Maths notation is needlessly complex [video]

https://aeon.co/videos/maths-notation-is-needlessly-complex-it-can-and-should-be-better

by ricksplat • 10 years ago

195 138 10 years ago

ricksplat• 10 years ago

My own personal opinion on this, is that it's all well and good but it's not that students are "Made" to learn three different types of notation but that these are three standard forms that you need to know to work in the mathematical space.

antoineMoPa• 10 years ago

Is there a quick way to use this triangle in LaTeX?

vlasev2• 10 years ago

A variation of this should do the trick:

\newcommand[3]{\triangle}{{}_{#1}\overset{#3}{\Delta}_{#2}}

overlordalex• 10 years ago

How is learning all the special cases of the triangle manipulation (O-plus when its bottom right and the top is missing but multiply in the bottom left, multiply when the top is given and bottom is missing etc) better than just learning the different notations?

Not to mention the existing mathematics you miss out on by using this.

That being said I think it's a fantastic tool to quickly explain the relationship between the notations (the first half of the video).

dzdt• 10 years ago

And we should all learn esperanto because it is so much more rational and regular than English. Somehow it doesn't work that way.

j2kun2• 10 years ago

The problem is not that the notation exists and that there are multiple ways to say the same thing, it's that students are forced to memorize all of it for its own sake.

In fact, there aren't just "three" ways to describe 2 * 2 * 2=8, there are infinitely many. Because 2 * 2 * 2=8 shows up in so many different contexts, and notation in each of those contexts highlights a different (hopefully useful) feature for that context, you'll never be able to have "just" one way to say a thing. You can have your favorite, sure, but all notational preferences are aesthetic.

FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation.

You see the video maker say, as an afterthought, that once the students are fluent in this beautiful notation they can go about understanding why the it works, but the same problem as before! They're memorizing arbitrary symbol shuffling, maybe reducing cognitive load but also introducing random extra facts along the way like parallel resistance (which algebra students care about that, again?), and the connection between the true idea and the work they're doing is thin.

noobermin• 10 years ago

FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation.

Yes, this times 10^100.

One thing to remember is that all notation is yes, tradition, but that tradition is somewhat subject to natural selection, which I'd argue is selection for ease of use. This is for example why we use symbolic notation in the first place, because it is more convenient once it's been learned.

I'm not sure how to deal with this. Perhaps one way is to teach the long, tedious way, just do a bazillion multiplications and then say, hey, exponentiation makes it much easier to write! That's usually how we get students to appreciate short-cuts in the first place. Teaching them the short-cut without at least giving them a taste of the internals is usually a recipe for bad abstractions disconnected from the internals.

mnemonicsloth• 10 years ago

I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day:

all notational preferences are aesthetic

Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns?

Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra.

How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.

j2kun2• 10 years ago

The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit.

To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book!

I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.

ianai• 10 years ago

i feel like the usual best practice for studying math is to use the same concepts in as many different settings/with as many different terms for the same thing as possible. By this I mean, literally, knowing the subject so well that there's not a context that can confuse you. Notation is no exception to this. If you know 2 ways to correctly solve a problem all the better. If you know 3 that's even better, for instance.

_nalply• 10 years ago

This is futile. It's just another standard. xkcd said it best: https://xkcd.com/927

chriswarbo2• 10 years ago

I suppose what would be really nice is for computerised mathematics to be marked up such that these interchanges can be made automatically, with libraries of rule sets shared online.

You're reading a document and it has some funny triangle thing you don't understand? Click on it to get a menu of alternative representations, and you see it can be swapped to "log" notation. Further, you go to your reader preferences and add a rule "whenever you see this triangle thing, show me it as a log".

A student finds some crusty old document with funny "heart monitor" symbols, clicks on them and finds they're just a particular kind of power triangle. They update their preferences to replace those symbols with power triangles.

Of course, holy wars rage on mailing lists about whether the default rules should convert "tau" to "2pi" or "pi" to "tau/2" ;)

Seriously though, too much time is spent making computerised mathematics look right in PDFs (e.g. TeX), compared to telling the computer precisely what it is/means. Thankfully there are some attempts at this (e.g. OpenMath), but they don't seem to be very widely used.

jordigh5• 10 years ago

I really need to expand this into a blog post that has been brewing in my head for years, but my overall thesis is that notation is the least difficult part to learn in mathematics, yet because it's the first part that is encountered, it is also the most derided one.

Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century.

However, I don't think better notation is where we need to focus most of our efforts in order to make our mathematics easier to understand. Logarithms and square roots are very basic things, and if keeping the mainstream notation for them straight is someone's biggest problem, then there are far bigger things that are likely to be problematic to this individual. If you start reading, say, the following mathematical discussion of neural networks,

http://neuralnetworksanddeeplearning.com/chap1.html#eqtn7

you're baffled because you don't know what those symbols mean, there's likely far deeper things that are unfamiliar, such as differentials, rate of change, derivatives, and the multivariable chain rule. A couple of days ago we had someone come to ##math in Freenode asking for help with this, and I tried, but the guy had never had any calculus training whatsoever. Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university, and I don't think this path to enlightenment could be shortened much more.

I guess I am being very old school and reiterating that the royal road everyone's been looking for for the past couple millenia just doesn't exist.

https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2....

hellofunk• 10 years ago

Can you recommend an approach to learn high level math on one's own? Any particular books that are good for that purpose?

pacala2• 10 years ago

A lot of math is simple and logical. Notation is a major obstacle, because of a. the sheer amount of various notations which are rarely used by a non-expert in a particular subfield, thus easily forgotten and b. un-searchability. Reading a math text becomes an exercise in jumping around figuring out what the author meant with a particular notation, which usually leads to more notations, etc.

