Fancy seeing you here!

@Darwin Wilder um what does that have to do with the video

@Darwin Wilder are you a bot

@Marlon Lachlan are you a different bot or that person on an alt account you both have blank pfps and made up names like bots

It's just caley diagram of operation

Oh my god i did not even notice that until i read your comment. That's awesome.

can you explain please

@@carozpp "Radical" has a double meaning

Pragmatic extension of the presented concept!

Yes! Let's make this the new default! Oh wait. People are slow when it comes to changes. Even when it's really better... Ugh.

@@jaypaans3471 depressing truth :(

Yeah I thought the same thing about the equilateral aspect in the video

YES! Good notation can be written sequentially, on a single line. The only exception to this is fractions (and continued fractions), but there are ways to denote them on a single line already. If you were to be typing the power triangle out in online text of some kind, what order would you write the symbols in? And why get rid of the equals sign? New notation should not eliminate good old notation like the equals sign. A single, reorientable symbol to connect 2 values, and a standard equals sign to indicate the value of the relationship, is immeasurably superior to the triangle of power demonstrated here.

Bear in mind that // is used in CS to mean rounded division

Yea the current notation mostly tells you what it means perhaps with logs being a bit of an exception. Powers are raised higher in the notation, implying growth. Subscript implies reduction or foundation. Another commenter pointed this out and I agree. Although they are kind of weird to learn at first, they really aren't difficult, and having the 3 different ways of notation allows us to express things differently in different situations/types of mathematical investigation. In my opinion, the different notations actually HELPED me to understand the connection between the operations and how useful they all are. Maybe I'm an exception and others don't agree, idk. I tend to use both my left brain and my right brain even in maths, I don't believe I'm right brain or left brain "dominant" as they say.

hell

Yeah. And the bg music isn't the typical soothe.

Eduardo Cortez you forgot the question mark

Heck.

He's up in serifs about it

Like x3 plx

agree!

There are lots of topics in maths that need a 3Blue1Brown TED Talk

I was just thinking this

Every lecture is a TED Talk

+Reuben F I wholeheartedly agree

There is sort of a point to this, in that it allows us to see that all three of these concepts are basically instances of the same thing: function iteration. In the case of derivatives, the operation you're iterating is that of taking the derivative, and in the case of exponentiation, the operation you're iterating is multiplication by the number being exponentiated. I agree it gets confusing when it comes to things like sin^-1(x)=arcsin(x) vs. 1/sin(x)=csc(x), though, because there are two different kinds of function application is taking place at two different conceptual levels. For this reason, I like to write sin(x)^2 instead of the more commonly seen sin^2(x), because I think that in a perfect world the second version, where the squaring appears to be applied to the function symbol rather than the result of the function, would be reserved for sin(sin(x)).

I prefer Leibniz notation a lot. Diferential notation.

Don't forget inverses, like sin^[-1] That notation drives me nuts.

Inverse notation can be thought of as a negative of function iteration.

cool

pogger

Why would you need permission tho

@@totoshampoin It's been four year, I have no memory of this

@@mistycremo9301 do you still use it tho? XD

This is what i thought of aswell. The tringular concept can be applied to alot of things

@@spudmckenzie4959 Yup. Me too. It kind of leapt right out. Though it's one of a few things that instantly reminds me of my old fella. He was an electrical down the pit, and interested in allsorts. I remember him teaching me Ohms law, and the triangle representation, on the wheeled adjustable table we had that floated about downstairs. A basic points, condenser/capacitor and coil car ignition system is another. Though this is Fiftyish years ago, I still would draw it identically to how he did. He was awesome. I was lucky, he pretty made me what I am. I was most impressed with this. Just so visual and intuitive. This kind of thing tends to help me no end.

it's funny that I actually found about that in a book as a kid without knowing anything about electricity, but it then became the only way I could understand algebra back then, "if ab=c then a=c/b" looked like a hard formula, so I remembered the triangle every time...

I learned this with v, s and t but it works with most multiplication equations like F=ma

In our school we'd form a triagle like this for almost every physics formula and it made it so much easier for students to understand the relation

A weapon to surpass metal gear

One triangle to find them

@@wf4860 One triangle to bring them all and in the darkness bind them

@@Bnelen In the Land of Math where the Syntax lie.

Franz Luggin they do represent functions, simpler ones too. When you have constants (which you generally always have at least one) there are specific steps you must take to cancel out unknowns (multiplication, basic addition, “o-plus”, etc). It’s more that it has the ability to simply many concepts in an intuitive way, and presenting it in a visual and intuitive way.

