Maths notation is needlessly complex [video]

This presentation misses a more fundamental point. It&#x27;s not about the <i>notation</i> at all, it&#x27;s about understanding that EXPT, SQRT and LOG are <i>functions</i> with a particular relationship to each other. That relationship can be expressed using a two-dimensional spatial notation, but that doesn&#x27;t really help you understand the concept at all because there are a lot of <i>different</i> relationships that are naturally described by putting three things in a triangle.<p>What you <i>really</i> want students to understand is that expt, log, and nth-root are <i>functions</i> that are related in the following way:<p><pre><code> expt(b, n) = x log-x-base-b(x, b) = n nth-root(x, n) = b </code></pre> It&#x27;s really that simple. No fancy notation needed. In fact, fancy notation <i>always</i> 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&#x27;re writing math with pen and ink it&#x27;s faster and uses less paper to use Greek letters and spatial relationships than full words. But when you&#x27;re on a computer, it&#x27;s easier to write out the names of functions, and that is actually a better impedance match to people&#x27;s natural mental processes, which involve <i>language</i>.<p>[UPDATE] I would like to revise this: not everyone thinks in language. But everyone <i>communicates</i> in language. For <i>communicating</i> mathematical concepts, language is the best tool we have. There&#x27;s a reason that the symbology in math papers is invariably wrapped in natural language. It&#x27;s the same reason that the video has a narration. It wouldn&#x27;t make any sense otherwise.

The problem is not that the notation exists and that there are multiple ways to say the same thing, it&#x27;s that students are forced to memorize all of it for its own sake.<p>In fact, there aren&#x27;t just &quot;three&quot; 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&#x27;ll never be able to have &quot;just&quot; one way to say a thing. You can have your favorite, sure, but all notational preferences are aesthetic.<p>FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how &quot;mirroring&quot; 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.<p>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&#x27;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&#x27;re doing is thin.

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&#x27;s the first part that is encountered, it is also the most derided one.<p>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.<p>However, I don&#x27;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&#x27;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,<p><a href="http:&#x2F;&#x2F;neuralnetworksanddeeplearning.com&#x2F;chap1.html#eqtn7" rel="nofollow">http:&#x2F;&#x2F;neuralnetworksanddeeplearning.com&#x2F;chap1.html#eqtn7</a><p>you&#x27;re baffled because you don&#x27;t know what those symbols mean, there&#x27;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&#x27;t think this path to enlightenment could be shortened much more.<p>I guess I am being very old school and reiterating that the royal road everyone&#x27;s been looking for for the past couple millenia just doesn&#x27;t exist.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Royal_Road#A_metaphorical_.E2.80.9CRoyal_Road.E2.80.9D_in_famous_quotations" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Royal_Road#A_metaphorical_.E2....</a>

&quot;It&#x27;s like it&#x27;s a different language!&quot; (incredulously)<p>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&#x27;s misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 &quot;in the same way&quot; (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn&#x27;t give ourselves language to express both &quot;the difference between 8 and 2 is 6&quot;, and &quot;2 more than 6 is 8&quot; because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts.<p>The second example, the 8^(1&#x2F;3) is not even equal to 2, it&#x27;s equal to three values, two of them are imaginary. It&#x27;s important to have notation for &quot;the thing that when you cube it is equal to 8&quot; 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 &quot;triangle of power&quot;? The positive branch cut? Okay, now you have to explain that you&#x27;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&#x27;t even a function.<p>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.<p>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.

Math notation is an endlessly interesting subject, but I must say I wasn&#x27;t very impressed with that idea.<p>If we&#x27;re going to give up some notation to adopt another, it would need to have some serious and obvious advantages.<p>I waited for that throughout the video. The author consistently seems to assume that what he&#x27;s saying is &quot;intuitive&quot;. It isn&#x27;t.<p>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.<p>Certainly, using our visual apparatus to help us complete computations without having to think about it can be appealing, but that&#x27;s probably not the most convincing example.

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?<p>Not to mention the existing mathematics you miss out on by using this.<p>That being said I think it&#x27;s a fantastic tool to quickly explain the relationship between the notations (the first half of the video).

The video doesn&#x27;t give any examples of complex formulae written using their &quot;triangle of power&quot; notation. This may be useful for teaching mathematics, but it&#x27;s not clear that it scales. Much of the benefit of the notation could come from simply writing x⁽¹&#x2F;²⁾ instead of √x. (Annoying, unicode doesn&#x27;t have a superscript slash. &quot;(&quot;,&quot;)&quot;, &quot;=&quot;, and all the digits are available in superscript, but not &quot;&#x2F;&quot;.).<p>A huge hassle in moderately advanced mathematics, where new domain-specific operators are introduced, is ambiguous precedence of operators. There&#x27;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&#x27;s videos on machine learning.<p>It might be useful to always parenthesize in textbooks. Teach kids to always write &quot;log(n)&quot; instead of &quot;log n&quot;. After all, how does &quot;log n × m&quot; parse? Is there an official standard on that? If so, where?