Heck, coding CRUD apps I have access to better notation exploration tools than a mathematician. I can navigate to a definition or check the unit tests in seconds. I don't have to instantly remember the purpose of every function, I can simply look up the precise details on the fly. I can rely that the authors have not overused symbolic operators, so I can easily search StackOverflow for tips on the framework design. No more rote memorization of hundreds of tiny factoids that may or may not be germane to the problem at hand.

jordigh5• 10 years ago

A lot of math is simple and logical

And I suppose you are implying a lot of math is illogical. :-)

Heck, coding CRUD apps

Yeah, I hear the "bad notation" argument frequently from programmers, precisely for the reasons you describe: because you can't read a mathematical textbook with a programmer's IDE.

Some mathematical texts will do you the favour of having a list of some notation that they consider to be idiosyncratic to their own text, but few would go as far as to letting you look up the definition of plus and minus. In general, though, mathematics isn't programming, despite some similarities and analogies between the two. The symbol for the partial derivative is used nearly universally to mean partial derivatives of some kind, so there's a lot of tradition that writers of mathematical texts expect you to know.

Programming has similar traditions that are baffling to outsiders but as invisible to the programmers as water is to fish. For example, in programming it is understood that everyone is capable of easily handling plain text files. Have you ever seen a newcomer try to write source code in Microsoft Word? I have.

Wikipedia and other online mathematical texts can help a little by hyperlinking the text to explain new notation or terminology, and sometimes texts are just plain bad in that they use idiosyncratic notation without explanation. In the last case, it takes effort to work out from context the likely meaning of a symbol.

In general, though, I stand by my thesis that notation is not the biggest obstacle to mathematics just like learning to use a text editor or IDE is not the biggest obstacle to programming. The fundamental ideas behind the practices of each are much deeper than the superficial aspects of text and notation.

vlehto• 10 years ago

It sure doesn't help that often it's forgotten to teach notation in the beginning. Not in high school but at university level this is my experience. And it's incredibly hard to google on your own.

yorwba2• 10 years ago

It's like that for other skills too, like how to write a proof. I guess it is assumed that you will simply assimilate it with time, but I have met students in their third year who still struggled with e.g. properly introducing variables.

Double_Cast6• 10 years ago

According to the trivium [0], the path to learning consists of a sequence: grammar; dialectic; and rhetoric. In my head I think of it in terms of vocab, logic, and application (respectively).

Notation may not be the hardest part. It is however (as you say) the first part. If our goal is to make math more readily accessable to the masses, perhaps the greatest immediate gains entail improving the notation to better reflect the concepts -- exactly because it is the first barrier.

[0] https://en.wikipedia.org/wiki/Trivium

Cacti• 10 years ago

I honestly find the concepts far easier than the syntax, and I am used to weird syntax.

It's not just the notation though. It's using crappy notation in place of explanation or as a short-cut.

While some notation is "modern" in absolute linear terms, it's really not in any practical sense. The entire premise of most mathematical writing, exposition, and notation, is still that of thousands of years ago: writing takes significant time, writing surfaces are expensive, communication lag is long, and cultural/linguistic barriers between people are significant. That is simply not very true any more.

There is no reason we should be writing, teaching, and discussing math as though it was still a thousand years ago. We have computers, we have amazing printing presses, we have relatively cheap material. There should be no reason we're still pulling this "as the reader will clearly see" crap with this massively dense short-cut notation.

sedachv• 10 years ago

Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century.

Forget linear algebra, modern notation for regular algebra is from the 19th century. And the form of the notation absolutely does matter to learning and comprehension.

Ancient Babylonians knew how to solve quadratic equations, but you basically had to be a professional mathematician to understand the way they phrased problems and work with their notation.

More well known is the notoriety of Roman numerals for doing basic arithmetic. It takes more than 20 pages to explain fractions with Roman numerals: http://dmaher.org/Publications/romanarithmetic.pdf

iaw3• 10 years ago

Beautiful, elegant, and I disagree it should be taught.

∆ Is used for delta's so frequently that I could see some nasty issues arising in notes for higher level maths.

I think this could be an excellent teaching tool, to be honest I sometimes pull log notation back into exponential with a variable if I can't remember it immediately, but I think this guy didn't read Feynman's biography where he discusses the problems with creating notation.

mrob2• 10 years ago

δ could be used instead (lowercase delta). Root/exponent/log is so common that it should get priority for the best notation.

iaw3• 10 years ago

I agree in theory, but in practicality I don't.

∆ and δ have distinct and separate meanings in some of my coursework that can't be exchanged. The nomenclature/symbolism is so screwed up in some engineering fields that it would either require all students to revert to the existing symbolism or a level of investment in rewriting literature from research academics that created this screwed system in the first place.

I think Feynman had it right, he could invent the best symbols in the world for every operation but no one will use them.

mcphage3• 10 years ago

I don't think this notation would get confused with deltas; how it's used is pretty different, and overloading symbols (especially once you're in higher levels) is common enough.

iaw3• 10 years ago

Overloaded symbols became a nightmare in some of my later undergraduate work.

It is not the math classes where I'm worried that this would be a problem, it's in the engineering courses. There's a few situations in mechanics/structures/materials that I can think of where using ∆ instead of the existing 3 notations would become nearly untenable. In these circumstances each "line" of a solution/step usually took the form of more than one page.

lisper7• 10 years ago

This presentation misses a more fundamental point. It's not about the notation at all, it's about understanding that EXPT, SQRT and LOG are functions with a particular relationship to each other. That relationship can be expressed using a two-dimensional spatial notation, but that doesn't really help you understand the concept at all because there are a lot of different relationships that are naturally described by putting three things in a triangle.