It reminded me of the Multiplication Triangle. (For people who for know that one, let's use the same symbols as this video: y on the top, x on the left and z on the right. Then y/x = z, y/z = x and x*z = y).

Let me rephrase you: this notation sucks and is unnecessary for anyone who understands what the hell log, power and root mean.

You can just put a () around the variable you want to make the function depend on

Just imagine a tiny Δ under the ³s if needed

2/³\8

I really like the positioning, and I like Yevgeny's. 2/\8, hmm... One of the big problems we are facing is the fact that many of these symbols /, Δ, are used for other things already in math, but I think having the two together like in Yevgeny's helps dispel that problem, and is typable by most keyboard layouts I know of.

Just use /\ for everything. 2/3\=8 /3\8=2 2/\8=3

William Pereira Gomes This could be cool for coding

i am a horses

Tsskyx "I understand logs perfectly...this triangle really helped me understand logs more" Lol?

I think exponentiation, and radical notation is okay. I think log notation sucks and should probably be replaced.

Chickenfrend agree

Bro, one minute you say "I understand logs perfectly," and then next minute you say "it help me understand logs better." You are making no sense?

I love it!

I see a problem here: in standard mathematical notation we are used to using infix symbols to represent operations, because our numbers are made out of digits and using a symbol infix clearly separates the operands from one each other. Since 12^3 clearly means "12 to the 3rd power", your notation has a problem with ^123 and 123^, since these are ambiguous in that the operands could be 12 and 3, or 1 and 23. But nice thinking anyway, I liked the simplicity of your idea.

It is infix though, isn't it? ^123 and 123^ aren't valid.

Reciprocate, and we get a // b (the // notation is *for* parallel resistance, not for rounding in the CS context) = ab/(a+b)

Now we just need a triangle of wisdom(addition/substraction) and a triangle of courage(multiplication/division)

....and now I remember that if the exponent is negative its division

Except addition/subtraction and multiplication/division aren't triangular operations in the same way. x+y is the same as y+x and x×y is the same as y×x but x^y isn't necessarily the same as y^x.

@@StarTheTripleDevil I actually took note of that and tried to run with this idea while taking into account the symmetry of the symbol. The first thing to decide is how these diagrams should compose. I chose vertically, so that the "result" of an operation is always on top. This means the expressions form trees, which is good. Obviously, exponentiation cannot be horizontal mirror symmetric, but addition and multiplication must. The symbols also have to be asymmetric vertically, because the "result" is at the top and the "inputs" are on the bottom. I think I ended up with convex and concave "caps" for addition and multiplication, but had a tough time finding a good one for exponentiation. The other difficulty this syntax has is scaling to large expressions, and expressing equality, at least conventionally. Unlike the traditional symbols, these operation diagrams are relations, not functions. We get a value (or set of values) back when we leave one point unspecified. When all are specified, we get a logical proposition that is either true or false. And when we leave two unspecified, we get a function-ish thing that could take one value and spit out the results (because might be more than one answer satisfying the expression) or take two values and spit out true or false depending on if the relation holds. This is very foreign thinking, and the syntax also doesn't scale very well to 3+ input relationships. But I still think it is worth exploring if you're up for it.

You may only pick two.

How do you cope when you have to draw a graph? Do you try and fit that into a line of your notes?

I'm grateful to a random youtuber for telling me a very simple way of getting accented letters on my android keyboard and I'm wondering if maths could make more use of these already available characters for inline notation. Maybe aébėc to represent the triangle abc starting at bottom left? So 3é2 = 9 etc . It seems to me that the triangle of power is a good method of visualisation and that in-line notation may be solvable.

I'm grateful to a random youtuber for telling me a very simple way of getting accented letters on my android keyboard and I'm wondering if maths could make more use of these already available characters for inline notation. Maybe aébėc to represent the triangle abc starting at bottom left? So 3é2 = 9 etc . It seems to me that the triangle of power is a good method of visualisation and that in-line notation may be solvable.

You could even just write it as: (2,3,•) = 8 (2,•,8) = 3 (•,3,8) = 2

Don't use lined paper, use blank. Scenario imagined. You're welcome.

Yeah, I’ll have to rewatch/analyze/play with it to really grasp it though. It’s not like you watch the video and learned it all either.

I was also thinking the same.

Even though the original statement got so so many up votes I am slightly relieved you have at least 30 votes.

Benoit Avril Yeah but they make you spend 8 years learning it I think it may because there is more slow people than fast people in math

Several years of spelling taught* in 1 comment... oh wait.

