And it's because you need long division to do division in algebra

After looking at the English Wikipedia I found that

a) what I learned in school is long division

b) the notation with the dividend and the divisor separated by a parenthesis and the result on top of a horizontal bar is totally alien to me

c) I've never seen short division before

d) the German Wikipedia doesn't even have an article about short division

Edit: although if someone else can explain it better I'd really like to know.

From what I have searched,(quickly mind you), it seems to be a mind set.

https://www.youtube.com/watch?v=UIKGV2cTgqA

the "short" notation relies on the ability to squeeze the remainder in between the digits of the dividend. if you have a divisor greater than ten, it's possible to have a remainder with more than one digit. two or more remainder digits is a lot harder to write legibly between the digits of the dividend. you also end up subtracting larger numbers in your head, which is error prone. finally, most people only memorize the multiplication table up to 12x12 or so. once they can't simply do a lookup into their memorized table, they switch to more of a "guess and check" approach, where they will inevitably have to cross out or erase their previous work.

in the end, it introduces a lot of unnecessary opportunities to make mistakes and hides stuff that people can't do reliably in their heads.

I went to school in Sweden where we do teach all this things, but in a strange order.

I have a PhD in computer science but I cannot understand long division.

I think you're trolling us a little: you could sit down with pencil and paper and think about how it must work and 20 minutes later you'd understand it, surely.

I have a similar mental block for standard deviation - I always seem to have another ‘but why?’ question that eventually people answer with ‘because!’ and then we’ve both given up.

This means the quantity of mean/variance depends on your choice of units, whereas the quantity of mean/standard deviation. Is constant no matter the units.

Why do we care about variance? Because it is a nice and linear property. This means it is easy to calculate and manipulate.

Moreover, the variance / standard deviation very nicely describe a Gaussian distribution (bell curve). This is a very important distribution because of the law of large numbers. Because we see bell curves so often, it is nice to have tools (std.dev) that work well with these curves.

It should be noted that, in optimization problems, there can be reasons to try and minimize something other than variance. We tend to pick the variance / std.dev because it is familiar, easy to work with, and very efficient. Notably, the derivative of variance tends to be linear, which makes it pretty efficient to use in gradient descent.

I think the OP's frustration in how computation of standard deviation as a quantity is just taught by rote in a cookbook manner is quite understandable. If one's not thought more in depth about distributions and estimation of their parameters, it really makes little sense.

Whether the statistics should be taught more in depth is another question. The current state of the mathematics of statistics and probability is such that it needs quite heavy tools to do rigorously. And countless of hours of getting to know the quirky inconsistent notation.

But until it is taught more rigorously, more intuitive measures of dispersion like MAD make a lot more sense for most applications. And I don't think it would be an unreasonable ask for the statisticians to lay out the "porcelain" for working with the more intuitive, if somewhat more analytically inconvenient measures. A bit like we don't require programmers to understand how transistors work.

https://web.stanford.edu/class/ee486/doc/chap5.pdf

Edit: standard deviation is just a fancy name for root mean squared over the sample offsets. The square is there to deal with the fact that there are positive and negative offsets. The mean is the whole point of the exercise, and the root is to negate the fact that we squared earlier.

This is how I always find maths and particularly statistics - I always just want to answer ‘but why’ and then people lose interest in explaining after a while.

You can. It's called the mean absolute deviation. Which one you use depends on your application. If you just want to give a summary of how spread out the data is then either would work. Lots of people say that we should use the mean absolute deviation as the default rather than the standard deviation.

I too have a problem with math tools passed down without any surrounding context - without telling why are we using this formula, instead of any other variant from the family of formulas that would satisfy the same goals. I'd have much easier time dealing with statistics in school if someone told me that a) deviation with ABS instead of root-square is also a thing, and b) we use the root-square one because it amplifies offsets from the mean.

Also one of my peeves. However over the years I've come to realize that in addition to teachers who omit the context, there is also a class of student that actively doesn't want to hear the context. I'm not sure exactly why this is, but I see it in my immediate family quite a bit. A sort of <cover ears, lalalala...too much detail> kind of thing perhaps due to difficulty taking on too much information?

