Zero - Charles Seife [91]
* When Newton was three, his mother remarried and moved. Newton didn’t accompany his mother and stepfather. As a result, he had little contact with his parents after that, unless you count the time he threatened to come over and burn their house down with them inside.
* If you multiply two numbers together and get zero, then one or the other must equal zero. (In mathematical terms, if ab = 0, then a = 0 or b = 0.) This means that if a2 = 0, then aa = 0, thus a = 0.
* Poncelet’s projective geometry brought about one of the oddest concepts in mathematics: the principle of duality. In high school geometry, you are taught that two points determine a line. But if you accept the idea of a point at infinity, two lines always determine a point. Points and lines are dual to each other. Every theorem in Euclidean geometry can be dualized in projective geometry, setting up a whole set of new theorems in the parallel universe of projective geometry.
* One thing that sometimes helps is thinking of the wave function (technically, the square of the wave function) as a measure of the probability about where a particle will be. An electron, say, is smeared out across space, but when you make a measurement to determine where it is, the wave function determines how likely you are to spot the electron at any given point in space. This very smeariness of nature was what Einstein objected to. His famous statement, “God does not play dice with the universe,” was a rejection of the probabilistic way that quantum mechanics works. Unfortunately for Einstein, the laws of quantum mechanics work incredibly well, and you can’t successfully explain quantum effects with traditional classical physics.
* To be precise, the Heisenberg uncertainty principle deals not with a particle’s velocity but with momentum, which combines speed, direction, and information about the particle’s mass. However, in this context, momentum, velocity, and even energy can be used almost interchangeably.
* Yes, mathematics can be “beautiful” or “ugly.” Just as it’s hard to describe what makes a piece of music or a painting aesthetically pleasing, it’s equally difficult to describe what makes a mathematical theorem or a physical theory beautiful. A beautiful theory will be simple, compact, and spare; it will give a sense of completeness and often an eerie sense of symmetry. Einstein’s theories are particularly beautiful, as are Maxwell’s equations. But for many mathematicians, an equation discovered by Euler, ei? + 1 = 0, is the paragon of mathematical beauty, because this extremely simple, compact formula relates all the most important numbers in mathematics in a totally unexpected way.
Table of Contents
Cover
Copyright
Contents
Chapter 0 Null and Void
Chapter 1 Nothing Doing
The Origin of Zero
Chapter 2 Nothing Comes of Nothing
The West Rejects Zero
Chapter 3 Nothing Ventured
Zero Goes East
Chapter 4 The Infinite God of Nothing
The Theology of Zero
Chapter 5 Infinite Zeros and Infidel Mathematicians
Zero and the Scientific Revolution
Chapter 6 Infinity’s Twin
The Infinite Nature of Zero
Chapter 7 Absolute Zeros
The Physics of Zero
Chapter 8 Zero Hour at Ground Zero
Zero at the Edge of Space and Time
Chapter 9 Zero’s Final Victory
End Time
Appendix A Animal, Vegetable, or Minister?
Appendix B The Golden Ratio
Appendix C The Modern Definition of a Derivative
Appendix D Cantor Enumerates the Rational Numbers
Appendix E Make Your Own Wormhole Time Machine
Selected Bibliography
Acknowledgments
Index