I learned a notation that could possibly be more intuitive: A || B (at my university it's called parallel sum, well because it's used for parallel resistances)

I think that's the standard that all electrical engineers use, and it's indeed quite intuitive for one of its more popular uses.

I'll see about using it next time i'm doing circuit analysis.

Coming from a programming context, A || B immediately reads to me as logical or. Bitwise xor has different notations depending on the language, so I think I usually just spell it out, (logical xor is just !=) though in the main language that I use, it's represented as x ^ b, because that's not confusing at all. (exponentiation is only available through functions like x.pow(b))

+Luis Carlos Thanks! The genius here goes to the originator Alex Jordan. I like the name you suggested, though of course it's too late to make any changes.

While both sound cool, the triangle of power gives a better connection, name-wise, to what it's talking about, at least as far as how we explain these topics currently, i.e. we raise 2 to the power of 3 to get 8, which can be represented using the triangle of power, which kinda gives the same feeling of using summation notation, since the notation's name tells you what's going on much more directly than some other notations. "Operational Triangle" makes it less connected to what it's talking about.

@@Naokarma I think the idea of using a general name like "operational triangle" is to suggest using the triangle notation (perhaps with an operator drawn inside the triangle) to three-way relations other than just x^y=z.

Love it! Are you a tutor?

unpaid, and only coincidental due to proximity, but this symbol makes me sad that i dont have more reasons to use it. Also: try replacing the values in the a^b=c triangle with their equivalents in a fractal, what is intuitive in the triangle form looks like the most ridiculous mess ever in "standard" form. lol, i actually laughed when i first came up with it.

You mean in a Sierpinski triangle kind of pattern? I can only imagine.

look at this: postimg.org/image/5y5w60563/ then try to convert it into "standard", you'll see the astoundingly fast progression of complexity

Hmmm, that does indeed seem absurd.

I just even realize when I see your comment :)

Just like fractions. So I don't think its a problem

@@wallacesantos0 Fractions can be written as x/y

Y. Z. Or x • y^-1

Exactly

​@@wallacesantos0 Just like fractions, that's the problem. Fractions mess up calculations. If I have more than one I always just write (x + blah blah )/(y + blah blah) to keep it on the same line. Having to do the same for power,log and root would just destroy any semblance of readability in my notes, and I bet I'm not the only one :/

It does in power engineering. Represents the relationship among active power, apparent power, and real power.

Exactly, the real problem is the horrible inflexibility of today's computer systems. Check out: "Drawing Dynamic Visualizations" from Bret Victor vimeo.com/66085662 which is strongly related.

Best reply.

ACoral i mean, writing a number on the top-right of another isn't very linear either but we managed to adapt it to computer language using the operator ^

@@metachirality how would it be any different than?

People in another comment were suggesting something along the lines of a/b\c, where a^b=c. Obviously not perfect, but something along those lines seems like it would work well.

Lol! For a challenge, see if you can figure out what this is: (-b+-/2\b/2\-4ac)/(2a). Good luck!

Which is something that sadly very few artists understand.

TheSpec90 I missed that joke genius blue

And art is math (y) golden ratio, etc ;)

beauty alone doesn't make something art

TheSpec90 I'd argue that math lacks the subjectivity that is rather important to most forms of art. It's definitely beautiful tho.

Vimal Gopal it might*

Same thinking here.

Degrees still feel easier to understand to me than radians

​@@thehiddenninja3428 Tau-based radians are, IMO, a lot easier to understand than pi-based radians. Radians are a measure of (relative) length. 1 radian is the length of the radius of the circle. If you phrase it in terms of tau (since tau is all the way around the circle), the coefficient is the percentage around the circle you are. So .5t = halfway around the circle = 180 degrees, and so on. If you instead phrase radians in terms of pi, you need to remember that ugly factor of 2 that comes out because pi is a diameter-based constant, not a radius-based constant. (Or, you could alternatively define a unit based on diameter. Let's call it the diameteran, and set it equal to the diameter of the circle. In this system, all the way around the circle is pi diameterans, and so 180 degrees is 0.5 * pi diameterans.)

Neural networks learn

K van der Veen that's actually an operator its called gradient I think

Grin Reaper Of Trolls no.

The Triforce is what happens when you combine the Triangle of Power with the Triangle of Wisdom and the Triangle of Courage.

..and may the force be with you

I decided to fix an error from the first one, which I decided was significant enough to warrant a second upload.

