right answer then?”
Alice took out six tarts she bought from the Queen of Hearts. She said, “There are six tarts and three of us. If I shared them with you, that is 6 divided by 3, isn't it?”
“Yes,” the twins nodded eagerly, licking their lips.
“The correct answer,” Alice said, “is that I get four tarts and each of you get one.”
“But that ain't fair!” they exclaimed.
“I say it is fair, because they are my tarts, and I am a lot bigger than you are. Anyway, what is 1 plus 1 plus 4?”
“That is 6, isn't it?” asked Tweedledee.
“Of course it is. Now is 1 + 1 + 4 = 6 fair?”
They had never thought about that. After a while, Tweedledum said, “Well, fairness doesn't come into addition.”
“Does it come into subtraction or multiplication?”
“No,” said Tweedledee. “So division is somewhat special, isn't it?”
“Indeed it is. There is a tacit assumption that it is supposed to be fair. Then 6 divided by 3 must be equal to 2. Even better news for you, 6 divided by 2 is equal to 3.”
“Hooray!” they exclaimed, and three tarts disappeared down each chute in an instant.
Basic Properties of Divisibility
The binary operation of division is defined as follows. Let a and b be positive integers. Then their quotient b ÷ a is a positive integer c such that b = ac.
The set of positive integers is not closed under division. For instance, 6 ÷ 4 is not a positive integer. A remedy is to expand our set so that it will include more than just the positive integers. This will be done in Chapter 6. Another way is to allow for remainders and stay within the set of positive integers. This will be our approach in Chapter 2. In this chapter, we consider only divisions that are exact, that is, the quotient is a positive integer.
![](http://static.cambridge.org/content/id/urn:cambridge.org:id:book:9781614441199/resource/name/9781614441199i.jpg)
Arithmetical Wonderland
- Andrew C. F. Liu
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- Published by:
- Mathematical Association of America
- Published online:
- 27 April 2017
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Arithmetical Wonderland is intended as an unorthodox mathematics textbook for students in elementary education, in a contents course offered by a mathematics department. The scope is deliberately restricted to cover only arithmetic, even though geometric elements are introduced whenever warranted. For example, what the Euclidean Algorithm for finding the greatest common divisors of two numbers has to do with Euclid is showcased.
Many students find mathematics somewhat daunting. It is the author's belief that much of that is caused not by the subject itself, but by the language of mathematics. In this book, much of the discussion is in dialogues between Alice, of Wonderland fame, and the twins Tweedledum and Tweedledee who hailed from Through the Looking Glass. The boys are learning High Arithmetic or Elementary Number Theory from Alice, and the reader is carried along in this academic exploration. Thus many formal proofs are converted to soothing everyday language.
Nevertheless, the book has considerable depth. It examines many arcane corners of the subject, and raises rather unorthodox questions. For instance, Alice tells the twins that six divided by three is two only because of an implicit assumption that division is supposed to be fair, whereas fairness does not come into addition, subtraction or multiplication. Some topics often not covered are introduced rather early, such as the concepts of divisibility and congruence.
2 - Congruence
- Andrew C. F. Liu, University of Alberta
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0 - Review of Arithmetic
- Andrew C. F. Liu, University of Alberta
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Summary
“How much basic education do you boys have?” Alice asked the twins.
“Not much,” said Tweedledum.
“Just the three Rs,” said Tweedledee, “Reeling, Writhing and Arithmetic.”
“So you have had some physical education,” said Alice, looking at the boys’ rotund bodies, “but you are not in shape.”
“We are in shape,” the twins protested together, “even though you may not like the shape we are in.”
“Oh dear,” said Alice, “it makes me wonder about how much arithmetic you have studied.”
“Just the four basic operations,” said Tweedledum.
Tweedledee elaborated, “Ambition, Distraction, Uglification and Derision.”
“What in the name of heaven are these?” exclaimed Alice. “You boys must start all over again. I will teach you some proper arithmetic.”
“We are listening,” said the twins.
Counting Numbers
“Let us start with the very basic. Can you boys count?”