What you really want students to understand is that expt, log, and nth-root are functions that are related in the following way:

  expt(b, n) = x
  log-x-base-b(x, b) = n
  nth-root(x, n) = b

It's really that simple. No fancy notation needed. In fact, fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships. Mathematical notation was invented not because it aids understanding, but because when you're writing math with pen and ink it's faster and uses less paper to use Greek letters and spatial relationships than full words. But when you're on a computer, it's easier to write out the names of functions, and that is actually a better impedance match to people's natural mental processes, which involve language.

[UPDATE] I would like to revise this: not everyone thinks in language. But everyone communicates in language. For communicating mathematical concepts, language is the best tool we have. There's a reason that the symbology in math papers is invariably wrapped in natural language. It's the same reason that the video has a narration. It wouldn't make any sense otherwise.

zeven7• 10 years ago

fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships.

I'm pretty sure this is false for me. I think thinking in spacial relationships comes much more naturally to me than words. Words immediately start making my brain hurt. The notation you wrote is incredibly more painful for me to look at and understand than the triangles in the video.

lisper7• 10 years ago

Let me rephrase that: people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language. There's a reason that the symbology in mathematical papers is invariably wrapped in natural language.

monk_e_boy• 10 years ago

I didn't even know that powers, logs and roots were that related. My math is bad :( I struggled through some of the Khan Academy but I find it really hard to remember all the rules. The triangles really helped me. But maybe the video was overly simplistic? These triangles (visual items) can be placed in the mind palace. I can picture them now and could probably figure out where the log2(x) fits in.

Practicality3• 10 years ago

"is actually a better impedance match to people's natural mental processes, which involve language"

This is false when applied to all people. And this is the reason you disagree with the demonstration. Not all people think in language. I would wager that only half the population thinks in language.

I think visually. It's work for me to translate my thoughts into words. I have to do a translation process to write this comment. It's easy work, but my natural thoughts are pictures of concepts.

Richard Feynmann liked to talk about this a lot, but the internal mechanisms of thought vary drastically from person to person. Ironically, it seems common for people who think in words to suppose everyone else is that way. I am not sure why.

lisper7• 10 years ago

Not all people think in language

Yes, you're right. But everyone communicates in language. I've updated the original comment.

jordigh5• 10 years ago

I think visually.

I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects:

http://i.imgur.com/HCfGOYp.gif

Better notation is certainly not worthless, and having a certain degree of symmetry or geometry to our notation is a good thing. I insist, nevertheless, that notation is not the biggest impedance towards conveying ideas.

mannykannot3• 10 years ago

It seems common for people who think in words to suppose everyone else is that way. I am not sure why.

Perhaps because that's how you get to know what someone else is thinking? Even when you draw pictures (which is quite common in technical fields) they usually need some explanation.

Practicality3• 10 years ago

In response to your update, I believe that visual methods of communicating are superior to words. Ever hear how a picture says 1000 words?

Is a description of a triangle really superior at communicating the concept when compared to a picture of one?

All that said the understanding of functions you point is helpful. Especially for someone from a computer science background.

jewbacca• 10 years ago

Counterexample:

In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. [...] The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be a rather arcane and abstract formula. As David Kaiser writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations", and as such "Feynman diagrams have revolutionized nearly every aspect of theoretical physics".

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

----

This is the first example that came to mind, but there are certainly very many others. Because the idea that written flat text is inherently a global maximum for representing/communicating/working with all concepts is absolutely absurd.

This is immediately self-evident if you've ever tried to teach anything complex to anyone who wasn't already a domain expert in an immediately adjacent area.

arcanus• 10 years ago

This is immediately self-evident if you've ever tried to teach anything complex to anyone who wasn't already a domain expert in an immediately adjacent area.

I am a practising computational scientist, and I work in an interdisciplinary field. I work with the sort of folks you mentioned above daily.

And while I often do not start with an equation, that does not mean they are not useful. I don't think anyone is suggesting that to a non-specialist they are. Typically, I start with thought experiments or plots. People are often visual learners.

However, that does not mean that mathematical notation is a problem. Mathematics is a language, and like any language one must spend years (or in my case, decades) learning how to 'speak' it. While I would not suggest it is inherently a global maximum, I'm skeptical it is not at least a pretty good local max.

posterboy5• 10 years ago

the idea that written flat text is inherently a global maximum

It's quite the opposite, text is a global minimum. Sequential text is best suited to represent some first order logic, a recursive enumeration of nouns. Or better yet a zeroth order logic, a simple enumeration of nouns. That is a desirable logic to work in, because it's complete. But we do need second order logic to ... beats me :)

lisper7• 10 years ago

Diagrams are good for representing and reasoning about physical things, and for mathematical concepts that have straightforward physical models, like numbers. But this is because there's a natural correspondence between these diagrams and the things they represent. There is no natural correspondence between a triangle and logs/exponents/roots beyond that fact that triangles have three sides and three vertices, and the set {log, expt, root} has three members.

mjfl• 10 years ago

This would have made self learning much easier for me. I remember when I was in high school I tried reading "The Road to Reality" by Roger Penrose and the gradient symbol seemed like some magical incantation only sorcerers could understand. If I had been told it was just a simple symbolic manipulation, I probably could have gotten through that section (now I would probably just Google it).

I bet if Professor Penrose rewrote his book in code-language it would be much easier to learn, at least for software engineers!

tnecniv• 10 years ago

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race"

-- Whitehead

jxy• 10 years ago

  > not everyone thinks in language. But everyone communicates 
  > in language. For communicating mathematical concepts, 
  > language is the best tool we have.

You definition of language is really narrow. Perhaps you really want to say that people are only comfortable in their native tongue? Do you consider 'expt' is a language? Actually I have no idea what "expt" is, which means we are not using the same language. I guess that was

  pow(3)

b*n

The inverse function 'log-x-base-b' also confuses me, is it

  b⍟x

  x⍟b

By the way, I'm speaking in unix man and a programming language there.

