If you like this, there is a whole book full of visual proofs [1]. See also wikipedia [2].<p>A few years ago I re-drew a bunch of these in latex with my PhD advisor and another colleague [3]. We planned to print them as posters and hang them for a Pi day event that unfortunately never happened because the pandemic broke out.<p>[1] <a href="https://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006" rel="nofollow">https://www.amazon.com/Proofs-without-Words-Exercises-Classr...</a><p>[2] <a href="https://en.m.wikipedia.org/wiki/Proof_without_words" rel="nofollow">https://en.m.wikipedia.org/wiki/Proof_without_words</a><p>[3] <a href="https://www.antonellaperucca.net/didactics/proof-without-words.pdf" rel="nofollow">https://www.antonellaperucca.net/didactics/proof-without-wor...</a>
This reminds me of this video about why we need to be very careful when inspecting visual proofs: <a href="https://www.youtube.com/watch?v=VYQVlVoWoPY" rel="nofollow">https://www.youtube.com/watch?v=VYQVlVoWoPY</a> (it includes a "proof" that pi equals exactly 4).<p>In this case, as someone else pointed out below, this proof has unjustified assumptions in it (at least it assumes that b < a).
Here is a visual proof for the Pythagorean theorem:<p><a href="https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras3.gif" rel="nofollow">https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras...</a><p>I find this much more "useful" since the Pythagorean theorem isn't immediately intuitive to me.<p>As for the proof in the original post, it seems really redundant to me. it follows from a (b+c) = ab + ac.<p>And while building intuition for this distributive property of multiplication is <i>extremely essential</i> when teaching maths, I feel that the intuition for why this is true is better built without leaning on geometry.
Be careful: with visual ‘proofs’ you can end up believing something like this: <a href="https://en.wikipedia.org/wiki/Missing_square_puzzle" rel="nofollow">https://en.wikipedia.org/wiki/Missing_square_puzzle</a>.
A similar method is handy for some mental arithmetic involving squares, e.g. it's easy to calculate 1005² because it's 1000² plus two added blocks of 5 x 1000, plus a small 5² block, so 1,010,025. Going the other way, 995² is 1000² minus those same two 5 x 1000 blocks, plus 5², so 990,025.<p>(Edited due to formatting fail!)
As one of the bad at geometry good at algebra people this blows my mind. I cannot even begin to comprehend how this shows, even for these specific boxes, the math works. But I can very clearly feel the relatedness of multiplication which makes the algebra work.<p>That’s not to say the example is bad, or good, more to marvel at how differently people think.
I enjoy some of the Mathologer YouTube videos and they often show some great visual proofs:<p><a href="https://www.youtube.com/watch?v=DjI1NICfjOk" rel="nofollow">https://www.youtube.com/watch?v=DjI1NICfjOk</a> (Fermat's sum of two squares)<p><a href="https://www.youtube.com/watch?v=rr1fzjvqztY" rel="nofollow">https://www.youtube.com/watch?v=rr1fzjvqztY</a> (Ptolemy's theorem)<p><a href="https://www.youtube.com/watch?v=yk6wbvNPZW0" rel="nofollow">https://www.youtube.com/watch?v=yk6wbvNPZW0</a> (Irrational numbers)
Have a look at <a href="https://www.matematicasvisuales.com/english/index.html" rel="nofollow">https://www.matematicasvisuales.com/english/index.html</a> .<p>It shows many nice visualizations. Amongst others it shows my my favorite proof of the Pythagorean Theorem<p><a href="https://www.matematicasvisuales.com/english/html/geometry/triangulos/pythagorasbaravalle.html" rel="nofollow">https://www.matematicasvisuales.com/english/html/geometry/tr...</a>
This is so beautiful! I could have never imagined this. I learnt this formula by rote when I was in school. Didn’t realize that it had a geometric equivalent. Same thing with differentiation and integration. Couldn’t understand. Learnt that too by rote. Is there a geometric equivalent for most formulas if not all? Is there a website?
Contrasting takes of Sophie Germain's quote from the page <i>Who invented Diagrammatic Algebra</i><p><a href="https://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra" rel="nofollow">https://mathoverflow.net/questions/168888/who-invented-diagr...</a><p>E.g.<p>>6 points: <i>I'd say God invented it.</i> –
Fernando Muro
Commented Jun 3, 2014 at 6:38<p>Approved answer:
Arthur Cayley. (Inspired by Sophie?)<p>><i>I believe diagrammatic algebra started with 19th century invariant theorists</i>
I love this, giving an intuition about something that is usually taught by rote memorization. I also love how it makes all the math nerds and pedants uneasy xD Man like, of course you need to be careful. You need to be careful about unintentional assumptions in logical proofs too. It’s a cool little creative visualization.
This same identity can be used to provide geometric intuition as to why i*i must equal -1. This is shown in the diagrams at the bottom of <a href="http://gregfjohnson.com/complex/" rel="nofollow">http://gregfjohnson.com/complex/</a>.
I'm finding this kind of strange because when I first came across this identity, I couldn't make any sense of it until I'd mentally visualised this exact sequence of images. Now I'm wondering how everyone else did it.
A fun book I have is Proofs Without Words (<a href="https://www.amazon.com/gp/aw/d/1470451867" rel="nofollow">https://www.amazon.com/gp/aw/d/1470451867</a>). Tons of neat diagrams like this.
This is best considered an illustration, rather than a proof, as it fails for any of the following:<p>a=0 and b is non-zero<p>b>a (result is negative. If you swap the variables, you have to imagine the area is a negative area)<p>either variable is negative
would think the point of algebra is to mechanize quantitative reasoning … trying to cast algebra operations in geometry may not be the most productive …
I'm a little disappointed at the focus here on how it's hard or impossible to visualize when b<a or when either a or b is negative.<p>That's not the point. To me, it's useful to already know that an algebraic proof of that equation exists, but to see it work out visually. I don't need to see it worked out visually for every single possible value for this to be helpful for understanding.<p>It also nicely illustrates how algebra and geometry are linked. And that multiplication is geometrically taking you from 1 dimension to 2.
that's a bit ridiculous - isn't it. also there are similar magic math proofs designed to lead the onlooker astray by miniscule inaccuracies and suddenly it follows that 1 = 0.