Most youtubers wouldn't do something like this, and I respect you for this.

+3Blue1Brown thats understandable. thanks for clearing that up haha

i just wonder what the mistake was? i think it was around the 6 identities... right? as i remember they were in 3 pairs, not 4 and 2, right?

I had put all the sub-triangles of the inverse-composition triangles on the wrong corner. Basically, when you unraveled what they meant, they all said things like x^(x^y) = y and log_(log_x(y))(y) = x, and other such nonsensical things.

Start with an unambiguous translation to the standard notation and tell them to deal with it :) You should be allowed to define your terms, right?

@@BudgieInWA Right...

It is very cool, though.

+Daniel Keriazis I'll poke around with it tomorrow, but I don't think Unicode does arbitrary superscripts or subscripts.

Well, all bets are off with math notation when you're in a text editor anyway. Radicals, existential quantifiers, special set symbols, etc. For the sake of programming, I see no problem with defining functions like "log", or "exp", but in a perfect world the image that would come to the programmers mind while writing these would be the symbol, in much the same way that a LaTeX writing saying "\dfrac{a}{b}" sees the fraction a/b in his mind.

+3Blue1Brown Funny you should mention that. Swift lets you use math symbols as operators, but it doesn't support sub or superscripting text.

+Daniel Keriazis Those change the color of a glyph... This would need to change its position. I'm not saying it can't be done, I just haven't come across it. Maybe the text system could be tricked into using an arbitrary string as an accent.

Could work too, but could be more limiting

Check out the Harmonic mean en.wikipedia.org/wiki/Harmonic_mean

Not helpfull

Isn't it called like the 'harmonic' sum?

1/((1/x)+(1/y))=1/((y/xy)+(x/xy))=1/((y+x)/xy)=xy/(x+y) You're welcome!

It's a special case of a^n + b^n = c^n. If n = 2, then it's euclidean distance. If n = 1, it's just normal addition (or the taxicab metric). If n = -1, it's the "o-plus" operation. Even more generally (Sₙ(v⃗))ⁿ = ∑ᵢ (vᵢ)ⁿ. Of course you can go even _more_ crazy if you want with different coefficients or powers for each term. There's a lot you can do with adding powers.

But it would have the benefit of making it look more like you are trying to summon an elder god

integral symbol is 3-line ⌠ │ ⌡ so, that would mean triangles of power are also 3-line?

It doesn't make things any simpler...

Yoav Shati no but it normalises the notation

The triangle both normalizes and simplifies

Yoav Shati I wouldn't say simplifies.

It can be more intuitive

This needs to be pinned!

they already do, just look at all the uses of the greek letter delta in stem stuff

Even though those are beautiful symbols...

I recognize the bird by his beak. :)

Usually, this can be remedied with clever notation. I have seen a (very general) formula in a book (which is sitting next to me as I type this) that they managed to fit into two lines that, when expanded could easily fills pages for even fairly simple examples (in case you're wondering what it is that's being expressed: it is a total homotopy operator for the variational complex). The layers of compactifying notation in use in that formula is honestly amazing.

Katzen33 Even though it can be annoying, there's something so graceful about the integral sign

The integeral signe is very meaningful !!! It is a curve and you take the equation which is under the curve (ie right to). Clearly why would you change?

What about making it look like this? triangle(a,b,?) = a^b triangle(a,?,c) = log base a of c triangle(?,b,c) = b root of c Or, to not completely change how computers interpret functions, make a function where the first argument tells you what part of the triangle is missing. 0 = left, 1 = top and 2 = right: triangle(2,a,b) = a^b triangle(1,a,c) = log base a of c triangle(0,b,c) = b root of c Altough that'd be harder to read for humans. Perhaps we could just write it the top way, and have computers internally write them as the second way.

L4Vo5 but that's exactly one of the points he dislikes about our current system (which I would choose over the triangles every time): text

@@L4Vo5 Performance? Parsing an extra function doesn't look good, if I need to call it a trillion times.

@@y.z.6517 The compiler could handle it since it'll already know which of the 3 functions to call

@@L4Vo5 Pre-compiling or meta-programming is a good idea. However, triangle() is a very bad function name, since coder may need to create a geometrical function called triangle(). I suggest ^(a,b,c). This is different from ^(a,b)=a^b

essentially*

English has its own share of inconsistencies...

omg youre so smart!1!!

You mean fractional exponents

And that's why. Fractional exponents make written expressions more difficult to read unambiguously unless you drown them in parenthesis which makes them overly busy and also difficult to read. In carefully type set expressions standing on their own line that's not the end of the world. In notes taken on paper with a pencil it's a complete and unmitigated disaster.

