Geometric multiplication: an explanation

Now for an explanation/proof of that weird method of multiplication that I talked about in a previous post. You’ll recall that it’s really quite simple: to multiply a and b, draw a line from (-a,a^2) to (b,b^2) and see where it crosses the y-axis. But why on earth should that cross the y-axis at height ab? Well, I’m afraid I still don’t know how to answer the why in a way that makes intuitive sense, but we can certainly show that it is true.

The first step is to figure out the equation of the line passing through the points (-a,a^2) and (b,b^2). To do that, we first need the slope. Slope is “rise over run”, that is, the change in y divided by the change in x, which in this case is

\displaystyle m = \frac{b^2 - a^2}{b + a}.

We can simplify this, however, by noting that b^2 - a^2 factors as (b+a)(b-a), so the (b+a)‘s cancel… as long as b+a \neq 0. In that case, we’d have b = -a and we’d be trying to draw a line from a point to itself, which doesn’t make sense. This just means that we can’t use this method to compute a product like 8 \cdot (-8), but that’s OK since we can always compute 8 \cdot 8 and negate the result. So, we now have

\displaystyle m = \frac{b^2 - a^2}{b+a} = b - a.

Now, how can we find the equation of the line? Since the slope of a line is the same everywhere, the slope between (b,b^2) and any point (x,y) on the line should always be equal to the slope m that we computed. We can write this requirement as

\displaystyle \frac{y - b^2}{x - b} = m = b - a.

This is already a perfectly good equation for the line, but let’s simplify it a bit:

\begin{array}{rcl} \frac{y - b^2}{x - b} & = & b - a \\ y - b^2 & = & (b - a)(x - b) \\ y - b^2 & = & bx - b^2 - ax + ab \\ y & = & bx - ax + ab \end{array}

Now that we have an equation for the line, we want to know where it crosses the y-axis. But this is simple: it crosses the y-axis when x is zero; plugging 0 into the above equation for x yields… y = ab!

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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