“Of course,” said Tweedledum.
Tweedledee said, “Zero, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, and so on.”
Alice said, “Very good. These are the counting numbers. We use the Hindu-Arabic symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and so on.”
“Why is each of ten, eleven and twelve represented by two symbols while each of the others is represented by only one symbol?” asked Tweedledum.
“A very good question,” said Alice. “We cannot have a single symbol for each number, as otherwise we will need infinitely many symbols to represent all the numbers. So we settle for a finite number of symbols, and represent larger numbers with more of the same symbols. We call a single symbol a digit. How many different digits are there?”
“Ten,” said Tweedledee, “namely, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.”
“Very good,” said Alice. “We probably settle for ten digits because we have ten fingers that we often use for counting. The word digit does mean fingers, or toes.”
“Why is ten represented by 10?” asked Tweedledum. “Ten is not equal to one plus zero.”
Contributors
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. 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Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
- Edited by Robert Audi, University of Notre Dame, Indiana
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- Book:
- The Cambridge Dictionary of Philosophy
- Published online:
- 05 August 2015
- Print publication:
- 27 April 2015, pp ix-xxx
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Contributors
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- By Ted Abel, Antoine Adamantidis, Karla V. Allebrandt, Simon N. Archer, Amelie Baud, Michel Billiard, Carlos Blanco-Centurion, Diane B. Boivin, Ethan Buhr, Matthew E. Carter, Nicolas Cermakian, Jennifer H.K. Choi, S.Y. Christin Chong, Chiara Cirelli, Marc Cuesta, Thomas Curie, Yves Dauvilliers, Luis de Lecea, Derk-Jan Dijk, Stephane Dissel, Annette C. Fedson, Jonathan Flint, Marcos G. Frank, Paul Franken, Ying-Hui Fu, Thorarinn Gislason, David Gozal, Devon A. Grant, Hakon Hakonarson, Makoto Honda, Hyun Hor, Christer Hublin, Peng Jiang, Takashi Kanbayashi, Jaakko Kaprio, Andrew Kasarskis, Leila Kheirandish-Gozal, RodaRani Konadhode, Michael Lazarus, Meng Liu, Michael March, Mark F. Mehler, Keivan Kaveh Moghadam, Valérie Mongrain, Charles M. Morin, Benjamin M. Neale, Seiji Nishino, Allan I. Pack, Dheeraj Pelluru, Rosa Peraita-Adrados, Giuseppe Plazzi, David A. Prober, Louis J. Ptáček, Irfan A. Qureshi, David M. Raizen, John J. Renger, Till Roenneberg, Elizabeth J. Rossin, Takeshi Sakurai, Paul Salin, Karen D. Schilli, Eva C. Schulte, Laurent Seugnet, Paul J. Shaw, Priyattam J. Shiromani, Patrick Sleiman, Mehdi Tafti, Joseph S. Takahashi, Matthew S. Thimgan, Katsushi Tokunaga, Giulio Tononi, Fred W. Turek, Yoshihiro Urade, Hans P.A. Van Dongen, Juliane Winkelmann, Christopher J. Winrow
- Edited by Paul Shaw, Mehdi Tafti, Michael J. Thorpy
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- The Genetic Basis of Sleep and Sleep Disorders
- Published online:
- 05 November 2013
- Print publication:
- 24 October 2013, pp xi-xiv
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Genome-Wide Association Study of Height and Body Mass Index in Australian Twin Families
- Jimmy Z. Liu, Sarah E. Medland, Margaret J. Wright, Anjali K. Henders, Andrew C. Heath, Pamela A. F. Madden, Alexis Duncan, Grant W. Montgomery, Nicholas G. Martin, Allan F. McRae
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- Journal:
- Twin Research and Human Genetics / Volume 13 / Issue 2 / 01 April 2010
- Published online by Cambridge University Press:
- 21 February 2012, pp. 179-193
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Human height and body mass index are influenced by a large number of genes, each with small effects, along with environment. To identify common genetic variants associated with these traits, we performed genome-wide association studies in 11,536 individuals composed of Australian twins, family members, and unrelated individuals at ∼550,000 genotyped SNPs. We identified a single genome-wide significant variant for height (P value = 1.06 × 10–9) located in HHIP, a well-replicated height-associated gene. Suggestive levels of association were found for other known genes associated with height (P values < 1 × 10–6): ADAMTSL3, EFEMP1, GPR126, and HMGA2; and BMI (P values < 1 × 10–4): FTO and MC4R. Together, these variants explain less than 2% of total phenotypic variation for height and 0.5% for BMI.
