# Das Miscellany

## Musings on software, math, guitars and more ...

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 …

Note that $b^2 - a^2 = (b + a)(b - a)$ and so …

By canceling, we get $m = (b - a)$ and then substituting back in equation #2 above we get …

So the value at which the line crosses the y-axis is equal to $ab$.

QED.