Here’s a graphical way to multiply two positive numbers a and b using the parabola y = x².
- Start at the origin, move a units to the left, then go up vertically to the parabola, and draw a point.
- Go back to the origin, move b units to the right, go up vertically to the parabola, and draw another point.
- 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.