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:<p><pre><code> &#x2F;U\ &#x2F;R*I\ </code></pre> (Where U = voltage, I = current, R = resistance.)<p>I&#x27;m not sure it helped <i>me</i>. Anecdotal evidence: Just now, as I was trying to figure out which letter goes where, I was actually thinking in terms of what&#x27;s going, as in &quot;if at constant voltage I increase the resistance, the current should drop, ok, so I = U&#x2F;R?&quot;.<p>So, in conclusion, I don&#x27;t know which one is &quot;better&quot;. Most likely, different people think in different terms and require different methods of learning, so if there&#x27;s another way of explaining things I think that&#x27;s good, isn&#x27;t it?

Most of the way through the video, the narrator is saying things like &quot;the student can easily see that...&quot; No, she can&#x27;t! Not if &#x27;see&#x27; 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.<p>The video is also plagued with distracting visual effects (there are some that are worthwhile, but most are not.)

I like the problem statement, I like the use of a 2D representation, but the triangle symbol as drawn in that presentation is <i>gargantuan</i>. I think the symbol needs more work.<p>One thought that comes to mind is that I&#x27;m not convinced the bottom part of the triangle should be there. It implies a direction connection that I&#x27;m not sure exists. Removing that overlaps with ∧, logical and, though.<p>Another that comes to mind is that I can&#x27;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&#x27;s still not like leaving those blank turns the integral into a differentiation (the opposite), depending. I&#x27;m sure there&#x27;s something else somewhere up in math, too, but nothing your average student will hit.<p>Similarly, note that filling in all three corners of that symbol is actually an <i>equation</i>. I&#x27;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&#x27;s not add to that. Again, somewhere up in higher maths than I&#x27;ve gotten to there may be symbols that constitute entire equations, but it won&#x27;t be something most students see.<p>Also I think once that symbol is being shown with full expressions rather than cute little single-digit numbers or single-letter variables, it&#x27;s going become very difficult to deal with.<p>I think there&#x27;s something to this, though. I&#x27;m criticizing in the spirit of continuing to move forward. (I&#x27;m aggressively hostile to the idea that math notation is perfected and debating better alternatives is some sort of betrayal or something.) Personally I&#x27;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&#x27;t literally draw it out in the formula.

While HN has definitely identified the problem with adding yet another notation, I think it&#x27;s fairly straightforward to to remove the &quot;root&quot; operator. Simply use the exponent notation with a power of &quot;1&#x2F;3&quot;.<p>This gets directly at the core concept of what a &quot;root&quot; really is, and it&#x27;s straightforward to manipulate with addition&#x2F;multiplication of groups of roots using the normal arithmetic methods.

Beautiful, elegant, and I disagree it should be taught.<p>∆ Is used for delta&#x27;s so frequently that I could see some nasty issues arising in notes for higher level maths.<p>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&#x27;t remember it immediately, but I think this guy didn&#x27;t read Feynman&#x27;s biography where he discusses the problems with creating notation.

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&#x2F;n) - but it does make some equations extra clear because there are so many roots that often need to be taken.<p>So, don&#x27;t complain about the notations - the triangle would be far more annoying to write equations with if that&#x27;s the only thing you use. The notations are like &#x27;helper&#x27; 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&#x27;s all the same thing (or all related, anyhow) but keep the notation all the same.

This is futile. It&#x27;s just another standard. xkcd said it best: <a href="https:&#x2F;&#x2F;xkcd.com&#x2F;927" rel="nofollow">https:&#x2F;&#x2F;xkcd.com&#x2F;927</a>

In one of my (middleschool?) science classes, my teacher explained the density equation in terms of a triangle. Of course, we didn&#x27;t adopt it into our notation. But I think that just having shown the class the diagram helped a lot of my peers.<p>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 &quot;Thanks. I was probably going to get a zero on the next quiz. I might actually pass now.&quot;<p><pre><code> d = m &#x2F; v v = m &#x2F; d m = v * d . &#x2F;m\ &#x2F;---\ &#x2F;v | d\ .-------.</code></pre>

It&#x27;s even worse than shown in the video. Consider:<p>sin^2(x) = (sin(x))^2<p>sin^-1(x) = arcsin(x)<p>Switching to triangle notation would help remove this confusing overloading of the superscript operator. The only drawback I see with triangle notation is there&#x27;s no obvious way to type it on a single line.

And we should all learn esperanto because it is so much more rational and regular than English. Somehow it doesn&#x27;t work that way.

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.<p>You&#x27;re reading a document and it has some funny triangle thing you don&#x27;t understand? Click on it to get a menu of alternative representations, and you see it can be swapped to &quot;log&quot; notation. Further, you go to your reader preferences and add a rule &quot;whenever you see this triangle thing, show me it as a log&quot;.<p>A student finds some crusty old document with funny &quot;heart monitor&quot; symbols, clicks on them and finds they&#x27;re just a particular kind of power triangle. They update their preferences to replace those symbols with power triangles.<p>Of course, holy wars rage on mailing lists about whether the default rules should convert &quot;tau&quot; to &quot;2pi&quot; or &quot;pi&quot; to &quot;tau&#x2F;2&quot; ;)<p>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&#x2F;means. Thankfully there are some attempts at this (e.g. OpenMath), but they don&#x27;t seem to be very widely used.

