This link has been bookmarked by 47 people . It was first bookmarked on 08 Mar 2010, by Robert Talbert.
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Historically, this step was the most painful of all. The square root of –1 still goes by the demeaning name of i, this scarlet letter serving as a constant reminder of its “imaginary” status.
This new kind of number (or if you’d rather be agnostic, call it a symbol, not a number) is defined by the property that
i2 = –1.
It’s true that i can’t be found anywhere on the number line. In that respect it’s much stranger than zero, negative numbers, fractions or even irrational numbers, all of which — weird as they are — still have their place in line.
But with enough imagination, our minds can make room for i as well. It lives off the number line, at right angles to it, on its own imaginary axis. And when you fuse that imaginary axis to the ordinary “real” number line, you create a 2-D space — a plane — where a new species of numbers lives.
These are the “complex numbers.” Here complex doesn’t mean complicated; it means that two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 + 3i.
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f a complex number and the result will still be a complex number.
Better yet, a grand statement called The Fundamental
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rary positive number, say 3, by i. The result is the imaginary number 3i.
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KPI_Library BookmarksSixth article in a series on math by Steven Strogatz in the New York Times
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ecmerrillHubbard was interested in problems with multiple roots. In that case, which root would the method find? He proved that if there were just two roots, the closer one would always win. But if there were three or more roots, he was baffled. His earlier proof no longer applied.
So Hubbard did an experiment. A numerical experiment.
He programmed a computer to run Newton's method, and told it to color-code millions of different starting points according to which root they approached, and to shade them according to how fast they got there.
Before he peeked at the results, he anticipated that the roots would most quickly attract the points nearby, and thus should appear as bright spots in a solid patch of color. But what about the boundaries between the patches? Those he couldn't picture, at least not in his mind's eye.
The computer's answer was astonishing.
The borderlands looked like psychedelic hallucinations. The colors intermingled there in an almost impossibly promiscuous manner, touching each other at infinitely many points, and always in a three-way. In other words, wherever two colors met, the third would always insert itself and join them. -
08 Mar 10
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Dewey RochesterImaginary numbers
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