People need to understand notation is a language. In you three lines of functions, you've used (),= four notational symbols, without which your statement would be uselessly long. People do communicate in languages. Math notations happen to be the native tongue of people using math (they were taught at first in schools).

For people speaking in that different language, the following are the same:

  bⁿ=x
  b^n=x
  pow(b,n)==x
  b*n=x

Edit: reformat because HN eats my power function.

mcphage3• 10 years ago

It's not about the notation at all, it's about understanding that EXPT, SQRT and LOG are functions with a particular relationship to each other.

The notation being proposed is for those three specific functions. It's just replacing the existing notations for those functions with a new notation for those functions. Specifically, the author claims that this notation will make the relationship between those functions clearer.

because there are a lot of different relationships that are naturally described by putting three things in a triangle.

He's not proposing this as a generic functional notation, but rather as a notation for these three specific functions. So yes, many other functions might benefit from a similar sort of notation, but this proposal isn't for any of them.

But everyone communicates in language. For communicating mathematical concepts, language is the best tool we have.

When teaching math, teachers use a combination of natural language, formulae, tables, and diagrams. Teaching math without words would be (as you point out) nonsensical; but teaching math only using words would be incredibly hard to understand & follow.

nnq3• 10 years ago

Nope... The triangle notation is actually awesome because it does NOT put all the focus on the fact that EXP, SQRT an LOG are functions, and it actually promotes thinking at a higher level than first order functions: it's about an expression than can generate multiple functions or a higher order function.

Let me explain:

Let we define

    EX(b, e, x) := { pow(b, e) == x }

And then the notational rule saying that "substitution a parameter for a star/asterix in an expression is defined as the value obtained by solving that expression for the substituted parameter", so now we have:

    EX(b, e, *) == pow(b, e)
    EX(a, *, b) == log(a, b)
    EX(*, e, x) == root(e, x) # or sqrt(x) if e == 2

And here you go: "triangles notation" in "words" :)

And from this on you can go much easier to much deeper questions than the ones you get from staring at the graphs of 3 different functions that don't seem to relate much to each other.

Classic mathematical notation is great for doing very basic physics and engineering... but horrible for anything deeper. Only thing is that people doing anything deeper are smart enough to be able to tolerate a few horrible notations and probably even like them because they scare the "peasants" away :)

But we should be prioritising what's best for education, not for the macho-ism of uber-physicists or uber-mathematicians, so go triangles go!

Chinjut7• 10 years ago

Yup. Frequently, in mathematics, we have some ternary relation r(a, b, c) such that fixing values for any two arguments uniquely determines a corresponding value for the third. This encapsulates not only the relationship between a^b = c, log_a(c) = b, and b-th root of c = a, but also the relationship between a + b = c, c - a = b, and c - b = a, the relationship between a * b = c, c/a = b, and c/b = a (as well as the non-commutative version of this where we have separate notions of left- and right-division), and myriad other such ternary relations.

In all of these cases, a triangular notation is apt for expressing and emphasizing the structure that any two arguments fix the third.

That having been said, that is all the triangular notation does; it does not help with any further structure that may be around for that particular relation (e.g., rules like a^(b * c) = (a^b)^c, which the triangular notation does little to simplify, though perhaps further notational choices could make these also immediate and clear). Still, when that ternary structure is what one cares to emphasize (and why shouldn't it often be of interest?), one should go ahead and emphasize it.

drostie• 10 years ago

If people around here are interested in this idea, I would just like to point out that this is basically what embedding miniKanren (a super-simple logic programming DSL) in your programming language lets you do. You write "give me all q such that 2^q = 8 with an expression like `find q. PowEq(2, q, 8)." Also since the asterisk is sort of "loaded" it may be more helpful notation-wise to use an underscore, PowEq(2, _, 8) standing for the above.

The most obvious problem here is that you have to think about the composability of relations the way you do with functions, because PowEq(3, PowEq(2, _, 8), _) makes sense (find me the number that you get when you raise 3 to the [whatever number you'd have to raise 2 to, to get 8]), but the very similar-looking PowEq(3, PowEq(2, 3, 8), _) is actually a type error, the function is being called on a statement rather than a value.

lisper7• 10 years ago

I can't quite figure out which side you're arguing here. On the one hand you say that the triangle notation is awesome, but then you go on to use linear strings of symbols in exactly the manner that I was advocating. The whole point of the triangle notation (and all spatial notations in math -- subscripts, superscripts, the various arguments to sums, products, and integrals, etc) is that it's two-dimensional. That makes it really hard to render into ascii, which is one of the many reasons I think it's a bad idea.

I am certainly not arguing against teaching higher level concepts.

adrianratnapala• 10 years ago

Whatever the relation between them, the fact is the EXP, LOG and ROOT really are distinct functions with their own personalities. EXP grows fast, and is single-valued. LOG and ROOT grow slowly and are multi-valued, but in different ways.

Triangles and the star notation above are useful in cases where the ternary relation is very salient. And that might include teaching, but they are only part of the picture.

posterboy5• 10 years ago

people naturally think in words, not in spatial relationships

Not at all. There was a prior post titled "The link between language and cognition is a red herring" (https://aeon.co/ideas/the-link-between-language-and-cognitio...). The brain follows a spatial layout that, as a graph, is at least three-dimesional.