Using roots helps avoid ambiguity ³√(-1) = -1, but ⁶√(-1)² = 1. However, they are still not perfect. In the complex domain, you come up with other problems. ³√(-1) = (1 + i√3)/2, and ⁶√(-1)² = 1. In the reals, odd roots keep sign, even roots are always positive, and do not exist if the argument is negative In the complex, the roots are always the principle root (divide the complex angle measured from 0 to 2pi by the root for the new complex angle). It would really create the least confusion if only the complex variant was used. However, so many teachers act as if complex numbers don't exist except when teaching specifically about them. Then they ignore them again.

It doesn't help that historical motivation for developing complex numbers in the first case was due to the Casus Irreducibilis (literally: case irreducible), which occurs with certain Cubic polynomials and their general solution via Cardano's method, which is not taught in the standard curriculum at all, let alone before the introduction of complex numbers (and the process and resulting formula that leads to the relevant cases is quite complex). I only learned of Cardano's method and this historical motivation for developing complex numbers from reading wikipedia. And yet, we have real world applications for the seemly esoteric complex number system in the theory of electromagnetism from physics.

Radicals are good for cleaning up the expression. Turning radicals into fractional exponents is basically commonplace when doing integrals (since a cube root is a 1/3 power, converting it for the Power Rule gives you 4/3 which is a cube root of x^4... unless you can do them in your head)

I disagree. Notation can have a big impact on our proficiency. Just switching from roman to arabic numerals, allowed Europe's mathematical knowledge to really advance. The concepts are the same, but it's far easier to do them.

I know, it also would be a problem to implement across countries with people who've been ingrained with the first three formulas throughout their life. It's honestly much better to stay with the normal three which can be used in with each other to solve problems more fluidly.

it just looks like one tbh it is just three times more complex than each of the others in isolation

And what about reading papers that already use the traditional notation? You have to learn the traditional notation, or the work of your predecessors will be closed to you.

you would have more luck getting people to use tau over 2pi than this

Saurabh Verma you forgot the question mark

yeah, it wastes space in a very bad way. Although complex formulas have more obvious pattern in them this way, they also grow in very asymmetric ways. This makes it very unintuitive when marshalling it into computer. I mean, fractions are bad enough, when they turn into monsters when written as (a*b*c)/(d*e*f).

"You could even just write it as: (2,3,•) = 8 (2,•,8) = 3 (•,3,8) = 2" You could follow this idea, originally made by "JNCressey"

There's no sense trying to write more complex math following the lines of notebook paper. How do you write summations inline?

Mutantcy1992 You just use taller lines for summations, or multiple lines, but it still follows a horizontal pattern.

that's why everyone will keep typing things like 2 ^ x even when this triangle symbol gets official a^x for forward operation a ^ (1/n) for nth root log x / log a for logarithm of x at base a

To me, I think the reason this feels unintuitive at first is because I've spent all my life (well, minus the first few years) working with the traditional system, and all of my knowledge of higher-level math, of course, uses the traditional log-exp-rad notation. However, I genuinely think that if I had been taught to use this, that using it (or short-hands deriving from it) in higher math would actually be more intuitive than our current standard.

