A visual proof that a^2 – b^2 = (a + b)(a – b)

https://www.futilitycloset.com/2024/12/15/tidy-2/

by beardyw • 2 years ago

731 192 2 years ago

futilitycloset.com

ujikoluk• 2 years ago

That seems to show that there exist a and b such that the equality holds. But not that it holds for all a and b.

wrsh074• 2 years ago

Which constraints on a,b (besides positive) does this proof require?

davrosthedalek10• 2 years ago

b<a Edit: and b,a element of R, but ok...

phoe-krk2• 2 years 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• 2 years ago

a and b have to be real numbers, whereas the identity works for any commutative ring.

notorandit2• 2 years ago

Really? It even holds true for either a=0 or b=0.

Scarblac2• 2 years ago

But not for b > a.

notorandit2• 2 years ago

We need to introduce these simple demonstrations in math courses.

Someone2• 2 years ago

Aren’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• 2 years 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.

konschubert4• 2 years ago

The 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• 2 years ago

Visual 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).

FartyMcFarter6• 2 years ago

This reminds me of this video about why we need to be very careful when inspecting visual proofs: 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• 2 years ago

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

FartyMcFarter6• 2 years ago

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.

foxglacier• 2 years ago

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.

kleiba2• 2 years ago

at least it assumes that b < a

But you can fairly assume that wlog.

im3w1l3• 2 years ago

karmakaze• 2 years ago

For those thinking this doesn't hold, note that "wlog" means without loss of generality (having numerous examples in proofs).

colejohnson66• 2 years ago

Yes, but that’s because it relies on infinite repetitions of the perimeter adjustment. Rearranging some boxes here does not involve infinity.

amelius• 2 years ago

Here's a similar one:

https://en.wikipedia.org/wiki/Missing_square_puzzle

blablabla123• 2 years ago

I vaguely remember from an Analysis lecture when Mathematical rigor wasn't such a thing, the professor mentioned a famous Mathematician (maybe Euler?) who had a notebook with hundreds of proofs where many of the visual ones didn't turn out right. Definitely nice for intuition though since this is oftentimes lacking

TheRealPomax3• 2 years ago

No, it assumes "one side is smaller than the other": the labels are arbitrary, so there iss no case where b is larger than a as we can just swap the labels and get a>b again. Even though you might think there are two cases that we need to look at because we're using two letters, there's actually only on case to demonstrate because we're demonstrating a property of the inequality.

The only "but what if..." would be if a=b, which has no geometric proof, but also doesn't need one because "zero = zero times anything" is (by definition) true for fields.

davrosthedalek10• 2 years ago

The problem is not symmetric between a and b (because a-b isn't). You cannot swap them. (And even if, this swapping is not part of the "proof", so in any case the proof is incomplete.)

petermcneeley• 2 years ago

Blasphemy! The value of pi is 3 https://religions.wiki/index.php/Biblical_value_of_pi

ComplexSystems• 2 years ago

There is no problem with the proof except the assumption that the value in the limit is the same as the value at infinity. If we simply define pi(n) as a function from N U {inf}, which gives the value that "pi" takes at the nth step of the process, and pi(inf) as the value that it actually takes for the circle, then we simply have a function where lim n->inf pi(n) ≠ pi(lim n->inf). For all finite n, it equals 4, and then at infinity it equals 3.1415... .

There are ways to reformulate the above so that "infinity" isn't involved but this is the clearest way to think of it. It isn't much different than the Kronecker delta function delta(t), which is 1 at t=0 and 0 elsewhere. We have lim t->0 delta(t) ≠ delta(lim t->0 t).

saghm2• 2 years ago

My geometry teacher in 9th grade was adamant that we couldn't assume lengths and angles in diagrams except where explicitly labeled. In particular, he said that we should never assume that a diagram is drawn to scale; if the diagram happened to depict a quadrilateral as a square, unless something stated that it was a square (or we had sufficient information to determine that it was), we should treat it as an unidentified quadrilateral and proceed only with the actual information provided or else he would "take more points off than the question was worth" on tests. On one of our quizzes, he did provide a diagram that looked like a kite (two pairs of adjacent sides each with the same length but different length than the other pair) but listed the angles in a way that could only work for a parallelogram that was not a kite, and he made good on his word to take off extra points for people who misidentified it as a kite.

brookst• 2 years ago

I like this teacher.

mmooss• 2 years ago

we need to be very careful when inspecting visual proofs

This is not a visual proof but a nice visualization, like a written explanation that is not a proof.

