Eclectikus told us virtually this 3-week quondam video
proposing a novel notation for the exponentiation based on triangles. Influenza A virus subtype H5N1 similar number was discussed 5 years agone at Mathematics Stack Exchange. Well, it's amusing in addition to sometimes useful to invent option notations in addition to I had to do it many times inwards the past times (especially whenever I needed to bargain with lots of algebra in addition to hundreds of price of closed to kind). But I just disagree with this particular i – in addition to and so does Eclectikus.
I think that this theme is a proficient illustration of the proficient in addition to bad thinking in addition to relevant facts inwards many proposed reforms of this kind. Also, I recollect inwards particular how my agreement of the powers was evolving since the historic menses of 5, what was right virtually it in addition to what was not. So let's start.
The primary justification for the triangle notation of powers is the dissatisfaction virtually the fact that at that spot are iii real unlike ways how to write downwardly a basically equivalent fact:\[
\Large
\eq{
2^3 &= 8 \\
\sqrt[3]{8} &= 2 \\
\log_2(8) &= 3
}
\] If the statements are equivalent, shouldn't the ways how nosotros write them live similar every bit well?
I think that the respond is a resounding No. The argue is that we're using these symbols non just to write the particular declaration of the shape \(a^b=c\). We're using these iii symbols to write the actual iii functions of ii variables – functions that may live used every bit parts of to a greater extent than complicated, composite functions in addition to expressions in addition to equations. And they're real unlike functions.
The video suggests to write all iii functions using an equilateral triangle with closed to symbols sometimes written side past times side to the 2 vertices:\[
\begin{align}
x^y &={} \stackrel{y}{_x\triangle_{\phantom{z}}}&&\text{$x$ to the $y$}\\
\sqrt[y]{z} &={} \stackrel{y}{_\phantom{x}\triangle_{z}}&&\text{$y$th beginning of $z$}\\
\log_x(z)&={} \stackrel{}{_x\triangle_{z}}&&\text{log base of operations $x$ of $z$}\\
\end{align}
\] The lines with the triangles are pretty, similar, in addition to symmetric. But the inquiry you lot should inquire is: Should they be? The right respond is No (at to the lowest degree to 2 of the 3 adjectives). They shouldn't live similar or symmetric because the functions are qualitatively different in addition to a proficient organization of notation makes their qualitative divergence every bit obvious every bit possible.
In particular, the exponent \(3\) plays a real unlike role inwards \(2^3=8\) than the other ii numbers \(2,8\). The triangle notation – particularly because it looks similar an equilateral triangle – indicates that at that spot is closed to shape of an \(S_3\) symmetry betwixt the iii numbers. But there's no such symmetry. The iii numbers play real unlike roles. And that's the asymmetry that a faithful notation should brand obvious.
I suspect that lots of people who back upwards such triangles just don't "feel" how internally unlike the exponentiation in addition to logarithms are. For them, \(\exp\) in addition to \(\ln\) are just ii buttons on a figurer that may live pressed in addition to that modify a number to another. But what's happening within is a dark box for them. They aren't genuinely familiar with the huge differences inwards the identities involving \(\exp\) in addition to \(\ln\), respectively, in addition to and so on.
The base of operations in addition to the resulting ability play unlike roles, too. With the triangle notation, at that spot would live constant confusions in addition to mistakes caused past times a permutation of ii or iii of the vertices. To confuse the base of operations in addition to the exponent is a serious mistake, i that completely changes the grapheme of the expression. The real persuasion that you lot "should" receive got a similar symbol for the exponential in addition to the logarithm is analogous to the claim that the verbs "giving a birth" in addition to "murdering" should receive got similar words because they're inverse in addition to thus analogous to i another, too.
Why don't nosotros say "to durmer" instead of "to plow over a birth" inwards monastic say to brand the symmetry with "to murder" to a greater extent than explicit? Well, nosotros don't in addition to I think that rational people know that it would live counterproductive, non great, to receive got similar or symmetric words for births in addition to murders. They're just unlike things – also acts that require a real unlike sort of piece of work (just similar \(\exp\) in addition to \(\ln\) do), despite the results' beingness reverse to each other – in addition to it's real of import to distinguish them. In the same way, nosotros could say "positive in addition to sopitive" instead of "positive in addition to negative". Again, these ii words would acquire confused often. Creators of usable languages receive got deliberately used unlike words for these "inverse" operations because the divergence – the take away to distinguish them – is in addition to so important.
