9 ms·
A visual proof that a^2 – b^2 = (a + b)(a – b)
- amai 2y agoHave a look at https://www.matematicasvisuales.com/english/index.html https://www.matematicasvisuales.com/english/index.html . It shows many nice visualizations. Amongst others it shows my my favorite proof of the Pythagorean Theorem https://www.matematicasvisuales.com/english/html/geometry/triangulos/pythagorasbaravalle.html https://www.matematicasvisuales.com/english/html/geometry/tr...
- ujikoluk 2y agoThat seems to show that there exist a and b such that the equality holds. But not that it holds for all a and b.
- wrsh07 2y agoWhich constraints on a,b (besides positive) does this proof require?
- davrosthedalek 2y agob<a Edit: and b,a element of R, but ok...
- phoe-krk 2y ago> (besides positive) You can chart a and b on a 2D coordinate system, where they're allowed to be negative. Even positivity is not strictly required here.
- xigoi 2y agoa and b have to be real numbers, whereas the identity works for any commutative ring.
- martin293 2y agoAnd how exactly did you come to the conclusion that this is relevant here?
- davrosthedalek 2y agoBy the fact that the geometric proof in the link wants to proof the formula, but only does so for a small subset of all a,b for which the formula is correct. This makes it a partial proof, at best.
- martin293 2y agoOk nvm I can't resist wasting my time and typing stuff on the internet again, probably gonna regret it later. How is it not obvious to the dullest of the dull that this visual proof is not supposed to work for goddamn commutative rings lmao It's probably not even supposed to work for negative reals, 0 or the case b>a. It's supposed to demonstrate the central idea of the visual proof. Also yes, by choosing suitable ways to interpret the lengths shown in the diagrams it's absolutely possible to extend the proof to all reals but I'm not convinced it's meant to be interpreted like that. But bringing commutative rings into this... man you're funny
- notorandit 2y agoReally? It even holds true for either a=0 or b=0.
- Scarblac 2y agoBut not for b > a.
- deleted 2y ago[deleted]
- wcrossbow 2y agoJust rename a to b and b to a.
- kleiba 2y agoWhy the downvote? That's a correct argument.
- supernewton 2y agoIt is not. a and b are not symmetric in this equation, you can't just swap them.
- olddustytrail 2y agoOf course you can. What do you mean?
- Scarblac 2y agoThe answer is going to be negative regardless of the names, so this geometric proof won't work.
- davrosthedalek 2y ago3^2-2^2 =!= 2^2-3^2. (You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)
- deleted 2y ago[deleted]
- notorandit 2y agoWe need to introduce these simple demonstrations in math courses.
- Someone 2y agoAren’t they already used in early high-school math books? They aren’t used later on because, for more complicated expressions, manipulating the algebraic formulas is easier than the geometric method, or completely breaks down (e.g. when the powers aren’t integers or are variables) For example, I think anybody who can visualize what (a+2b)³ or (a+b)⁴ looks like geometrically can also, and easier, do the expansion algebraically.
- pwg 2y ago> Aren’t they already used in early high-school math books? No, or at least they were not in the high-school math books I was assigned. There were no explicit referrals in any way to visual representations of the algebra involved.
- konschubert 2y agoThe expression immediately follows from a ( b+c) = ab + ac. I agree it's important that students intuitively feel that a ( b+c) = ab + ac is true. But once we are at that, why would we even bother to teach or "proof" all variations of that expression?
- GuB-42 2y agoVisual proofs can be deceptive. For example, you have the famous missing square puzzle, where rearranging pieces of a triangle causes a square to vanish, something that should be impossible. It doesn't mean it is not possible to have rigorous visual proofs, the one in the article is, but doing so can be deceptively hard. It is also common for visual proofs to be less complete, for instance, here, it only covers the cases where a>0, b>0 and a>b, there are no such limitations when doing so with algebra. I guess you can tweak the visual proof to account for these cases, but it will become far less elegant. So I understand why teachers avoid visual proofs in math classes, they actually want you to stay away from them. They have entertainment value, that's why you see them so much in pop-science, and I think it is important, and also some historical value, as they were much more present in ancient times, but to actually learn maths, not so much. That is unless you are studying visual proofs specifically, but I think it is an advanced topic you would tackle well after you can master simple equations like a^2 – b^2 = (a + b)(a – b).
- FartyMcFarter 2y agoThis reminds me of this video about why we need to be very careful when inspecting visual proofs: https://www.youtube.com/watch?v=VYQVlVoWoPY https://www.youtube.com/watch?v=VYQVlVoWoPY (it includes a "proof" that pi equals exactly 4). In this case, as someone else pointed out below, this proof has unjustified assumptions in it (at least it assumes that b < a).
