The Babylonians asked how to find two numbers if you know their sum and their product. Their product they saw as the area of a rectangle.
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And their sum is half the perimeter of that same rectangle. But this is geometrically rather unnatural. The Babylonians found a better way.
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They took two copies of the rectangle, rotated one by 90 degrees, and juxtaposed them to make a shape whose base is the sum of the sides.
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But this shape is unappealingly asymmetrical. So they made a second copy of it, rotated it by 180 degrees, and assembled both into a square.
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Or rather, into a square with a square hole at its center. The whole now consisting of four copies of the original rectangle.
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The area of this new almost-square is 4 times the product of the two unknown numbers. Its side length is their sum.
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The crucial final point is that the side length of the missing square at its center is the difference between the two unknown numbers.
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The sum of these unknown numbers is known, so its square is as well. And this is 4 times their product, plus the square of their difference.
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So the square of their difference is known, and the difference itself is the square root of its square -- up to an ambiguous overall sign.
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And once you know the sum and the difference of two numbers, the numbers themselves are easy to determine.
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The great thing about this approach is that it generalizes. One "completes the cube" in essentially the same way, to solve cubic equations.
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Oh, and it is also very pretty. The usual picture, described in the video above, comes from dividing the Babylonians' square in quarters.
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