I feel like the standard deviation is a black box to basically everyone without statistics degree. It also bothers me that it is so widespread, because almost nobody know how to interpret this value. It seems to me like most scientists treat the standard deviation as a magical number that allows to compare spread between datasets and otherwise doesn't mean anything on it's own. I feel like a lot more insight could be gained from the mean absolute deviation.

I'm also curious how are scientific insights affected by defaulting to something that is extra-sensitive to outliers. The effect can't be big per single research, but in collective?

Also, consider that the correlation coefficient consists of similar formulas. They all look like the theorem of pythagoras. This ties in nicely with the fact that observables can be seen as having angles in between them where the correlation coefficient is the cosine of these angles.

You can read up on the central limit theorem if you want to understand more.

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

Put simply, standard deviation is arbitrary for most people.

This is also where I get frustrated.

I'm taught about the CLT but every place I think I might use it, the samples aren't really IDD when you look hard enough, so it doesn't apply.

Computer benchmarking (which is where I'm usually trying to apply statistics) is the classic example. One iteration changes the state of the computer for the next iteration. They aren't remotely IDD, but people still try to claim the CLT. Seems like all their stats beyond that point is broken.

I'd assume that computer benchmarks have very low correlation between runs, unless you leak ram or your components don't keep a stable temperature. So in that case the CLT seems to apply just fine.

And they are not in a computer program. The state of, to just give one example, the processor cache depends on what happened in the previous run.

We have all this rigour in how statistical techniques are applied, and we make black or white decisions based on the numbers that come out of them, but at their very foundation we're saying 'whatever I'll just pretend this rule applies it's probably fine'. How can we build on that flawed foundation? I don't get it!

"All models are wrong, but some models are useful." Just remember that there's some imprecision in the answer a model gives you, and keep that fact in mind when applying that answer to make real-world decisions.

e.g. https://socratic.org/questions/what-is-the-difference-betwee...

There are many "right" metrics - this is one of them. The big picture is we want a measure/idea of how spread out the data is. One can come up with many ways to do this, and absolute value and root mean square are two of the most common. They correspond to L0 and L2 norms. There is also the L-infinity norm (just use the farthest point).

The L2 norm is the most complicated of the 3, but as people pointed out, it is one of the easiest to use analytically because of its nice mathematical properties.

Have a look at https://en.wikipedia.org/wiki/Statistical_dispersion for other measures.

To illustrate, perhaps an example closer to home:

If someone asked you to tell them "how many lines of code in this project?" you'd start wondering things like "do I count comments?", "do I count dependent libraries?", "perhaps some metric that is equivalent of a statement count but not strictly counting lines?" and on and on.

Statistical measures are like that. Someone with a decent knowledge of the problem space came up with the best stab at how to characterize the data.

Speaking of language, have you ever thought about how all names and grammatical structures are arbitrary as well? There are parallels.

My larger point is that, something can be arbitrary and still be meaningful. If people agree on a common meaning, it formalizes a communication protocol that allows for much better information sharing.

This means that any distribution with standard deviation of X will behave as a normal distribution with standard deviation X when you have enough of them. Or in other words, any distribution with standard deviation of X will behave like any other distribution with standard deviation of X when you have enough of them. There is no other measure like this as there is no other equivalent to the central limit theorem. Therefore it makes sense to have this as a universal measure of random processes.

Now that we have a normal distribution, the mean is an obvious metric to pick because it captures the notion of 'middle' in a useful sense. Then we have proofs that we can characterise the normal with the mean and 1 other parameter (normal normal) or a matrix (multivariate normal). We call that the standard deviation^.

The exact formula wasn't a coincidence. The normal can obviously be characterised by the mean and a statistic from another formula. There were a bunch of experiments tried (eg, using |x| instead of sqrt[x^2]) but it turned out that sqrt[x^2] had some other nice property that minimised some sort of error so they went with it as a standard. I forget what one, might be error of estimating the true parameters from a sample or similar.

We could characterise the the normal as an infinite sum or something quirky, but when people say 'easy to work with' they mean instead of a function or something quirky we can simply pick a number.

Standard Deviation isn't as important when working with non-normal distributions, although I think it still turns out to be useful. But its importance is that it characterises a normal apart from the information captured in the mean. I'm not a mathematician, YMMV, could be wrong, standard disclaimers.