- 2^3, log_2(8) ... why do we have 3 different ways ? Because, playing with tautologies is the whole point of mathematics (and language) !<p>- The new &quot;notation&quot; is essentially expressing the same language in a different alphabet.<p>I was hoping for some smack talk about partial derivatives (see SICM) or about the absolute proliferation of symbols in Differential Geometry...

My own personal opinion on this, is that it&#x27;s all well and good but it&#x27;s not that students are &quot;Made&quot; 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.

It&#x27;s a work in progress <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Zenzizenzizenzic" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Zenzizenzizenzic</a>

I agree that the notation can be a problem, but it has to serve a number of different uses:<p>1. Communication between mathematicians.<p>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.<p>2. A shorthand for key concepts and relations to other math users.<p>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&#x27;s a large hurdle, but then you&#x27;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.<p>3. A method of communication between non-mathematicians (like physicists or engineers).<p>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&#x27;t a large problem at the start, but then you&#x27;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 &quot;weeded out&quot;.

While this post limits itself to basic mathematics, the notations in group theory and advanced linear algebra JUST DON&#x27;T MAKE SENSE. Please read the &quot;Brakerski’s Homomorphic Cryptosystem&quot;, section 3.3, of this ( <a href="https:&#x2F;&#x2F;eprint.iacr.org&#x2F;2015&#x2F;137.pdf" rel="nofollow">https:&#x2F;&#x2F;eprint.iacr.org&#x2F;2015&#x2F;137.pdf</a> ) paper.<p>I had to read it 10 times to truly understand what the notation was trying to say! Fail.

i feel like the usual best practice for studying math is to use the same concepts in as many different settings&#x2F;with as many different terms for the same thing as possible. By this I mean, literally, knowing the subject so well that there&#x27;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&#x27;s even better, for instance.

Presumably people have been rightly complaining about this for 100&#x27;s of years? I&#x27;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

Let&#x27;s see how this notation holds up to nesting...

This also brings to mind other attempts to clarify notation, such as <a href="https:&#x2F;&#x2F;mitpress.mit.edu&#x2F;sites&#x2F;default&#x2F;files&#x2F;titles&#x2F;content&#x2F;sicm_edition_2&#x2F;chapter009.html" rel="nofollow">https:&#x2F;&#x2F;mitpress.mit.edu&#x2F;sites&#x2F;default&#x2F;files&#x2F;titles&#x2F;content&#x2F;...</a> and <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Geometric_algebra#Relationship_with_other_formalisms" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Geometric_algebra#Relationship...</a>

Not this crap again.<p>The notation is the way it is because it works for mathematicians. Works well. It&#x27;s not that complicated, but it does take some time.<p>Just learn the god damn notation and quit whining about it. Ugh.

I don&#x27;t like it for a number of reasons.<p>One relatively simple one is that it doesn&#x27;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&#x27;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?<p>Second, it&#x27;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&#x2F;n). All of the normal rules for roots follow from the rules for exponentiation (and fractions) immediately. You don&#x27;t need a third side to the triangle, that&#x27;s just adding extra complications - all you really need is exponentiation and logarithms and a way of representing that they&#x27;re inverse operations in a certain sense.<p>So there&#x27;s actually a simpler way to express all this in a notation that&#x27;s much more similar to what we&#x27;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&#x27;s just really obvious.<p>Most of the basic arithmetic operations can be written using binary infix operators, e.g. +, * and &#x2F;. It turns out, exponentiation and logarithms can be too. In fact, if you&#x27;re typing, exponentiation already is.<p>Let x raised to the nth power be written as (x ^ n) (note this is essentially exactly the way it&#x27;s already typed out, only I&#x27;m using some unusual extra white space to emphasize that we&#x27;re treating ^ as an infix operator). It&#x27;s a little upward arrow that says scale x up by n (exponentially).<p>And let log base n of x be written as (x v n) or possibly (x \&#x2F; n). It&#x27;s a little downward arrow that says scale x down by n (logarithmically).<p>This makes the two operations work in a way that&#x27;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.<p>Lots of familiar relationships carry over, e.g. Note that ((x * n) &#x2F; n) = x. Similarly ((x ^ n) v n) = x. And where (n * (x &#x2F; n)) = x, it&#x27;s also the case that (n ^ (x v n)) = x.<p>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&#x27;re structurally almost exactly like the rules for operations you&#x27;re familiar with. You already know those when you learn about exponentiation, so you don&#x27;t have to learn new and weird geometric relationships.<p>All of those nice properties there follow from the fact that exponentiation is just the next operation in the sequence of hyperoperations (see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hyperoperation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Hyperoperation</a> ) after multiplication.<p>Introducing this weird three place operator actually masks the underlying simplicity of exponentiation and its inverse.

One practical concern I have for this is that you&#x27;re going to need bigger triangles to avoid things bumping into eachother around the triangle.<p>I <i>really</i> like the point he&#x27;s making, just not sure if this triangle notation is the most practical.

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

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?

:( <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=12125756" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=12125756</a>

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

I can&#x27;t help but wonder if the triangle of power is a Zelda triforce reference.