I'd guess that language can be encoded in higher dimensions, too, but I assume that the sequential nature of words in a sentence would require an ordering in a zero order logic to be logically complete. As I've learned from the RegEx discussion about the stackoverflow bug today, backreferences don't have to be done with recursion and that recursion and backtracking have suboptimal space and time complexity (https://news.ycombinator.com/item?id=12131909) (or sumsuch, I'm sure I misunderstood some).

it's easy to transform your examples into one line, in some sort of differential equation. At least representing log and root as inverse(exp(x, n)) for bound x or n.

omaranto• 10 years ago

As Gauss wrote: "In our opinion, however, such truths should be extracted from notions rather than from notations."

(Nowadays most technical math is written in English, but back in Gauss's time people used Latin. He really wrote: "At nostro quidem iudicio huiusmodi veritates ex notionibus potius quam es notationibus hauriri debebant.")

sullyj3• 10 years ago

"people naturally think in words, not in spatial relationships"

I think you may be committing the typical mind fallacy. I often find I have to consciously shove words out of my phonological loop so that I can focus on visualizing a concept in order to understand it.

jerf4• 10 years ago

I like the problem statement, I like the use of a 2D representation, but the triangle symbol as drawn in that presentation is gargantuan. I think the symbol needs more work.

One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists. Removing that overlaps with ∧, logical and, though.

Another that comes to mind is that I can't think of another mathematical symbol that has such divergence in meaning depending on what is left blank like that. The closest I know of is integral, where you can leave the from and to parts blank for a symbolic integral, but that's still not like leaving those blank turns the integral into a differentiation (the opposite), depending. I'm sure there's something else somewhere up in math, too, but nothing your average student will hit.

Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own. In fact hiding away an = symbol is probably a big strike against the idea as if anything standard math education underplays and abuses that most fundamental of symbols; let's not add to that. Again, somewhere up in higher maths than I've gotten to there may be symbols that constitute entire equations, but it won't be something most students see.

Also I think once that symbol is being shown with full expressions rather than cute little single-digit numbers or single-letter variables, it's going become very difficult to deal with.

I think there's something to this, though. I'm criticizing in the spirit of continuing to move forward. (I'm aggressively hostile to the idea that math notation is perfected and debating better alternatives is some sort of betrayal or something.) Personally I'd seek out a smaller, inline symbol that may visually reference a richer presentation (which may not be this literal triangle) for a nicer didactic experience, but doesn't literally draw it out in the formula.

Double_Cast6• 10 years ago

One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists.

The operator in question is actually quite common.

Buying stocks. Suppose you buy $1000 worth of stocks each month, no matter the price (dollar cost averaging). You pay $25/share in Jan, $30/share in Feb, and $35/share in March. What was the average price paid? It is 3 / (1/25 + 1/30 + 1/35) = $29.43 (since you bought more at the lower price, and less at the more expensive one). And you have $3000 / 29.43 = 101.94 shares. The “workload” is a bit abstract — it’s turning dollars into shares. Some months use more dollars to buy a share than others, and in this case a high rate is bad.

https://betterexplained.com/articles/how-to-analyze-data-usi... (harmonic averages section)

The concept describes various processes which contribute towards identical workloads at different speeds (or rates).

jerf4• 10 years ago

I'm not sure how to relate what you said to my post? Wrong reply button?

jpfed• 10 years ago

I am sympathetic to the aims here. I think it makes a lot of sense to represent the relation (a,b,c) where a^b = c in a unified way. Of course, it may also make sense to represent other ternary relations in similar ways, too. You could put something inside the triangle to represent which relation is intended.

Double_Cast6• 10 years ago

Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own.

Multiplication and division can also be put into a triangle, such that filling in each corner constitutes an equation. See my comment [0] about the density equation. I suspect such triangles reflect some deeper relation of abstract algebra.

[0] https://news.ycombinator.com/reply?id=12138817&goto=threads%...

jerf4• 10 years ago

Well, sure, we can put whatever we like in explanatory diagrams. I'm not aware of recognized common mathematical symbols that constitute equations on their own, though. This video did not seem to be proposing a diagramming method, because if that's all it was there wouldn't be any controversy to speak of; it appeared to be proposing a new family of operators. That's what I was criticizing.

loup-vaillant• 10 years ago

The triangle being an equation by itself is a bit of a bummer. The obvious followup to yours removing a line would be to remove two. Exponentiation would be:

    e
   /
  x

Logarithm would be:

  e
   \
    b

Root would be:

  x—b

Now a simple line may be too little. And it's not clear we should be using 3 different symbols ('/', '\', '—') either. But you get the idea. It's less gargantuan, and expressing the entire equation without the '=' sign, while possible, is so baroque nobody would do this except to make a point.

But what I like best with this idea is the possibility of replacing the line by some arbitrary symbol (including no symbol at all). That way, other triangular relations could be expressed without stepping on each other's namespace.

By the way, there is an obvious direction we haven't used yet:

  a
  |
  b

I'm not sure in which cases it could be useful, tough.

---

Now that I think of it, hiding the `=` sign may not be so bad in some cases. Relations don't have to be limited with the identity and orders (<, >, ≤, ≥). They don't have to be limited to relations with 2 variables either. Other relations would have to justify their usefulness to warrant a special symbol of their own however.

mrob2• 10 years ago

It's even worse than shown in the video. Consider:

sin^2(x) = (sin(x))^2

sin^-1(x) = arcsin(x)

Switching to triangle notation would help remove this confusing overloading of the superscript operator. The only drawback I see with triangle notation is there's no obvious way to type it on a single line.

Mickydtron• 10 years ago

You might be able to approximate the triangle in line by using slash and backslash. 2/3\ = 8. 2/\8 = 3. /3\8 = 2. Not nearly as pretty, but if you're familiar with the triangle in a hand written context, it might get the point across.

Retra• 10 years ago

Just use the connecting edge: 2(/)3 = 8, 2(-)8 = 3, 3(\)8 = 2.

mcphage3• 10 years ago

The only drawback I see with triangle notation is there's no obvious way to type it on a single line.

