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Comment by orangepanda | Hacker News Reader

It still holds if you’re not bothered by negative-area rectangles

But I definitely am bothered.

A visual proof is supposed to appeal to our visual intuition - I don't know about you, but negative areas are not something that is visually intuitive to me.

Negative area's can be the only practical solution. Imagine the worst case: a 10*10 room with a pillar in the middle. How would you calculate that?

I don't see that as negative area. I see it as subtracting two areas. Both the room and the pillar have a positive area, none of them has negative lengths or areas.

Kids learn subtraction before they learn negative numbers - once you learn negative numbers, you know that addition and subtraction are almost interchangeable, but this is not necessarily intuitive to begin with.

Subtracting two areas is like adding two areas where one is negative…right?

Yes , but see my second paragraph. Just because something is true it doesn't mean it's intuitive.

That's interesting. To me it seems intuitive. It's a real area that can be drawn like any other. The sign is an operator describing what function to visualize, not a property of the measured area. So thinking of it in that way eliminates any need for the term "negative area."

But, intuition is subjective, so you may need to adjust the terminology to fit the visualization.

It is intuitive on some higher level, now that I know about signs, operators, negative numbers and all that. But when talking about visual information (i.e. "visual proof"), it isn't intuitive in that context.

In addition to that, for all I know there could be some pitfalls involved with negative areas which I'm not aware of. Even if there aren't any pitfalls, this isn't immediately obvious to someone who isn't familiar with the concept of negative area.

If I'm willing (or forced) to think in such abstract terms, I would much prefer an algebraic proof to this visual proof.

Your intuition is irrelevant if they’re mathematically equivalent.

There are no serious pitfalls with oriented areas. Adding them to your arsenal of geometric proof tools will greatly simplify many proofs. Not having such a concept makes ancient geometry books much more complicated than they need to be, often requiring lots of detailed case analysis where the separate cases are essentially the same, just oriented opposite ways.

It’s intuitive, just in more dimensions. People have different ideas and abilities to imagine/think dimensions, but on top of that we rarely train them to do that.

I think that build up to tensor fields should be in every school program. If you can’t think of a field, you’re mathematically disabled and too many basic ideas about real world are inaccessible to you. This limits the ability to vote on a set of topics and participate in non-local decisions that involve systemic understanding. Same for formal logic and statistics.

Once familiarized with that, you can easily start thinking of nonlinearly signed areas, complex areas and areas simultaneously positive and negative by an attribute.

The concept of negative area still feels like it'd get messy in a hurry. For a square pillar, the side lengths should be the same, suddenly giving you imaginary lengths just for the eventual area subtraction to work out. For a negative volume though, you need cubic roots of unity for the side lengths, throwing off your area calculations. Has anyone actually put together a system where the sort of concept you're describing is cohesive?

You're in for a nice trip, the concept is called Geometric Algebra:

https://youtu.be/60z_hpEAtD8?si=HHs_9m0IJ43nfI3S (~50m video)

TLDW: Yes, the concept is there, makes much more sense than a cross product (which is just an oriented area) and generalizes really nicely.

Alternatively, read: https://en.m.wikipedia.org/wiki/Bivector

Gotcha; it's less that my complaint doesn't apply, but more that it isn't relevant (i.e., "squares" and "cubes" aren't especially interesting constructs which need to coexist nicely, and if you relax that constraint then directional geometry can be very interesting). Does that sound right?

It seems like an elegant addition to the understanding.

Maybe if you viewed an animation where `a` starts larger than `b`, and steps till it's smaller.

Then you could see where the negativity happens. And seeing is nice.

If you're bothered by the idea of negative-area rectangles, there's no need to justify the assumption that b < a, because that's the only way you can assign any meaning to either side of the equation.

The equation isn't about rectangles at all, so I disagree. The rectangles only exist for the purposes of this proof. Any resulting complications are on the proof, not the reader's burden.

One easy place to get some intuition for signed areas is in the context of integrals.

You know position is the integral of velocity right? So say you walk in a straight line from your starting point, then you keep slowing down until you come to a stop and walk backwards past your starting point.

If you were to graph your velocity vs time at some point it would dip below the t axis because your velocity would be negative. Ok cool. If you integrate from the point you came to a stop and started walking backwards you’re calculating the area above the negative velocity curve(between it and the time axis). You’ll find it is a negative area. You know it has to be negative because you walked backwards past your starting point so it gets so negative that it cancels out all the positive area from when you were walking forwards.

This sort of thing is more useful for teaching than being an actual proof. You teach the same concept in different ways and the students can form a more solid understanding of the underlying concept independent of symbols or shapes, or at least they may understand it one way if they don't get the other ways.

You can do all sorts of factorizations the same way and handle negative areas by drawing them in a different color.

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