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

^ I'm not going to edit this but it occurs to me that we call it the Variance. Same thing as std. dev in my opinion.

' There is a fairly subtle observation to make - Normal is characterised by mean and std. dev, but the most efficient ( https://en.wikipedia.org/wiki/Efficiency_(statistics) ) unbiased estimator of the std. dev of the population is the adjusted std. dev of the sample. Therefore, accounting for mean, you can't get a more efficient characterisation of the normal than mean & std. dev. Ie, if you picked a formula other than std. dev then the most efficient estimators to characterise it would still be mean and std. dev. Don't recall if there are equally efficient choices but I think that proves there are none better.

That might have been the logic for why std. dev was chosen. Just a guess.

Unless you try to solve more problems yourself, you won't understand the advantages or disadvantages of some methods in solving the problems. So the real answer is: construct the examples, compute and compare, don't approach it "philosophically."

All the methods used today survived because they were the solutions to some problems. It doesn't mean that everybody applies them properly: for that you have to get some experience yourself.

Learning about the historical development of the methods is also for me satisfying experience: e.g. logarithms aren't a concept devised for philosophical purposes, Napier developed them to save astronomers time and limit "slippery errors" of (their) calculations:

https://www.thocp.net/reference/sciences/mathematics/logarit...

"I have a PhD in computer science but I cannot understand long division"

For that, personal experience in devising the examples, evaluating them and comparing the results is the only way to get the understanding. One can't complain that one doesn't understand Greek texts if one personally never tried to learn the Greek letters. One could have learned "about" Greek but at the end one hasn't done the inevitably necessary preconditions to actually read Greek texts.

The error in thinking is believing that because one already learned about something else somewhere else he should somehow "understand" something without doing the necessary work on that something. The solution is "make your own homework" to fill the gaps.

The topic of what, how and why children learn in schools is much more complex.

Edit: Also, now reading the Wikipedia article: https://en.wikipedia.org/wiki/Long_division I do understand why people don't understand it: the taught notation for the process, used in many countries, is utterly confusing to me, so my reason for confusion when looking at various notations is not that I don't understand the nature of the method but that I haven't spent time analyzing what was built on top of the basically simple idea -- I can also "understand" all the notations but I'd also need more time for that. Still, having enough experience, I can claim that I understand the idea and the process even if I never learn all the notations and conventions in different countries (which part of the process you write where, what you write and what you don't when etc). The idea is simple, the conventions enforced don't have to be. Those who only learned the conventions maybe never invested any time to figure out what's behind the conventions: what is actually being done.

Standard deviation is mathematically easier to work with though and has nice properties e.g. you can compute the standard deviation of combining a bunch of independent things by combining their individual standard deviations.

6240 / 5

= divide(6240, 5)

= divide(1240, 5) + 1000

= divide(240, 5) + 1000 + 200

= divide(40, 5) + 1000 + 200 + 40

= 1000 + 200 + 40 + 8

= 1248

----------

2. Regarding standard deviation, one of these bullets might help:

- The normal distribution has exactly one shape, centered at x=0. But it's useful to apply two transformations to it: translation and horizontal stretch. To translate a distribution left/right, change its mean. To horizontally stretch a normal distribution, change its standard deviation.

- Mean has units of length. Standard deviation has units of length. They tell you where the normal distribution is offset, and how wide it is. Mean and standard deviation are just measuring sticks/rulers for normal distributions.

- When people talk about the standard deviation with any arbitrary data, they're usually assuming the data is normally distributed. If the data is not normally distributed, standard deviation no longer refers to the width of the normal distribution, so we lose that visualization.

- With non-normally distributed data, the standard deviation is still useful as an analytical tool because taking the (sqrt of the) summed squares still gives us a number that grows as the data spreads further apart or if the distribution grows wider. There's a center point for the data (the mean), so to ensure you're measuring the overall spread of the data, you subtract the center point off each data point before squaring them. In other words, you're squaring deviations from the mean. And unlike the sum of absolute differences, the sum of squared differences (variance) is differentiable. Differentiability is a great property, so this is the standard way to compute a sum of deviations from the center.

also maybe they are specific about they way they do long division in the US? it was a surprise to me in college too and i have a french background. it's not how the french system does long division...

