I visited the National Museum of Mathematics in New York City with my family just recently. It’s a neat little museum that can be experienced in a few hours and I highly recommend it. One of the exhibits there is called “String Product”. It’s a large model of a paraboloid that sits in the middle of a spiral staircase between floors and illustrates an interesting property of the simple parabola \(y = x^2\). If you take two positive numbers \(a\) and \(b\) and draw two vertical lines, parallel to the y-axis, from \(x = -a\) and \(x = b\), and then draw a line through the two points where these vertical lines cross the parabola, then that line will meet the y-axis at the value \(a * b\). So this gives a nice geometric trick for multiplication.
Try it out …
a: , b: , a * b =
Proof
Why is this so? Let’s see.
Points on the parabola are parameterized by the coordinates \((x, x^2)\). So if we take the two positive integers \(a\) and \(b\) and then look at the vertical lines \(x = -a\) and \(x = b\), the points where these cross the parabola are \((-a, a^2)\) and \((b, b^2)\).
The general equation of a line is \(y = mx + c\) where \(m\) is the slope and \(c\) is the value at which the line crosses the y-axis. To find \(m\) and \(c\) we can use the two points that we know are on the line, namely \((-a, a^2)\) and \((b, b^2)\). So …
1) From point \((-a, a^2): a^2 = -ma + c\)
2) From point \((b, b^2): b^2 = mb + c\)
We can eliminate c from these two equations by subtracting them …
\[b^2 - a^2 = mb + ma + c - c\] \[b^2 - a^2 = m(b + a)\]Note that \(b^2 - a^2 = (b + a)(b - a)\) and so …
\[(b + a)(b - a) = m(b + a)\]By canceling, we get \(m = (b - a)\) and then substituting back in equation #2 above we get …
\[b^2 = (b - a)b + c\] \[b^2 = b^2 - ab + c\] \[c = ab\]So the value at which the line crosses the y-axis is equal to \(ab\).
QED.