Chaitanya's Random Pages

January 15, 2011

The parabola as an analogue multiplier

Filed under: mathematics — ckrao @ 2:10 am

Suppose you had the parabola y = x^2 at your disposal. Here is a neat way of using it to multiply two positive numbers a and b. I saw this at the Wild About Math blog.

Locate the points (-a,a^2) and (b,b^2) on the parabola y = x^2. Then the line joining these two points has y-intercept ab. 🙂

Why is this true? Try to figure it out for yourself before reading on.

One way of seeing it is in observing that the y-intercept will be a weighted sum of the heights a^2 and b^2. Given the points’ distances from the y axis are a and b, the weights are in the ratio b:a and since they sum to 1, they must be b/(a+b) and a/(a+b). Hence the y-intercept is

\displaystyle a^2.\frac{b}{a+b} + b^2.\frac{a}{a+b} = \frac {ab(a+b)}{a+b} = ab.

1 Comment »

  1. I am reminded of one of those good old compass and straight edge constructions. Given line segments of length a and b, how do you construct sqrt(ab). You should do a series on constructions, with illustrations to boot!

    Comment by Radhika — January 15, 2011 @ 2:29 am | Reply

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