I think that the wishing to brand symbols symmetric fifty-fifty though the underlying objects patently don't receive got the symmetry is i of the signs of the widespread fanatical egalitarianism in addition to leftism. Influenza A virus subtype H5N1 huge component subdivision of the leftist contamination of the contemporary Western fellowship may live described every bit examples of the aggressively spread prevarication that things in addition to people are the same fifty-fifty though they self-evidently in addition to importantly aren't. And closed to people wishing to spread this prevarication to the basis of bases, exponents, in addition to powers, too.
Even the Byzantine missionaries sent to Great Moravia (Czechoslovakia) inwards 863 were obsessed with counterproductive symmetric symbols. Saints Cyril in addition to Methodius developed a novel glagolitic alphabet to faithfully reproduce the Slavic sounds. Using this alphabet, the Bible was translated to the Old Church Slavonic – the start regular people's natural language legitimized every bit a religious linguistic communication after the "classic" religious languages such every bit Hebrew, Greek, in addition to Latin. Fair enough. But you lot may regard that an unnaturally high fraction of the symbols receive got a left-right or up-down mirror symmetry. They're genuinely besides redundant in addition to to a greater extent than complex (and thus to a greater extent than time-consuming) than they could live (an example of a text). Couldn't they receive got used the Greek alphabet with closed to diacritics, for example? Within a century, this script disappeared from Czechia, anyway, along with the Eastern Christianity. They were replaced past times the Catholic belief in addition to the Latin alphabet. And it was shortly realized that the Latin alphabet is just fine for the Slavic sounds, with closed to extra two-letter symbols. Diacritics were introduced inwards the early on 15th century. Meanwhile, inwards the ninth century, Eastern Slavs did start to write inwards Cyrillic script which was a variation of the Greek alphabet. Some Cyrillic letters were also besides symmetric in addition to besides redundant but most of them went extinct in addition to at the end, the Greek, Latin, in addition to Cyrillic scripts appear comparably simple (complex) in addition to comparably (a)symmetric today.
But the symmetry vs asymmetry dispute isn't the only employment of the triangle notation. Another employment is the real proposition that the functioning has fundamentally iii players. Why?
When I was 5, I learned virtually exponentiation for the start fourth dimension in addition to it looked great. At that moment, I persuasion that at that spot were 5 operations. I also learned virtually complex numbers in addition to sometimes when I was a first-grader, I was obsessed past times the wishing to know how much is \(a^b\) for whatever \(a,b\in\CC\). I was already convinced that the complex numbers allow you lot to calculate anything, that the annoying "E" final result that my figurer sometimes returned doesn't hap inwards the paradise-like basis of complex numbers, in addition to and so on.
(I also established a math-teaching fellowship when I was a start grader, working difficult to part the joy of \(\pi\), factorials, powers, in addition to complex numbers with my friends. We generally did non-mathematical things, however, similar physically facing a gang of gypsies. Also, my friend Ota K. taught me that a adult woman inwards the street had had sex; in addition to that when people wishing to receive got children, they receive got to utilization a condom.)
With hindsight, it looks foreign that I was in addition to so obsessed with such a technicality. But the reasons are similar to the reasons why I always found it of import to memorize the start thirty digits of \(\pi\) (three years later, I memorized the next lxx digits only because of fun – I saw that it is a mode to print closed to people inwards a mode that looked slow to me; but I already knew it was pretty much pointless): I just had oversimplified in addition to narrow-minded ideas virtually "how much of import mathematics is out at that spot in addition to what is genuinely important". So learning an of import number such every bit \(\pi\) or the methods to calculate the ability \(a^b\) for whatever ii allowed (complex) numbers looked similar a "big chunk of mathematics", to say the least. It was rubbish, of course, but I couldn't receive got quite known it was rubbish earlier I learned much to a greater extent than virtually mathematics in addition to physics etc.
But why I am maxim these things virtually the childhood? Some ii years later, I did acquire how to (numerically) compute \(a^b\) for specific numbers \(a,b\in\CC\). As you lot know, the exponentiation may live reduced to the measure exponential:\[
\Large
a^b = \exp(b\cdot \ln(a))
\] The funny affair is that you lot don't take away a full general complex business office of ii variables, the full general exponentiation. One business office of i variable (the exponential) in addition to its inverse (the natural logarithm) are enough!