Take a course in ring theory if you want to see how all of this machinery gets built up.

    func Longdiv(A, B) {
        // Assume A > 0, B > 0.
        N = floor(log10(A))
        Q = 0
        R = A
        while (N >= 0) {
            // Invariant: A = B * Q + R
            // Invariant: B * 10^(N+1) > R
            C = 0
            while (B * (C + 1) * (10 ^ N) <= R) {
                C += 1
            }
            // C is the largest C such that  B * C * 10 ^ N <= R
            R -= B * C * (10 ^ N)
            Q += C * (10 ^ N)
            // R is still positive.
            // Invariant A = B * Q + R is maintained.
            // We know B * 10^N > R because otherwise we
            // would have picked a larger C.
            N -= 1
            // Invariant B * 10^(N+1) > R is maintained.
        }
        // By loop invariants, we have
        // A = B * Q + R and B * 10^0 > R.
        return (Q, R)
    }

I mean your very first line has a logarithmic operation that was not mentioned anywhere in your text or your comment! Just pops in there out of nowhere. I guess it's about the number of decimal digits? But why? Why are we doing things in decimal?

A dense ten-line algorithm like this seems far more complicated than other algorithms we try to get school children to memorise.

I think this is maybe the crucial part that you might be missing. All this long division (and long multiplication) stuff works on base-n represenations of numbers. I.e. if you have an number like 3376, it is actually a short hand for

  3*10^3 + 3*10^2 + 7*10^1 + 6*10^0.

  (3*10^3 + 3*10^2 + 7*10^1 + 6*10^0) / 4 ==

  3*10^3/4 + 3*10^2/4 + 7*10^1/4 + 6*10^0 / 4 ==

  (3*10^3 + 3*10^2)/4 + 7*10^1/4 + 6*10^0 / 4 ==

  (33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==

  (33)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==

  (32+1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==

  (32)/4*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==

  8*10^2 + (1)/4*10^2 + 7*10^1 /4 + 6*10^0 / 4 ==

  8*10^2 + (17)/4*10^1 + 6*10^0 / 4 ==

  8*10^2 + (16+1)/4*10^1 + 6*10^0 / 4 ==

  8*10^2 + 4*10^1 + (1)/4*10^1 + 6*10^0 / 4 ==

  8*10^2 + 4*10^1 + (16)/4*10^0 / 4 ==

  844

The idea of long division is just that to solve A/B, we can find another problem C/B we know the answer to and break it into the smaller problem A/B = (A-C)/B + C/B.

For example, to solve 742/13, the normal way is to observe that 13×5=65 is as close as we can get to 74 without going over, so we can reduce this to

    742/13 = (742 - 13×50)/13 + 13×50/13
           = (742 - 650)/13 + 50
           = 92/13 + 50

But you could also use 13×4=52 and work out

    742/13 = (742 - 520)/13 + 40 = 222/13 + 40
    222/13 = (222 - 130)/13 + 10 =  92/13 + 10
     92/13 = ( 92 -  91)/13 +  7 =   1/13 +  7

We’re subtracting off multiples of 10^N because that’s easy when you write your numbers in base 10. Try subtracting off multiples of 9^N, or multiples of N!, or some other choice, and you’ll see why.

That could be meant shallow or deep. The shallow answer is because that's all you need to do to function in society (and conversely, while arguing with a police officer about the presumed base on speed limit signs may be fun, it is also pointless).

The other answer probably needs to explore the question a bit further. Perhaps a good starting point is the fact that we almost universally share the physical characteristics of having ten fingers.

His method for polynomials/derivatives is also interesting if you watch until the end, dead simple calculus.

https://commons.wikimedia.org/wiki/File:LongDivisionAnimated...

In essence, repeated subtractions. The blank areas to the right of each number are really just zeros.

35/350 (to simplify) Once is 325, twice is 300, X, X, X, we end up at 10 times is 350 divisible by 35.

I'm wondering if I'm missing something (and if I am I'll own up to it lol)

NN = (10d + e)(10d + e) = 100dd + 20de + ee

Group the digits of the square by two, and add extra values to the "divisor" so that the extraction of the square root in effect undoes the squaring above.

