Charles Seife - Zero: The Biography of a Dangerous Idea

* The Greek word for ratio was (logos), which is also the term for word. This translation is even more rational than the traditional one.

* 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 cant successfully explain quantum effects with traditional classical physics.

* To be precise, the Heisenberg uncertainty principle deals not with a particles velocity but with momentum, which combines speed, direction, and information about the particles mass. However, in this context, momentum, velocity, and even energy can be used almost interchangeably.

* Yes, mathematics can be beautiful or ugly. Just as its hard to describe what makes a piece of music or a painting aesthetically pleasing, its 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. Einsteins theories are particularly beautiful, as are Maxwells equations. But for many mathematicians, an equation discovered by Euler, e i? + 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.

* The early Babylonians were apparently unaware of the difficulty in trisecting an angle. In the Epic of Gilgamesh, the narrator states that Gilgamesh was two-thirds god and one-third man. This is as impossible as trisecting an angle with a straightedge and compassesunless gods and mortals are allowed to have an infinite amount of sex.

* This is a necessary, but not sufficient, condition. If the terms go to zero too slowly, then the sum of the terms doesnt converge to a finite number.

* One dating system had the year 1 based upon the founding of the city of Rome, and the other was based on the accession of the emperor Diocletian. To the Christian monk, the birth of his Savior was a more important event than the foundation of a city that had been sacked by Vandals and Goths a few timesor, for that matter, the beginning of the reign of an emperor who had an unfortunate penchant for maintaining his menagerie of exotic animals on a diet of Christians.

* When a computer programmer makes a program do something over and over, hell more than likely make the computer count from, say, zero to nine to make the computer take ten steps. A forgetful programmer might make it count from one to nine, yielding only nine steps instead of ten. More than likely a bug like this was what ruined an Arizona lottery in 1998. In drawing after drawing, a nine never appeared. They hadnt programmed it in, admitted a spokeswoman sheepishly.

* Tally sticks caused no end of trouble. The English Exchequer used to keep accounts on a variant of the tally stick until 1826. Charles Dickens told of the outcome of that long-outdated practice: In 1834, it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However, they never had been useful, and official routine required that they should never be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost thereof.

* When Newton was three, his mother remarried and moved. Newton didnt 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 a 2 = 0, then aa = 0, thus a = 0.

* Poncelets 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.