Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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11. Nine Zulu Queens Ruled China
11. Nine Zulu Queens Ruled China
Pages 169-183
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From page 169... ...
= 0, and so on. That gets us a certain way toward understanding the Riemann Hypothesis, which, just to remind you, says that The Riemann Hypothesis All non-trivial zeros of the zeta function have real part one-half.
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From page 170... ...
There have been determined attempts to reject irrational numbers, even in modern times, and even by important professional mathematicians: Kronecker in the late nineteenth century, Brouwer and Weyl in the early twentieth. For some further remarks on this topic, see Section V in this chapter.
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From page 171... ...
Many others can be expressed with"close(1 forms" like 5~7 + ~ or ~ 2 / 6. Failing ad else, or to give an idea of the actual numerical value of a real number, we write it as a decimal, generally with three trailing dots to mean "this isn't the whole thing, I could supply more (ligits if I really had to:" -549.5393169816448223....
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From page 172... ...
It is an interesting thing about rational numbers that if you write a rational number in decimal form, the decimal digits always repeat themselves sooner or later (unless they just come to a dead stop, like 87 = 0.875~. The rational number 265163, for example, if written as a (lecimal, looks like this: 2.4156088560885608856088....
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From page 173... ...
They are very occasionally referred to collectively as A There is a very important subset of (C caped the aigebraicnumbers, sometimes also given a hollow letter of its own, A
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From page 174... ...
You can't subtract a greater numberfrom a lesser one. As technology developed, this became a stumbling block.
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From page 175... ...
History never ends; as soon as one chess game has been won, another begins immediately. My little bogus history (lees show how the Russian (lolls fit together, though, and I hope it offers some insight into why mathematicians do not regard imaginary and complex numbers as anything
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From page 176... ...
So ~ = ~ x ~ . Now, ~ is, of course, a perfectly ordinary real number, with a value of 1.732050807568877....
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From page 177... ...
If you square an imaginary number, you get a negative real number. Note that this is true even for negative imaginaries.
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From page 178... ...
For example, ~ is a little way east of 1, not quite half way to 2, - Fez is just slightly west of -3, and 1,000,000 is off in the next county somewhere. I can, of course, show only part of the line on a finite sheet of paper.
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From page 179... ...
For the amazing thing is that not only is there an infinity of ~rrationals, and not only are they, too, everywhere dense; but there is a precise mathematical sense in which there are far more irrationals than rationale. This was shown by Georg Cantor in 1874.
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From page 180... ...
is Am~z) .57 Positive real numbers have amplitude zero; negative real numbers have amplitu(le Fez; positive imaginary numbers have amplitude ~z/2; negative imaginaries have amplitude -;z/2.
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From page 181... ...
with its modulus, amplitude, and conjugate. Finally, the complex conjugate of a complex number is its mirror image in the real line.
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From page 182... ...
Analysis in action, an infinite series closing in on its limit. Notice that while we lost the simplicity of one dimension when we moved to complex numbers, we gained some imaginative power.
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From page 183... ...
, laid out as elegant patterns on the complex plane.
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