See, for example, https://sciencing.com/calculate-square-root-hand-5081134.htm...

https://www.solipsys.co.uk/new/SquareRootByLongDivision.html

https://en.m.wikipedia.org/wiki/Newton's_method

[1] https://en.m.wikipedia.org/wiki/Hero%27s_method

https://en.wikipedia.org/wiki/Singapore_math#Bar_modeling

For anyone interested in going down the rabbit hole of those infinite levels.

Afaikr it's just division, how man X are in y.

       083.28
         2.26...
       ___
    7 |583.00

Long division:

       083.28
         2.26...
       ___
    7 |583.00 [<-dividend]
       56     [=8x7]
       --
        23    [2 from subtracting answer to 8x7 {8 is put in answer line}, 3 from the dividend]
        21    [=3x7]
        --
         2.0  [2 is remainder, 0 from dividend]
         1.4  [=2x7 is remainder, 0 from dividend]
         ---
          .60
          .56 [=8x7]
           --
            4 [is the remainder in 100ths]

I'd do an algebra example but it's a pain in the arse just using ASCII.

Try it with 1/7?

The same process still works if you want to calculate the decimals though. Just pretend you're doing (1000000/7) * (1/1000000) or however many digits you want.

My 17 year old was struggling with the division algorithm they were teaching in Elementary school at the time (late 2000s). I can't remember the details but it was something I had never seen before that was a step up from throwing poop at the wall.

I showed her how to do long division and she still uses it to this day. It's simple and there is a history of work to easily identify errors.

The homework scene from The Incredibles 2 really hit home for me. I feel like book publishers development new methods or ways to teach the methods as a means to sell new editions of books, not because the new ways are better. This causes a rift in society because people no longer share a common understanding of a subject.

It’s completely different than the way I was taught so I have to invest in figuring it out but it’s led them extremely quickly through multiplication & division and into algebra.

I’d characterize it as “multi-algorithmic”. They come at it from a variety of algorithms, assumedly so that each student can find one that works for them.

As someone who had to teach themselves a different algorithm than I was taught many painful years later this resonates.

Perhaps the rift was always there and you ended up in the lucky side of it?

But in reality the common core way explained the actual algorithm and the reason it worked so much better, and further, the only reason it appeared more complicated was because the people who were targeted by the meme already knew long division. If you took someone completely new to both methods, they’d find the more “complicated” algorithm far easier, because a large part of its complexity was explaining why the division worked, whereas long division was about memorizing a process.

If you actually understand the math picking up the common core methods is minutes of work at worst.

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

Just show long division for “proofs” while introducing the concept of division, and let kids use a calculator for division after that (supplemented with calculator-free drills). That way there will be more lesson time for advanced topics. Same thing with long multiplication. Show the proofs using long multiplication, then have kids do progressively harder multiplication drills where all the math is done in their head. In fact, kids might grok first principles better that way.

https://apps.apple.com/us/app/dragonbox-algebra-12/id6344441...

https://news.uchicago.edu/story/female-teachers-can-transfer...

He got stuck on a level where he had to solve a simple equation, and tried to do it in the “wrong” order. He eliminated an unknown from one side first, and then the system wouldn’t allow him to do the type of operation necessary to complete the other side, maybe because that particular technique hadn’t been taught yet. It would have been a perfectly natural way to solve the equation on a blackboard. I could see they probably wanted him to do the other side first.

My nephew tried the same basic approach 3 times in a row, and then gave up and threw the tablet down and went off to play with Lego or something.

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

Children are mostly idiots. Education is the process of painfully breaking them out of their idiocy.

Bonus points if you implement a long division algorithm that works on strings in your favorite language.

You're right that it could be a fun programming exercise, though

I can understand if most people shared your frustration, but not a PhD holder. Your degree basically says that you're capable of doing long, intensive research on arcane topics.

What am I missing here?

I have impaired executive functioning and this is basically my experience in tying shoelaces, I struggled to accommodate the mental model of the process myself so I sought out others so manage it to try and learn from them.

What instead happened was exasperation on their part and frustration on mine, so after repeating that a few times a year until I was 22 I just stopped and bought shoes without laces.

When you go at it for long enough without succeeding it can make the experimentation not very fun or interesting.

