3 ms·
What is a is positive and b is negative? How do you extend the visual proof?
by cowsandmilk 2y ago
What is a is positive and b is negative? How do you extend the visual proof?
- jacobolus 2y agoYou can do this if you extend your concepts of length and area to signed quantities. You have to be clear to explain how the setup of the drawing works, but instead of cutting away a square of side b out of the corner of the square of side a, you might end up appending a negatively-signed square of side −b.
- lizzas 2y agoI am not convinced since in this world you can have the following rects: 1 x 1 1 x -1 -1 x -1 Is just the middle one a cut out? I reckon you need the axiom from the field to do generalize this to negative numbers. https://en.m.wikipedia.org/wiki/Field_(mathematics) https://en.m.wikipedia.org/wiki/Field_(mathematics)
- chisquared 2y agoThere’s no need for this extension: https://news.ycombinator.com/item?id=42425681 https://news.ycombinator.com/item?id=42425681
- jacobolus 2y agoWell the whole point here is that we're doing geometry. If you just want to do symbol twiddling you don't need the picture at all.
- chisquared 2y agoIf b is negative, define c = -b, and use the linked result to show that a^2 - b^2 = a^2 - (-c)^2 = a^2 - c^2 = (a + c)(a - c) = (a + b)(a - b)