Contributors
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- By Waiel Almoustadi, Brian J. Anderson, David B. Auyong, Michael Avidan, Michael J. Avram, Roland J. Bainton, Jeffrey R. Balser, Juliana Barr, W. Scott Beattie, Manfred Blobner, T. Andrew Bowdle, Walter A. Boyle, Eugene B. Campbell, Laura F. Cavallone, Mario Cibelli, C. Michael Crowder, Ola Dale, M. Frances Davies, Mark Dershwitz, George Despotis, Clifford S. Deutschman, Brian S. Donahue, Marcel E. Durieux, Thomas J. Ebert, Talmage D. Egan, Helge Eilers, E. Wesley Ely, Charles W. Emala, Alex S. Evers, Heidrun Fink, Pierre Foëx, Stuart A. Forman, Helen F. Galley, Josephine M. Garcia-Ferrer, Robert W. Gereau, Tony Gin, David Glick, B. Joseph Guglielmo, Dhanesh K. Gupta, Howard B. Gutstein, Robert G. Hahn, Greg B. Hammer, Brian P. Head, Helen Higham, Laureen Hill, Kirk Hogan, Charles W. Hogue, Christopher G. Hughes, Eric Jacobsohn, Roger A. Johns, Dean R. Jones, Max Kelz, Evan D. Kharasch, Ellen W. King, W. Andrew Kofke, Tom C. Krejcie, Richard M. Langford, H. T. Lee, Isobel Lever, Jerrold H. Levy, J. Lance Lichtor, Larry Lindenbaum, Hung Pin Liu, Geoff Lockwood, Alex Macario, Conan MacDougall, M. B. MacIver, Aman Mahajan, Nándor Marczin, J. A. Jeevendra Martyn, George A. Mashour, Mervyn Maze, Thomas McDowell, Stuart McGrane, Berend Mets, Patrick Meybohm, Charles F. Minto, Jonathan Moss, Mohamed Naguib, Istvan Nagy, Nick Oliver, Paul S. Pagel, Pratik P. Pandharipande, Piyush Patel, Andrew J. Patterson, Robert A. Pearce, Ronald G. Pearl, Misha Perouansky, Kristof Racz, Chinniampalayam Rajamohan, Nilesh Randive, Imre Redai, Stephen Robinson, Richard W. Rosenquist, Carl E. Rosow, Uwe Rudolph, Francis V. Salinas, Robert D. Sanders, Sunita Sastry, Michael Schäfer, Jens Scholz, Thomas W. Schnider, Mark A. Schumacher, John W. Sear, Frédérique S. Servin, Jeffrey H. Silverstein, Tom De Smet, Martin Smith, Joe Henry Steinbach, Markus Steinfath, David F. Stowe, Gary R. Strichartz, Michel M. R. F. Struys, Isao Tsuneyoshi, Robert A. Veselis, Arthur Wallace, Robert P. Walt, David C. Warltier, Nigel R. Webster, Jeanine Wiener-Kronish, Troy Wildes, Paul Wischmeyer, Ling-Gang Wu, Stephen Yang
- Edited by Alex S. Evers, Washington University School of Medicine, St Louis, Mervyn Maze, University of California, San Francisco, Evan D. Kharasch, Washington University School of Medicine, St Louis
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- Anesthetic Pharmacology
- Published online:
- 11 April 2011
- Print publication:
- 10 March 2011, pp viii-xiv
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