Here’s a graphical way to multiply two positive numbers a and b using the parabola y = x².

1. Start at the origin, move a units to the left, then go up vertically to the parabola, and draw a point.
2. Go back to the origin, move b units to the right, go up vertically to the parabola, and draw another point.
3. Connect the points and see where they cross the y-axis. That point is ab.

Here’s an example multiplying 3 and 5.

Here’s why this works. The slope of the line is the change in y over the change in x which is

m = (b² − a²)/(b + a) = b − a.

Use the equation of a line

y − y₀ = m(x – x₀)

with x₀ = b and y₀ = b² to get

y − b² = (b − a)(x − b).

Stick in x = 0 and you get y = ab.

The post Multiplication via parabola first appeared on John D. Cook.