For an actual novel proof, nobody would imagine that they could eyeball it for a few minutes and conclude it was complete, correct, and consistent - maybe with the exception of professional mathematicians examining for simple proofs. You might eyeball it and follow its logic and not see any immediate flaws, but that's different.

trescenzi2• 2 years ago

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.

That’s not to say the example is bad, or good, more to marvel at how differently people think.

umanwizard4• 2 years ago

Which part are you struggling to understand?

Jabbles• 2 years ago

In which one of the five images would you say your comprehension fails?

dbsmith832• 2 years ago

In 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

trescenzi2• 2 years ago

I 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.

hi41• 2 years ago

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?

palsecam• 2 years ago

There is https://en.wikipedia.org/wiki/Proof_without_words / https://commons.wikimedia.org/wiki/Category:Proof_without_wo...

zozbot2342• 2 years ago

There 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.

wrsh074• 2 years ago

I 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/

mturmon• 2 years ago

The 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

layer85• 2 years ago

The geometric “proof” isn’t actually equivalent, because it assumes a > b, and doesn’t generalize geometrically to b > a. The algebraic proof, on the other hand, generalizes at least to commutative rings.

Geometric “proofs” like this are neat, but are no real substitute for the algebraic ones. I’d argue that in cases like the present one they also don’t provide any deeper insights. You’re just moving geometric shapes around instead of algebraic symbols. They might give you the feeling that the theorem isn’t as arbitrary as you thought, but it isn’t arbitrary in algebra either.

I’m putting “proof” in quotes here because there are many examples of incorrect geometric “proofs”, and there is generally no formal geometric way to verify their correctness.

zozbot2342• 2 years ago

there is generally no formal geometric way to verify their correctness.

There are formal models of synthetic (i.e. axiom-and-proof based) Euclidean geometry where proofs can in fact be verified. This is accomplished by rigorously defining the set of allowed "moves" in the proof and their semantics, much like one would define allowed steps in an algebraic computation.

kbutler4• 2 years ago

This 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

dbsmith832• 2 years ago

I 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.

xpe6• 2 years ago

A 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?

kbutler4• 2 years ago

1. 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)

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.

Ma8ee3• 2 years ago

Result is negative. If you swap the variables, you have to imagine the area is a negative area

And why is that a problem? Just reverse the equation before and after the proof.

kbutler4• 2 years ago

It means it isn't a "proof", it's an illustration of a partial domain of the original equation.

loadingcmd2• 2 years ago

A good reminder. This was taught in the elementary school along with (a + b)^2,

loadingcmd2• 2 years ago

Not sure how to edit, I think it may have been middle school, is been quite a few years :)

sahmeepee2• 2 years ago

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.

(Edited due to formatting fail!)

jgrahamc• 2 years ago

There are other fun tricks for squares like this: https://blog.jgc.org/2010/03/squaring-two-digit-numbers-in-y...

sahmeepee2• 2 years ago

Thanks - the comments are worth reading too

jfengel• 2 years ago

Futility Closet used to have a charming and fascinating podcast. I miss it. I'm glad that he's still writing the blog.

papercrane• 2 years ago

I really enjoyed their podcast. I haven't found anything quite like it to replace it.

fghorow• 2 years ago

s/proof/demonstration/ and be done with the arguments.