Suggesting that math notation needs an overhaul, takes courage. Dismissing it by saying that it would be like using toy hammers is just being lazy and ignorant. However, I'm not sure either that this exact way of doing it is the right way to go, but it is definitely a step in the right direction. The notation should be conducive to asking questions as he said in the video, and that's the big idea here. When you see a periodic table for instance, one of the first questions that pops up in your head is probably why does it have to stop at 118? Why are they grouped this way? What do they have in common? What's periodic about it? But I think the coolest/craziest thing about it has been through history where although no one knew for sure that a certain element existed (what if there are "proton gaps" in nature and you can't just assume a linear progression of the number of protons for each atomic number?) they could still get curious about it seeing that elements on either side of it in the table had been discovered and probably even guess what properties this new element should have based on where it "should" be inserted in the table. The periodic table, too, probably needs rethinking, but you get the point. A more general theme would be that notations are much more than a way of recording or doing calculations. They are a way of thinking. Most of the innovation in chemistry and material science has come from the fact that people started using the "atomic" way of thinking about things, instead of saying everything is earth water fire wind etc. The idea that everything we see in nature has just been naturally occurring compositions, begs the question what if you put the atoms together this other way? It is however quite natural that math has evolved this way and it is by no means a "mistake" or whatever that we've been doing it this way. In the beginning of any field, people are too busy finding out where things lead and what properties something or some idea has and what consequences it would have, that they don't have the patience/perspective/or even time to explore new notations. Feynman had famously his own symbols in the beginning but he had to give it up for the sake of practicality and mostly collaboration with others. It takes several intellectual/conceptual generations (not necessarily human generations) to get a good enough feel for the whole thing to start wondering whether the notations have been good enough. This is what essentially happened to programming and computing. In the beginning they were literally called machine instructions and you'd be mostly focused on telling the computer how to calculate things. This is analogous to the current mathematical notation. But as time went on, people came up with the object oriented and functional way of programming and programming languages started to become ways of thinking more than ways of instructing the machines directly. Hence higher level . I suspect that the same will be true of math, if it would still have a place when we have stronger AI assistants that would maybe make math irrelevant all together. Its notations, too, would evolve into more than one way of expressing your way of thinking and ultimately new sets of mathematical languages. There would then be something similar to the compiler for these different languages in order to convert them to more so called low level representations (which would be our current notation) where it would also try to pick the most efficient way of translating an expression (in the triangle notation for instance or more advanced ones) in the new language back to the lower level one (since as these newer languages evolve, they tend to abstract more and more nitty-gritty things about ways of calculating things) and this way keep everyone happy including the past people who have put so much work into math, using the low level language. This actually makes me think of a question... What if physics', math's and programming languages' way of expressing relationships and notations ultimately converges to a single language that could represent all these ideas and make thinking in each of them just as "easy" and inviting to questions and new ideas. What if these are all specific instances of a more general concept in the world that we're missing? Everything in the universe ultimately comes down to stuff that exist (= data) and the way they could change and interact with each other (= a set of operations on the data.) So maybe it wouldn't be so crazy to think that physics, math and programming are just different applications of the same underlying concept. Please let me know, anyone, if there has been more work done on the questions towards the end of my thoughts above. Thanks btw for making and sharing the clip. It's a great habit to get into. ;) Very nice to see that there are people out there curious enough to work on math on this deep level.

+miscibi that was such an excellent comment. very insightful!

The old notation is also more useful for programming/typing in ASCII . The old notation still has it's place, but I don't see why both can't exist, similar to how students are usually taught both leibniz and Lagrange's notation when learning calculus.

Just write it in display mode rather than inline mode.

Did you ever read a mathematical paper? Or physics or computer algorithms for that matter. I think all of them put formulas into separate lines already, because they do /not/ fit into normal text. Just take a look at the Wikipedia article for "Integral" and you’ll see what I mean.

Kissaki of course I do. Yes, but when you want to pinpoint something about logarithm or the power, writing putting that triangle into the line will increase the line size whereas classical notation is pretty much made for "inlining"... just look at the thumbnail and compare that cubic root sign to the triangle... or the power... it's a waste of space.

Peter Bočan, using superscripts to the right for powers and superscripts to the left with the root symbol for roots already makes those two kinda look like miniature versions of the triangle. So we could just make log look more like it does in the triangle, something like 2Λ8 to mean log_2(8). Then all three will have an inline version that looks kinda like the large triangle version. 2Λ8=3 2³=8 ³√8=2

You can consider it as a function of the two corners that are filled out, that evaluates to the one that is missing. I don't see what's stopping you from doing calculus with it

It can't be a function, it has an implied ' = ', you can't graph this

For example imagine the triangle with e in the bottom left, x at the top, and the bottom right blank. Would that not represent to you e^x? If you wanted to make it explicit that it was indeed a function of x, you could write f(x) = or use the shorthand notation here kzfaq.info/get/bejne/qbt8lJx9yJrUk58.html On the other hand if all three corners of the triangle are filled out, then I agree there is an implicit `=`. I don't see myself using the notation in that way though (the dual meaning makes me uncomfortable, but maybe it's just cause I'm not used to it)

TheBlehBlehBleh Well, that's one roundabout way of doing it, not elegant at all. It also becomes confusing when you are adding or multiplying different triangles. It tries to solve one problem no one is having by creating a million others.

It is not clear to me what these problems are. The whole triangle (together with its two filled out corners) is just an expression like any other. It might have variables in it, it might not. If it does have variables, they may be fixed or they may vary, just like in any other expression. You may make independent/dependent variables explicit it is a function by saying f(x) = ..., just like any other expression. To add or multiply you put the expressions side by side like any other. I've yet to find a use case where this makes things more ambiguous or complicated.