- orangepanda 2y agoIt still holds if you’re not bothered by negative-area rectangles
- FartyMcFarter 2y agoBut 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.
- Out_of_Characte 2y agoNegative 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?
- FartyMcFarter 2y agoI 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.
- ses1984 2y agoSubtracting two areas is like adding two areas where one is negative…right?
- trescenzi 2y agoAs 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. That’s not to say the example is bad, or good, more to marvel at how differently people think.
- School-Cotton 2y agoWhich part are you struggling to understand?
- Jabbles 2y agoIn which one of the five images would you say your comprehension fails?
- dbsmith83 2y agoIn the first image on the left, you can see the large square has length and width of a, which would have an area of a*a, or a^2. There is then a little square inside with length and width b, for an area of b^2. Essentially, the little square is getting removed from the big one (a^2 - b^2). In the last image on the right, you can see that the length of one side is (a-b) and the top side is (a+b), which would mean the area is equal to the product of (a-b)(a+b). This means that a^2 – b^2 = (a + b)(a – b). The intermediate steps just show how to move the area around visually
- trescenzi 2y agoI think this helps the most. The piece that doesn't "just click" for me is that area = product of multiplication. Manipulating symbols generally just feels more natural to me.
- lupire 2y agoWhat is area? Do you find the the basic concept of area (or the relationship between different dimensions in general) "doesn't click"?
- hi41 2y agoThis 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?
- palsecam 2y agoThere is https://en.wikipedia.org/wiki/Proof_without_words https://en.wikipedia.org/wiki/Proof_without_words / https://commons.wikimedia.org/wiki/Category:Proof_without_words https://commons.wikimedia.org/wiki/Category:Proof_without_wo...
- zozbot234 2y agoThere is some geometric equivalent for any formula that can be built using some arbitrary combination of the four operations of arithmetic, plus square roots. If you allow for either 3D geometry or origami folds, you can extend this allowing for cube roots too. Note that the geometric equivalent is not going to be necessarily trivial or intuitive in most cases, though.
- wrsh07 2y agoI have really enjoyed reading some of the better explained pages. I wouldn't recommend most people start on calculus on that site, but since you requested it specifically here's the overview: https://betterexplained.com/guides/calculus/ https://betterexplained.com/guides/calculus/
- mturmon 2y agoThe notion of “completing the square” is often done as an algebraic trick (“just add and subtract this magic term”). But it can also be viewed as translating the quadratic expression along the “X-axis” so that (at its new origin) it is left/right symmetric. That is, Q(x) = ax^2 + b x + c With the right substitution, x’ = (x - B), the linear term vanishes. So when you re-write in terms of “x”, you get: Q (x) = a (x - B)^2 + C So the intuition is that the linear term in the original quadratic is the thing that shifts the “symmetry axis” of the quadratic. I have found this helpful when “X” is a vector and you have a quadratic form. In this case, the coordinate shift centers the quadratic “bowl” about some point in R^n. * The chain rule for differentiation is another one with simple geometry but cumbersome notation. It’s like: we know[*] that f(x) = g(h(j(k(x)))) must have a linear approximation about some point x0. The only possible thing it could be is the product of all the little local curve slopes of k, j, h, and g, at the “correct” point in each. Thinking about little slopes also clarifies derivatives like f(x) = g(x^3, x^2) where g is an arbitrary function of two variables. [*] Because we read it in baby Rudin, ofc
- kbutler 2y agoThis is best considered an illustration, rather than a proof, as it fails for any of the following: a=0 and b is non-zero b>a (result is negative. If you swap the variables, you have to imagine the area is a negative area) either variable is negative
- dbsmith83 2y agoI think you could make it work for negative numbers. You would just shade the negative area differently to show it. For example, if you subtract 1000 sq ft from 100 sq ft, geometrically it means you are trying to remove more area than you have. So, to represent this, draw a rectangle of 100 sq ft. Conceptually "extend" this area by subtracting 1000 sq ft. The extra 900 sq ft that you try to subtract but can't "exists" as a negative representation. This could be represented by flipping the surplus 900 sq ft into a negative axis or shading it differently to denote that it's a deficit rather than an actual, positive area.
- xpe 2y agoA koan in four parts: 1. What constitutes a proof? 2. In what context is a "proof" embodied? 3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all? 4. If you came across a proof that was persuasive to alien intelligences -- and led them to conclude true things were true and false things were false -- but, alas, you did not understand it, does that make it less of a proof?
