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“I tell you, with complex numbers you can do anything.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“Published mathematical papers often have irritating assertions of the type: “It now follows thatâ€�,â€� or: “It is now obvious thatâ€�,â€� when it doesn't follow, and isn't obvious at all, unless you put in the six hours the author did to supply the missing steps and checking them. There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious thatâ€�.â€� Here he stopped, fell silent, and stood motionless with furrowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious thatâ€�.â€� If he”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“Mathematicians call it “the arithmetic of congruences.â€� You can think of it as clock arithmetic. Temporarily replace the 12 on a clock face with 0. The 12 hours of the clock now read 0, 1, 2, 3, â€� up to 11. If the time is eight o’clock, and you add 9 hours, what do you get? Well, you get five o’clock. So in this arithmetic, 8 + 9 = 5; or, as mathematicians say, 8 + 9 â‰� 5 (mod 12), pronounced “eight plus nine is congruent to five, modulo twelve.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“(which has inspired at least one novel, Apostolos Doxiadis's Uncle Petros and Goldbach's Conjecture29).”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“It was in 1742 that Christian Goldbach put forward his famous conjecture that every even number greater than 2 can be expressed as the sum of two primes.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“The rules of evidence can deliver very persuasive results, sometimes contrary to the strictly argued certainties of mathematics. […] Hypothesis: No human can possibly be more than nine feet tall. Confirming instance: A human being who is 8'11¾" tall. The discovery of that person confirms the hypothesis â€� but at the same time casts a long shadow of doubt across it!”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“If Weierstrass is a rock climber, inching his way methodically up the cliff face, Riemann is a trapeze artist, launching himself boldly into space in the confidence—which to the observer often seems dangerously misplaced—that when he arrives at his destination in the middle of the sky, there will be something there for him to grab. It is plain that Riemann had a strongly visual imagination, and also that his mind leaped to results so powerful, elegant, and fruitful that he could not always force himself to pause to prove them. He was keenly interested in philosophy and physics, and notions gathered from long, deep contemplation of those two disciplines—the flow of sensations through our senses, the organizing of those sensations into forms and concepts, the flow of electricity through a conductor, the movements of liquids and gasesâ€� can be glimpsed beneath the surface of his mathematics.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“Bernhard Riemann was a very pure case of the intuitive mathematician. This needs some explaining. The mathematical personality has two large components, the logical and the intuitive. Both are present in any good mathematician, but often one or the other is strongly dominant.”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

“Russell”
― John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics