Multiplication via parabola

Here's a graphical way to multiply two positive numbers a and b using the parabola y = x^2.

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 = (b^2 - a^2)/(b - (-a)) = b - a.

Use the equation of a line

y - y₀ = m(x - x₀)

with x₀ = b and y₀ = b^2 to get

y -b^2 = (b - a)(x - b).

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

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