Intuitive but formally wrong is still good for intuition.

xpe6• 2 years ago

Such 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.

layer85• 2 years ago

No, “formally wrong” means that it fails formal verification.

xpe6• 2 years ago

Analogously, one wouldn't call a mathematical proof "visually wrong" simply because it lacked visual diagrams.

lupire3• 2 years ago

Quod erat demonstrandum.

quantum_state• 2 years ago

would think the point of algebra is to mechanize quantitative reasoning … trying to cast algebra operations in geometry may not be the most productive …

taeric2• 2 years ago

A fun book I have is Proofs Without Words (https://www.amazon.com/gp/aw/d/1470451867). Tons of neat diagrams like this.

lupire3• 2 years ago

Modern version:

https://m.youtube.com/@MathVisualProofs

exe34• 2 years ago

Is 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?

taeric2• 2 years ago

Proofs is not really accurate. Just visualizing some equations. I suspect this is part of some intuitions.

jrmg• 2 years ago

Be careful: with visual ‘proofs’ you can end up believing something like this: https://en.wikipedia.org/wiki/Missing_square_puzzle.

gklitz• 2 years ago

What 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.

dagss5• 2 years ago

That 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.

dagss5• 2 years ago

And, if you intend to deceive, there are many examples of solid looking written algebraic proofs that 1 = 2. That doesn't disqualify written proofs...

konschubert4• 2 years ago

Here is a visual proof for the Pythagorean theorem:

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.

littlestymaar4• 2 years ago

I 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…

konschubert4• 2 years ago

I guess my intuition for vector algebra is much weaker...

nuancebydefault5• 2 years ago

So 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.

quietbritishjim7• 2 years ago

How does Pythagoras's theorem follow from the definition of the dot product?

Do you mean that x.y = x₁y₁ + x₂y₂ and x.x = |x|², so it follows directly from that? If you define the dot product to be the first of those identities then you need Pythagoras's theorem to prove the second, so your argument is circular.

(Or you can prove that x.y = |x| |y| cos θ but that's even further removed from the component-wise definition than Pythagoras's theorem. Or you can define the dot product that way, but then you still have to prove the component-wise formula from it.)

alberto_ol• 2 years ago

I think that the visual proof is not complete. You have to demonstrate also that the quadrilateral in the right figure is a square.

konschubert4• 2 years ago

Shouldn’t that be clear from considering angle properties?

EDIT: Yes, I think it’s clear considering that the sum of the non-rectangular angles in a triangle is 90 degrees.

axus• 2 years ago

One time I was really sure that splitting a line into three equal segments let you draw lines from a point and split a 60 degree angle into three 20 degree angles, but this wasn't actually true.

gregfjohnson• 2 years ago

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 http://gregfjohnson.com/complex/.

ndsipa_pomu• 2 years ago

I enjoy some of the Mathologer YouTube videos and they often show some great visual proofs:

https://www.youtube.com/watch?v=DjI1NICfjOk (Fermat's sum of two squares)

https://www.youtube.com/watch?v=rr1fzjvqztY (Ptolemy's theorem)

https://www.youtube.com/watch?v=yk6wbvNPZW0 (Irrational numbers)

dvh• 2 years ago

But weren't the multiplication and addition rules set such that this visual proof holds true? Not the other way around.

larodi• 2 years ago

this 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.

nuancebydefault5• 2 years ago

Also 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• 2 years ago

oh my heart! this is very pleasant

flufluflufluffy• 2 years ago

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.

sebtron• 2 years ago

If 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-Classr...

[2] https://en.m.wikipedia.org/wiki/Proof_without_words

[3] https://www.antonellaperucca.net/didactics/proof-without-wor...

Kostarrr• 2 years ago

Super cool, thank you!

petemir• 2 years ago

Thank 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 :).

kelnos2• 2 years ago

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.

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• 2 years ago

Contrasting takes of Sophie Germain's quote from the page 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• 2 years ago

Cayley is my suspect for whom to blame that CS trees/forests are Australian: with their roots up top and leaves underneath.

alex-lx• 2 years ago

aha, I tried teaching my son this thing by the exactly same method a few days ago.

Dban1• 2 years ago

Now I want to see visual proof of Euler's Identity: e^(iπ) = -1

geniium• 2 years ago

That's so amazing! I wish I saw this at school while learning these by heart like a dumb student.

tessierashpool9• 2 years ago

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.

dtj1123• 2 years ago

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.

bawana• 2 years ago

But why is spacetime distance (x^2 - t^2)^.5

tim333• 2 years ago

I wonder to what extent you can do that in physics?

amai• 2 years ago

Have a look at 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/tr...

A visual proof that a^2 – b^2 = (a + b)(a – b)

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