## Geometric multiplication

I’ve just learned about a fairly useless, yet utterly beguiling method for performing multiplication, and I’d like to share it!

Suppose you have two numbers you wish to multiply, call them a and b. Your first instinct is probably to write one under the other and start writing down partial products, or to reach for a calculator. But wait—there’s another way! All you need is a pencil, a ruler, and a Very Large sheet of graph paper. The first step is to draw the parabola which is the graph of $y = x^2$ (actually, this step is optional). Now, find the points on the parabola corresponding to $x = -a$ and $x = b$. In other words, plot the points $(-a,a^2)$ and $(b,b^2)$. So, for example, if we wanted to multiply 4 by 9, we would plot the points $(-4,16)$ and $(9,81)$. Now, use the ruler to connect the two points with a straight line, and look to see where it crosses the y-axis—and voila! You have your product.

Pretty amazing, huh? I have some questions for you:

• Can you explain why this works? Hint: you’ll need to dust off your knowledge on finding slopes and equations of lines (unless it isn’t dusty).
• Does it still work even if a and b have different signs?
• What about if a and/or b are equal to zero?

The ancient Greeks (who did all their arithmetic and algebra using geometry) would be proud.

This multiplication method comes courtesy of Sol Lederman at Wild About Math. Fans of The Math Less Traveled might want to check out Sol’s blog; he’s got some interesting posts on various math topics and math teaching. Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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### 4 Responses to Geometric multiplication

1. Sol Lederman says:

Brent,

Thanks for adding a bit of rigor to this and for adding some exploration questions.

I’ve been wondering if there’s some analogous kind of graph for adding a pair of numbers the way the y=x^2 parabola allows for multiplication of two numbers.

Sol

2. Brent says:

Sol: that’s an interesting question, and I wondered about that, too. It turns out that there isn’t! I’ll see about posting my analysis sometime soon.

3. Andrée says:

I’m thinking that I’m going to use both (the parabola and the ordered pair) graphing exercises for my middle schoolers as they learn to graph and see if they notice the intercept. It will also help with integers and squares. Great stuff. I’ll ask them your questions. Thank you.