Definitely a valid complaint, although nth-root and log-base-b are already poor for typing on a single line.

dasfasf• 10 years ago

Has this ever actually caused you confusion?

Pxtl2• 10 years ago

I have seen students get very confused by the ^-1 arcsin thing. Imho that one needs to go away and just use the word arcsin or develop a proper symbol for "inverse"

yoha• 10 years ago

There are just overloaded notations from two different contexts. sin² can mean either:

* x → sin(x)² (in multiplicative group)

* x → sin(sin(x))² (in composition group)

Similarly, sin⁻¹ can mean either:

* x → 1 / sin(x) (in multiplicative group)

* x → arcsin(x) (in composition group)

wfo3• 10 years ago

"It's like it's a different language!" (incredulously)

Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly, flexibly, creatively, and it works beautifully. Saying 2^3 = 8 and log_2 8 = 3 sort of get at the same fact but not really, that's misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 "in the same way" (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn't give ourselves language to express both "the difference between 8 and 2 is 6", and "2 more than 6 is 8" because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts.

The second example, the 8^(1/3) is not even equal to 2, it's equal to three values, two of them are imaginary. It's important to have notation for "the thing that when you cube it is equal to 8" distinct and understood so that when you begin understanding imaginary numbers (high school iirc) you have context. Then you can explain the definite article in that quoted sentence is actually inaccurate. What if he had selected an example with two real roots, like sqrt(4)? What should we put in the the "triangle of power"? The positive branch cut? Okay, now you have to explain that you're really doing a different operation now, that has multiple answers, but we are going to pick one of them and put it there, but we have to remember that it could be either. Which is best expressed using separate notation to explain the operation you are doing that isn't even a function.

The fact of the matter is that these ideas are distinct and relating them is a separate, worthwhile exercise that helps understand the structure of exponentiation and its inverses.

And in doing mathematics you will find that switching to equivalent but more informative or clean or applicable notation is one of the most valuable workhorses we have for solving simple problems.

apalmer• 10 years ago

cant upvote enough... the triangle notation is actually "wrong"

nemetroid2• 10 years ago

The second example, the 8^(1/3) is not even equal to 2, it's equal to three values

No, the expression "8^(1/3)" is equal to 2 and only 2, even though there are three values of x for which it holds that x^3 = 8.

What if he had selected an example with two real roots, like sqrt(4)?

The notation √a refers to the principal square root of a, which is defined to be positive. You might recall expressions such as "x = +/- sqrt(a)", which would be redundant if sqrt itself was a multivalued function. a^b, for rational b, is defined in a similar manner.

wfo3• 10 years ago

It depends entirely on the context -- mathematical notation, as always, depends upon the level of instruction, textbook, context, mathematician etc. In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. When you're teaching it for the first time (as is the context we are discussing right now) it's quite important to reinforce how many square roots there are. Context generally makes it clear (among people who already understand what is happening) which version you mean as is usually the case. Similarly with logs, though we have the nice log/Log distinction which I suppose has become standard-ish. There is literally no reason we select the positive branch other than notational convenience and one of the biggest mistakes I've seen with freshmen students trying to learn calculus is algebra mistakes like this -- taking a square root and only considering the positive branch. Because they were taught square root of 4 is 2. And that's true sometimes (when you are computing sqrt(x)), and not others (when you take the square root of something for the purpose of solving an equation). I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.

Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.

Practicality3• 10 years ago

I can't help but wonder if the triangle of power is a Zelda triforce reference.

damptowel2• 10 years ago

I'm sure if would capture imaginations if it were called the "triforce" :)

vlasev2• 10 years ago

Let's see how this notation holds up to nesting...

mankash6662• 10 years ago

While this post limits itself to basic mathematics, the notations in group theory and advanced linear algebra JUST DON'T MAKE SENSE. Please read the "Brakerski’s Homomorphic Cryptosystem", section 3.3, of this ( https://eprint.iacr.org/2015/137.pdf ) paper.

I had to read it 10 times to truly understand what the notation was trying to say! Fail.

GFK_of_xmaspast2• 10 years ago

That seems just fine and should be understandable by a reasonably sharp undergraduate.

mankash6662• 10 years ago

Would you care to elucidate the exact math that makes homomorphic search possible

davidivadavid• 10 years ago

Math notation is an endlessly interesting subject, but I must say I wasn't very impressed with that idea.

If we're going to give up some notation to adopt another, it would need to have some serious and obvious advantages.

I waited for that throughout the video. The author consistently seems to assume that what he's saying is "intuitive". It isn't.

Why should I put a particular number in a particular corner of the triangle? How does it help computation? I see triangles being nested within triangles and fusing together according to rules that seem completely arbitrary.

Certainly, using our visual apparatus to help us complete computations without having to think about it can be appealing, but that's probably not the most convincing example.

baby2• 10 years ago

Same here, it confused me more than it should have.

sixo• 10 years ago

Yeah. The notation should ideally have some obvious spatial symmetries/structures that correspond with the symmetries of the operation (sticking a smaller triangle in the corner doesn't LOOK like it cancels out, and it should.)

Double_Cast6• 10 years ago

In one of my (middleschool?) science classes, my teacher explained the density equation in terms of a triangle. Of course, we didn't adopt it into our notation. But I think that just having shown the class the diagram helped a lot of my peers.

Later in high school (in a different state), a mathematically-challenged friend was studying logarithms during study hall. Our curriculum used this bizarre, three-step arrow rule to transform logs into the familiar (y = b^e). Having remembered the density triangle, I showed my friend a similar diagram for logs. He said "Thanks. I was probably going to get a zero on the next quiz. I might actually pass now."

  d = m / v
  v = m / d
  m = v * d 
      .
     /m\
    /---\
   /v | d\
  .-------.

paulmd• 10 years ago

While HN has definitely identified the problem with adding yet another notation, I think it's fairly straightforward to to remove the "root" operator. Simply use the exponent notation with a power of "1/3".

