Quick Insight: Subtracting Negative Numbers

A math teacher recently asked how to explain the concept of subtracting negative numbers to her class. Why is 8 - (-6) = 14 the same as 8 + 6 = 14?

I've long internalized negatives as "opposite" and subtraction as "opposite of addition" so in my head, I had a notion of "opposite of opposite of addition" which simplifies down to "addition".

But that inner verbalization was still pretty abstract. After thinking of a better intuition, here was my reply:

Great question! I had to think about it for a bit. Addition and subtraction are related, but slightly different, than positive and negative numbers.

想象一下去散步。你面朝前，向前走8步。This is really:

0 + 8

0 is your starting point. The "+" means "facing forward" and "8" means "8 steps in the direction you're facing". Ok.

现在，假设我们想继续向前走6步。That'd be:

8 + 6 = 14

Which gives us 14 steps from our starting point. What if we had faced backwards and took 6 steps?

8 - 6 = 2

Which means we're pretty close to our starting point, just 2 steps away. What if we had faced backwards butwalked backwards6 steps?

8 - (-6) = 14

Ah! The addition/subtraction tells us which way to face, and the positive/negative tells us if our steps will be forward or backward (regardless of the way we're facing).

In a sense, the addition/subtraction acts as a verb ("face forward" or "face backward"), and the positive/negative acts as an adjective ("regular steps" or "backwards steps"). Or maybe it's an adverb, modifying how we walk (walk forwardly, walk backwardly). You get the idea.

对于年龄较大的学生来说，“减负”可以被视为“取消债务”。如果我有一个\ 30美元的债务，有人“减去它”，我实际上获得了\ 30美元。总的来说，如果你消除了一个缺点，你就改善了你的处境——这是积极的。

这些解释有点抽象，直接尝试走路更有趣。我实际上是边走边用直觉思考的。(如果你喜欢冒险，你可能会开始考虑侧身行走或跳跃，以及如何表现这些动作。)

Happy math.

Appendix

When doing simple arithmetic, we only track the final location, not orientation. Facing backwards and walking backwards might have us looking at 0 while we advance forward. But mathematically, our endpoint is the same: 8 - (-6) = 8 + 6 = 14.

If we care about the way we're facing, we need a more complex math object (a vector) to keep track of our orientation as well as position ("14, facing forward" vs. "14, facing backward"). Perhaps we'd use a line integral, moving along a path and tracking the direction we face as we go.

A fitting analogy leads to questions about what else is possible.

Appendix

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