### Can you have a negative length?

Preliminary confidential to the KMS middle schoolers who asked this question: I am continually impressed by your active minds; it was a delight to meet you and be your teacher for a day recently. And you should be very grateful for the teacher you get to have every day. Ms. O cares deeply about you, about keeping your minds active and about your success. Thank her for that now and again in 10 years when you realize the impact she has had on your lives.

**Back to the question**: Technically, no. “Length” is always positive. As is “Area”. As is “Speed”.

But if you are willing to expand your mind and imagine a world with negative lengths, you don’t cause any problems. But you will have to be ready to consider negative areas. And negative speeds.

In Physics, they deal with this by the difference between *speed* and *velocity*. Velocity can be negative, as it includes direction. Forward velocity is positive; backward velocity is negative. Speed is the absolute value of velocity. Whether my velocity is 10 mph or -10 mph; whether I am running forward or backward, my speed is 10 mph (positive).

And this requires a negative version of distance. In Physics, this is *displacement. *Displacements can be positive or negative. Distance is the absolute value of displacement.

In Calculus, it can be useful to imagine a world in which area (and length) can be negative. It’s not usually encouraged, but it causes no problems and in fact can make some concepts more intuitive.

How can negative areas and lengths be intuitive? Well, you’re an eighth grader and you thought of them-shouldn’t a college student be able to also?

Brilliant!

I hope your students get to read this post – it is as clear an explanation of the question’s answer as one could hope for.

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