- kbutler 2y ago1. What constitutes a proof? An irrefutable demonstration of a conclusion, possibly via sequence of steps or combination of elements. 2. In what context is a "proof" embodied? Do you mean what the range or domain of the proof are? Not sure on the "embodied". I think you mean the communication of the proof and the expected base knowledge to understand the proof. 3. Why is scribbling lines on a paper (that look like math to humans) 'more' of a proof than visual diagrams, if at all? You seem to be focusing on the representation of the proof in a particular notation, rather than the actual logic of the proof. The graphical demonstration leads to false conclusions. For example, if a=0, it implies that a^2 - b^2 is 0 (or it requires some unfamiliar graphical representation of negative areas) 4. If you came across a proof that was persuasive to alien intelligences -- and led them to conclude true things were true and false things were false -- but, alas, you did not understand it, does that make it less of a proof? Again, the representation is not the proof, it is a means to record or communicate the proof. If the representation implies that false things are true (e.g., if a==0), then it is not a proof.
- loadingcmd 2y agoA good reminder. This was taught in the elementary school along with (a + b)^2,
- loadingcmd 2y agoNot sure how to edit, I think it may have been middle school, is been quite a few years :)
- sahmeepee 2y agoA 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. (Edited due to formatting fail!)
- jfengel 2y agoFutility Closet used to have a charming and fascinating podcast. I miss it. I'm glad that he's still writing the blog.
- papercrane 2y agoI really enjoyed their podcast. I haven't found anything quite like it to replace it.
- fghorow 2y agos/proof/demonstration/ and be done with the arguments. Intuitive but formally wrong is still good for intuition.
- xpe 2y agoSuch a visual argument isn't "formally wrong" because it isn't trying or claiming to be a formal proof at all. "Formally wrong" would mean that formal mathematics is used incorrectly.
- layer8 2y agoNo, “formally wrong” means that it fails formal verification.
- xpe 2y ago>> "Formally wrong" would mean that formal mathematics is used incorrectly. > No, “formally wrong” means that it fails formal verification. The word "formal" has meaning in mathematics independent of formal verification. The latter builds upon the former. Agree? > In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. - Wikipedia https://en.wikipedia.org/wiki/Formal_verification https://en.wikipedia.org/wiki/Formal_verification Here is how I'm using the terms: - "Formal" in mathematics refers to rigorous logical reasoning with precise definitions and deductive steps. This is the older, more general meaning. - "Formal verification" emerged later as a specific term in computer science referring to automated/mechanical verification of system properties. This is now the standard meaning in software/hardware contexts. So, putting these terms into use... If a human verifies a formal proof "by hand", I think it is fair to say that does not comprise "formal verification". On the other hand, if an automated system verifies the proof, then I would say "formal verification" has happened. Perhaps you will agree this is how many, if not most, experts use the terms. Are we mostly playing language games -- or is there a key insight you think I don't understand?
- quantum_state 2y agowould think the point of algebra is to mechanize quantitative reasoning … trying to cast algebra operations in geometry may not be the most productive …
- taeric 2y agoA fun book I have is Proofs Without Words (https://www.amazon.com/gp/aw/d/1470451867 https://www.amazon.com/gp/aw/d/1470451867). Tons of neat diagrams like this.
- lupire 2y agoModern version: https://m.youtube.com/@MathVisualProofs https://m.youtube.com/@MathVisualProofs
- exe34 2y agoIs there by any chance a book that explains what the illustrations mean? I had a quick look and it looks like I might need to know the theorems to begin with?
- taeric 2y agoProofs is not really accurate. Just visualizing some equations. I suspect this is part of some intuitions.
- jrmg 2y agoBe careful: with visual ‘proofs’ you can end up believing something like this: https://en.wikipedia.org/wiki/Missing_square_puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle.
- gklitz 2y agoWhat do you mean “end up believing” are you insinuating that the shown example isn’t true? It very much is true. The reason it’s true might initially be confusion to you as a viewer because you have a difficult time telling the difference between 3/8 and 2/5 and assume the triangles have identical slopes, but the visual proof very much and truthfully shows that this is not he case.
- dagss 2y agoThat appears to be constructed in order to deceive though. Someone who was thinking about a problem and drawing something would always with a drawing like that either intend the angle to be the same, or otherwise highlight the fact that one triangles is 8/3 and the other is 5/2 so that the slope is obviously not the same. Good visual proofs simply use lines and figures to talk about actual algebra instead of symbols; but every outcome is still in a sense algebraic -- like the one linked and the popular about Pythagoras. Once you pull our your ruler and measure you are obviously lost. Every result should be algebraic, not visual, but it's fine to express the algebra in figures instead of letters.
- dagss 2y agoAnd, if you intend to deceive, there are many examples of solid looking written algebraic proofs that 1 = 2. That doesn't disqualify written proofs...