This gets directly at the core concept of what a "root" really is, and it's straightforward to manipulate with addition/multiplication of groups of roots using the normal arithmetic methods.

Terr_• 10 years ago

While I do prefer fractional exponents sometimes, what about the "asymmetry" of imaginary numbers?

There may be a benefit to capturing that sqrt(square(x)) != x

everyone• 10 years ago

Presumably people have been rightly complaining about this for 100's of years? I'd say the same is true for musical notation. Though similarly I doubt anything will ever be done to ameliorate these issues, the bad old system which has accreted over time has a tremendous amount of cultural momentum

serge2k• 10 years ago

Not this crap again.

The notation is the way it is because it works for mathematicians. Works well. It's not that complicated, but it does take some time.

Just learn the god damn notation and quit whining about it. Ugh.

jandrese• 10 years ago

I've always viewed it kind of like minimized and obfuscated source code. If you took a program and replaced all of the variables with single (greek) letters and created custom notation for every type of function you would get something that looks a lot like mathematical notation.

People can certainly learn it, but it's hard to argue that it isn't a barrier to entry and has no doubt helped to turn many students off from mathematics.

I mean the primary reason a lot of our notation looks the way it does is so that it can be written compactly (paper was expensive) using quill pens. Is it so crazy to consider a world where mathematical language is more self documenting?

htns• 10 years ago

It's not just math, but every other field as well. Notation has only become more succinct as the price of paper fell, so it's really programming that's held back by legacy limitations and ASCII.

Animats• 10 years ago

The video doesn't give any examples of complex formulae written using their "triangle of power" notation. This may be useful for teaching mathematics, but it's not clear that it scales. Much of the benefit of the notation could come from simply writing x⁽¹/²⁾ instead of √x. (Annoying, unicode doesn't have a superscript slash. "(",")", "=", and all the digits are available in superscript, but not "/".).

A huge hassle in moderately advanced mathematics, where new domain-specific operators are introduced, is ambiguous precedence of operators. There's a tendency to define operators in such a way as to minimize the number of parentheses required for the most popular uses of that operator. Such idioms make formulas hard to read. For an example, watch Andrew Ng's videos on machine learning.

It might be useful to always parenthesize in textbooks. Teach kids to always write "log(n)" instead of "log n". After all, how does "log n × m" parse? Is there an official standard on that? If so, where?

catpolice4• 10 years ago

I don't like it for a number of reasons.

One relatively simple one is that it doesn't naturally convert to a typed out version. Outside of category theory, where big commutative diagrams really do a ton of work, we should try to avoid introducing too much notation you can't type out, especially at the introductory level. Suppose two kids are trying to study together over facebook chat - how are they going to write these huge triangles out?

Second, it's actually unecessarily complicated. It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate. But actually, the easiest way to understand how to work with roots is to just define them in terms of exponentiation. E.g. the nth root of x is just x^(1/n). All of the normal rules for roots follow from the rules for exponentiation (and fractions) immediately. You don't need a third side to the triangle, that's just adding extra complications - all you really need is exponentiation and logarithms and a way of representing that they're inverse operations in a certain sense.

So there's actually a simpler way to express all this in a notation that's much more similar to what we've seen before. The trick is to draw out parallels between familiar operations like multiplication and division. Note: I made this up 10 minutes ago, apologies if there are very similar proposals, it's just really obvious.

Most of the basic arithmetic operations can be written using binary infix operators, e.g. +, * and /. It turns out, exponentiation and logarithms can be too. In fact, if you're typing, exponentiation already is.

Let x raised to the nth power be written as (x ^ n) (note this is essentially exactly the way it's already typed out, only I'm using some unusual extra white space to emphasize that we're treating ^ as an infix operator). It's a little upward arrow that says scale x up by n (exponentially).

And let log base n of x be written as (x v n) or possibly (x \/ n). It's a little downward arrow that says scale x down by n (logarithmically).

This makes the two operations work in a way that's fairly analogous to multiplication and division in a relatively neat way. For example, for positive integers, multiplication can be thought of as (linearly) scaling one number up by another, while division scales it (linearly) downward in an inverse way. As already noted The same holds for these two operators, only the scaling is non-linear.

Lots of familiar relationships carry over, e.g. Note that ((x * n) / n) = x. Similarly ((x ^ n) v n) = x. And where (n * (x / n)) = x, it's also the case that (n ^ (x v n)) = x.

And where multiplication and division interact with addition and subtraction, those operators interact similarly with multiplication and division, e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m). And so on. You can derive all the relationships you need from a very small number of rules that are easy to remember because they're structurally almost exactly like the rules for operations you're familiar with. You already know those when you learn about exponentiation, so you don't have to learn new and weird geometric relationships.

All of those nice properties there follow from the fact that exponentiation is just the next operation in the sequence of hyperoperations (see https://en.wikipedia.org/wiki/Hyperoperation ) after multiplication.

Introducing this weird three place operator actually masks the underlying simplicity of exponentiation and its inverse.

Chinjut7• 10 years ago

"It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate."

Hm? This is the exact opposite of what the notation is intended to do. It is intended to emphasize that there is just one underlying ternary relation [with all of these seemingly separate functions just being different ways of solving for one argument to this ternary relation in terms of the other two].

catpolice4• 10 years ago

I guess what I was trying to say is that treating that as a ternary relation is already overcomplicating the matter. Really, there's no need for nth root notation at all, because we already have fractional exponentiation. Why have a third place on the triangle putting any n there is the same as putting 1/n on the top?