I don't think I'd call it a red flag for a PhD to have fatigued themselves out of interest in a problem, it's something that happens to everyone at some point.

37>1, 37>12 but 37*3<=123.

Write down 3 (meant for 3000) and start again with 123456-37x3000 as the new dividend.

I wish the algorithm was taught by writing the full 3000 instead of only the digit 3; this appears to be a source of major confusion and to hide the underlying additions/multiplications.

Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!

I can distinctly remember learning long division as a kid, and I wonder why this isn't immediately taught afterwards as the logical progression

Have you heard of "chunking"?

This video shows old style long division (and you're right, the method is confusing). It also shows an easier method that is based on understanding numbers rather than just applying an algorithm by rote. https://www.youtube.com/watch?v=K2rBaSj2L74

Here's another explanation: https://www.theschoolrun.com/what-is-chunking

(All these methods had the same asymptotic complexity. Mine was slower by a small constant factor; on the other hand it actually made sense. It was just repeated subtraction!)

I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.

Multiplication = repeated addition

Addition = a form of counting

Maybe they need to explain it in those terms. Or use pictures.

I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.

Just another anecdotal data point but my experience was completely opposite yours.

I was taught long division in the 4th grade (age 9) but wasn't taught a proof of why it works until a college algebra course (age 19).

For whatever it's worth a different type of long division is taught now under common core. The common core math stuff that I've seen is impressive - there is more focus on teaching things in a way where comprehension of deeper concepts should be possible. It's a shame there's so much fear of it from parents who are upset kids are learning differently than they did.

Eg, say you are dividing 76543 by 5. The normal procedure is to say something like:

- 5 goes into 7 once. The first quotient digit is 1.

- 1 times 5 is 5, write that under the 7, and draw a short line under the five

- 7 minus 5 is 2; write that under the line you just drew

- the next digit is 6, so write that next to the 2 you just drew, ie, 26

- 5 goes into 26 five times; put that up in the quotient, etc

That minimizing the writing, but it hides what is really happening.

Really what is going on is this. How many times can you subtract 5 from 76543? One procedure would be to subtract five over and over until the remainder is less than five. The number of times you subtracted is the quotient. But in this example you'd have to subtract more than 15000 times. Instead of subtracting 5 at a time and incrementing a count, it would be faster to subtract 50 at a time and increment your quotient by 10 until the result is less than 50, and then switch to subtracting by 5s and incrementing by 1 as before. If you can see why that works, then you realize it would be even faster to subtract by 500 at a time and increment by 100, then switch to subtracting 50 at a time and increment by 10, then subtract 5 at a time and increment by 1. Etc.

So rather than minimizing the amount of digit copying, it would be better at first to write everything out. In this particular case:

- write the dividend: 76543

- remove groups of 50000. we can see we can subtract 50000 from this once; the leading digit of the quotient is 1

- write 50000 (150000) under the 76543

- subtract, writing 26543 (this is the current remainder)

- remove groups of 5000. now we see we can subtract 25000 from this, so the next digit of the quotient is 5.

- write 25000 (55000) under the 26543

- subtract, writing 1543 (current remainder)

- remove groups of 500. 500 goes into that 3 times, so the next digit of the quotient is 3

- write 1500 (3500) under 1543

- subtract, writing 43 (current remainder)

- remove groups of 50. it is larger than the current remainder, so the next digit of the quotient is 0.

- remove groups of 5. 5 goes into 43 eight times, so the next quotient digit is 8

- write 40 (85) under the 43

- subtract. the final remainder is 3.

The sequence of quotient digits was 15308.

The standard way long division is taught, the child skips writing down all the digits of the remainder and it is confusing as to why this all works.

$SIMPLE_CONCEPT that you’re trying your best to understand is easy if you think of it as a specific example of $MORE_COMPLICATED_CONCEPT that you surely can’t understand without understanding $SIMPLE_CONCEPT

E.g.: I distinctly remember that many of the original articles about the Amplituhedron spent like a page explaining what a triangle was, then jumped straight into functions over higher-dimensional analogues without explaining any of the intermediate steps. Something-something-logarithm-of-a-simplex.

I can grok what a simplex is, but they literally just hand-waved over the bit where they take a logarithm of it in some way.

The page of explanation of what a triangle is was of zero assistance.