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Parabolic Numbers (Posted on 2017-09-18) Difficulty: 2 of 5
Consider two points on parabola y=x2, (-a,a2) and (b,b2), where a and b are distinct real numbers.

If these two points are connected by a straight line, where does that line intersect the y-axis?

Inspired by an interactive sculpture at the Museum of Mathematics, NYC.

  Submitted by Charlie    
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Solution: (Hide)
The line has slope (b2-a2)/(b+a).

Starting at (-a,a2), the line rises a*(b2-a2)/(b+a) while going to the y-axis, where it therefore hits at

y = a2 + a*(b2-a2)/(b+a)

which simplifies to a*b.

Similar reasoning applies when the direction of travel along the line differs from that assumed above, as when a and b have opposite signs.

The sculpture at the Museum of Mathematics uses buttons to represent integers from 1 to 10 as the two multiplicands. In two dimensions it represents what is called a nomogram, where the alignment of two numbers shows on a third scale the result of some operation on those two numbers. The museum just rotated the plane of different problems to produce a sculpture that's more interesting by being 3-D, but hides the two-dimensional nature of the nomogram if you don't read the nearby computer's explanatory material, which most people don't.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: No Subjectchun2017-09-18 15:30:04
No SubjectJayDeeKay2017-09-18 13:07:49
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