Part of the argument for the triangle is that the relationships you'd have to learn are suggested by simple geometric manipulations.

But why learn whole new geometric manipulations when you've already learned the algebraic manipulations behind multiplication and division? By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking "it's exactly like multiplication and division, except [etc]"

Chinjut7• 10 years ago

"And let log base n of x be written as (x v n) or possibly (x \/ n)." "Lots of familiar relationships carry over, e.g. Note that ((x * n) / n) = x. Similarly ((x ^ n) v n) = x."

You must mean ((x^n) v x) = n. Note that ^ is not at all a symmetric operator, and neither is v.

"e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m)."

You must mean x ^ (n + m) = (x ^ n) * (x ^ m).

It seems your proposed notation does not make things so clear as that one cannot get lost in it, either.

catpolice4• 10 years ago

Ah bummer. Both of those were actually just transcription errors because I was writing formulas out in the course of a meeting, playing with whether the order of the v operator should be that way or the other way around - I don't have time to give it much thought at the moment but I'm admittedly less convinced I picked the right one. In the second case, that was a typo. In the first case, what I'd meant to write was: "Note that ((x * n) / x) = n. Similarly, ((x ^ n) v x) = n." You're right that neither is a symmetric operator but of course neither is division or subtraction. I sort of glossed over that intentionally, but the non-commutativity of the ^ operator would indeed probably be the main place that switching to infix operators could mislead.

mannykannot3• 10 years ago

Most of the way through the video, the narrator is saying things like "the student can easily see that..." No, she can't! Not if 'see' implies understanding. The narrator acknowledges this at the end, but by then, he has passed up the opportunity to show the usefulness of this notation (if that is so) by applying it in more complex situations.

The video is also plagued with distracting visual effects (there are some that are worthwhile, but most are not.)

ulkram• 10 years ago

Does anyone know the history of this notation? Like were the concepts discovered independently by three different people? Hence there are three ways to notate?

svckr• 10 years ago

When I learned the basics of electronics in middle school our teacher (and, actually, every teacher since) explained the relation between resistance, voltage and current using a triangle:

     /U\
    /R*I\

(Where U = voltage, I = current, R = resistance.)

I'm not sure it helped me. Anecdotal evidence: Just now, as I was trying to figure out which letter goes where, I was actually thinking in terms of what's going, as in "if at constant voltage I increase the resistance, the current should drop, ok, so I = U/R?".

So, in conclusion, I don't know which one is "better". Most likely, different people think in different terms and require different methods of learning, so if there's another way of explaining things I think that's good, isn't it?

baby2• 10 years ago

I always pick one at random, U=R/I

You really only have 1/2 chance to make a mistake.

vinchuco• 10 years ago

It's a work in progress https://en.m.wikipedia.org/wiki/Zenzizenzizenzic

chriswarbo2• 10 years ago

This also brings to mind other attempts to clarify notation, such as https://mitpress.mit.edu/sites/default/files/titles/content/... and https://en.wikipedia.org/wiki/Geometric_algebra#Relationship...

Kinnard• 10 years ago

:( https://news.ycombinator.com/item?id=12125756

ittekimasu• 10 years ago

- 2^3, log_2(8) ... why do we have 3 different ways ? Because, playing with tautologies is the whole point of mathematics (and language) !

- The new "notation" is essentially expressing the same language in a different alphabet.

I was hoping for some smack talk about partial derivatives (see SICM) or about the absolute proliferation of symbols in Differential Geometry...

agumonkey• 10 years ago

I strongly need to reread that book.

andreyk• 10 years ago

This seems like a good teaching tool, but seriously would anyone want to write out big equations with the triangle instead of x^n ? Powers are used incredibly often and the standard notation is clearly easier to write in both typing and handwriting, and the same is true for log. And as has been mentioned the n-th root symbol is technically not a true extra notation, as you could always write x^(1/n) - but it does make some equations extra clear because there are so many roots that often need to be taken.

So, don't complain about the notations - the triangle would be far more annoying to write equations with if that's the only thing you use. The notations are like 'helper' functions that make implementing a larger function easier, but also make things less clear to start out with because there are more forms of the same thing - so use the triangle to point out it's all the same thing (or all related, anyhow) but keep the notation all the same.

cttet• 10 years ago

I thought it would talk about lack of namespace, implicit overloading of symbols etc...

Pxtl2• 10 years ago

Lack of support from normal unicode text files, particularly with the frequent use of subscript and superscript.

nikdaheratik• 10 years ago

I agree that the notation can be a problem, but it has to serve a number of different uses:

1. Communication between mathematicians.

Like coding (or legal communication), the notation needs to be precise and unambiguous or you can cause misunderstandings the derail the point you are trying to make. This does not always make for easy to understand notation.

2. A shorthand for key concepts and relations to other math users.

This is something of a problem that is caused by the fact that mathematicians are often the people who teach the non-mathematicians. Who then in turn use the math or teach it to novices. It's a large hurdle, but then you're left with people who learned one notation as a novice, and are then forced to relearn how to communicate these concepts if you want to participate in the academic conversation.

3. A method of communication between non-mathematicians (like physicists or engineers).

There is actually a fairly large difference between how physicists communicate a key concept in their field, and how a mathematician might communicate the same idea. This isn't a large problem at the start, but then you're left with trying to move the ideas down the chain to novices and you have possibly several competing notations that eventually have to be sorted out. Which is why some notation is carried down and others eventually gets "weeded out".

damptowel2• 10 years ago

One practical concern I have for this is that you're going to need bigger triangles to avoid things bumping into eachother around the triangle.

I really like the point he's making, just not sure if this triangle notation is the most practical.

Maths notation is needlessly complex [video]

Related Stories