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 to and see where it crosses the y-axis. But why on earth should that cross the y-axis at height ? 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 and . 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
We can simplify this, however, by noting that factors as , so the ‘s cancel… as long as . In that case, we’d have 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 , but that’s OK since we can always compute and negate the result. So, we now have
Now, how can we find the equation of the line? Since the slope of a line is the same everywhere, the slope between and any point on the line should always be equal to the slope m that we computed. We can write this requirement as
This is already a perfectly good equation for the line, but let’s simplify it a bit:
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… !