- konschubert 2y agoHere is a visual proof for the Pythagorean theorem: https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras/pythagoras/pythagoras3.gif https://www.dbai.tuwien.ac.at/proj/pf2html/proofs/pythagoras... I find this much more "useful" since the Pythagorean theorem isn't immediately intuitive to me. As for the proof in the original post, it seems really redundant to me. it follows from a (b+c) = ab + ac. And while building intuition for this distributive property of multiplication is extremely essential when teaching maths, I feel that the intuition for why this is true is better built without leaning on geometry.
- littlestymaar 2y agoI don't feel like it's more redundant that Pythagorean theorem though, as we can say that the later directly follows from the definition of dot product…
- konschubert 2y agoI guess my intuition for vector algebra is much weaker...
- nuancebydefault 2y agoSo you prove something in the 2d space via a 3d space intermezzo? Not very intuitive to me. Distribution on the other hand, can be explained by counting a handful of the same objects.
- r0uv3n 2y agoThe dot product exists in any dimension
- quietbritishjim 2y agoIndeed I was even taught it in 2d before 3d (and higher). Even Pythagoras applies to any dimension, although admittedly it doesn't quite fit its usual statement in terms of triangles for higher dimensions: if a vector v has components (v₁, v₂, ...) then its length squared equals v₁² + v₂² + ...
- gregfjohnson 2y agoThis 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 http://gregfjohnson.com/complex/ http://gregfjohnson.com/complex/.
- deleted 2y ago[deleted]
- ndsipa_pomu 2y agoI enjoy some of the Mathologer YouTube videos and they often show some great visual proofs: https://www.youtube.com/watch?v=DjI1NICfjOk https://www.youtube.com/watch?v=DjI1NICfjOk (Fermat's sum of two squares) https://www.youtube.com/watch?v=rr1fzjvqztY https://www.youtube.com/watch?v=rr1fzjvqztY (Ptolemy's theorem) https://www.youtube.com/watch?v=yk6wbvNPZW0 https://www.youtube.com/watch?v=yk6wbvNPZW0 (Irrational numbers)
- dvh 2y agoBut weren't the multiplication and addition rules set such that this visual proof holds true? Not the other way around.
- larodi 2y agothis again opens the question what does multiplication actually mean. if you approach it geometrically the metric is something^squared, if you approach it as numbers - its not that straightforward on the numbers line.
- nuancebydefault 2y agoAlso it leans on the premise a x b (orthogonally measured) equals surface and that cutting pieces off surfaces leads to subtraction. Are those premises definitions or not, is a valid question.
- pn3k0 2y agooh my heart! this is very pleasant
- flufluflufluffy 2y agoI 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.
- sebtron 2y agoIf you like this, there is a whole book full of visual proofs [1]. See also wikipedia [2]. 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. [1] https://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006 https://www.amazon.com/Proofs-without-Words-Exercises-Classr... [2] https://en.m.wikipedia.org/wiki/Proof_without_words https://en.m.wikipedia.org/wiki/Proof_without_words [3] https://www.antonellaperucca.net/didactics/proof-without-words.pdf https://www.antonellaperucca.net/didactics/proof-without-wor...
- Kostarrr 2y agoSuper cool, thank you!
- petemir 2y agoThank you and congratulations for this incredible work. Maybe consider adding some type of reference/credits to the PDF? I think it would be useful for people downloading this and then not remembering where they got it from. It would be cool to be able to give credit where its due :).
- kelnos 2y agoI'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. 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. It also nicely illustrates how algebra and geometry are linked. And that multiplication is geometrically taking you from 1 dimension to 2.
- gsf_emergency 2y agoContrasting takes of Sophie Germain's quote from the page Who invented Diagrammatic Algebra https://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra https://mathoverflow.net/questions/168888/who-invented-diagr... E.g. >6 points: I'd say God invented it. – Fernando Muro Commented Jun 3, 2014 at 6:38 Approved answer: Arthur Cayley. (Inspired by Sophie?) >I believe diagrammatic algebra started with 19th century invariant theorists
- 082349872349872 2y agoCayley is my suspect for whom to blame that CS trees/forests are Australian: with their roots up top and leaves underneath.
- alex-lx 2y agoaha, I tried teaching my son this thing by the exactly same method a few days ago.
- Dban1 2y agoNow I want to see visual proof of Euler's Identity: e^(iπ) = -1
- geniium 2y agoThat's so amazing! I wish I saw this at school while learning these by heart like a dumb student.
- tessierashpool9 2y agothat'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.
- dtj1123 2y agoI'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.
- bawana 2y agoBut why is spacetime distance (x^2 - t^2)^.5
- tim333 2y agoI wonder to what extent you can do that in physics?