The to a greater extent than experienced physicist (or calculus-based mathematician) i becomes (and I was becoming), the to a greater extent than he or (less likely) she realizes that the natural "systemic" mode to speak virtually the full general powers is the exponential with the base of operations \(e=2.718\dots\). Of course, i has to empathise why the base of operations \(e\) is to a greater extent than natural, fifty-fifty relatively to widespread bases such every bit \(2\) in addition to \(10\). The exponential \(\exp(x)=e^x\) is the only business office of \(x\) alongside functions \(b^x\) whose derivative is just the same function, for example. Equivalently, the derivative of \(\ln(x)\) is \(1/x\) in addition to this simplest final result also plant for the base of operations \(b=e\) only.
Consequently, the Taylor expansions of functions \(e^x\) in addition to \(\ln(x)\) are simpler than the expansions of other full general powers, in addition to and so on. I do think that the "example with the involvement rates" was my start meet with the importance of \(e\) every bit a base. Take i Czechoslovak crown (to provide the currencies of the 1980s). After i twelvemonth in addition to 100% yields, you lot plow CSK 1 to CSK 2. If you lot acquire 50% twice a year, you lot volition ain CSK 1.5 x 1.5 = CSK 2.25 after the start year. Try to acquire 10% x times. You acquire CSK 1.110 = CSK 2.59 etc. If you lot acquire the smaller involvement to a greater extent than frequently, inwards the boundary of the high frequency, you lot volition only cease upwards with a finite amount of money, namely CSK 2.718...
OK, the base of operations \(b=e\) is the most natural base of operations alongside those that may hit "all" numbers at all – banking concern notation that the base of operations \(b=1\) is no proficient for logarithms because all the logarithms would live ill-defined. Once you lot realize that it's genuinely just the simple functions of 1 variable that are fundamental, you lot may write\[
\eq{
x^y &= \exp(y\ln (x))\\
\sqrt[u]{v} &= \exp(\ln(v)/u)\\
\log_c(d) &= \frac{\ln d}{\ln c}.
}
\] Nice. You regard that the full general results aren't quite well-defined for complex values of the declaration because the complex business office \(\ln(z)\) is only defined upwards to an arbitrary additive shift \(+2\pi i k\) where \(k\in\ZZ\) is whatever integer. So the full general ability (or beginning or logarithm) involving ii (finite, nonzero) complex numbers that I wanted to live able to calculate was calculable but the final result wasn't unique. That was a sort of a resolution I didn't quite await but this real fact was also an of import lesson: Questions sometimes receive got answers you lot don't expect, answers "in the middle", answers with disclaimers in addition to exceptions, in addition to and so on.
Incidentally, the realization that \(e^x\) in addition to \(\ln(x)\) are the only functions that you lot genuinely take away has affected me, too. I speedily realized that fifty-fifty multiplication in addition to partition may live eliminated from the listing of basic operations because \[
\eq{
u\cdot v &= \exp(\ln u + \ln v)\\
\frac{u}{v} &= \exp(\ln u - \ln v)
}
\] which agency that the addition, subtraction, exponential, in addition to natural logarithm are the only operations you lot need. Just to live sure, you lot may kill the subtraction every bit good because\[
u - v = u + \exp(\ln v + \pi i).
\] Well, assuming that I am allowed to write downwardly complex constants such every bit \(\pi i\). I don't wishing to move into besides ludicrous. Subtraction is in addition to so simple in addition to widespread that you lot don't wishing to supersede it past times complicated functions involving exponentials in addition to logarithms. In practice, I haven't fifty-fifty "used" the mode to rewrite the full general production inwards price of the exponential, logarithm, in addition to addition. It's an identity you lot sometimes take away but every bit a tool to bargain with full general products, it's just a curiosity that you lot may live without a standalone production operation. It's often overnice to trim back the number of basic operations in addition to concepts but if you're besides fanatical virtually it, you lot may do to a greater extent than damage than good, too.
But the dot I wishing to brand is that non only the triangle notation is a problematic mode to stand upwards for particular simple objects – an unjustifiably symmetric representation of a goal. What's worse: the finish itself is misguided. The real fact that it's a triangle that looks "fundamental" suggests that closed to complicated business office of ii variables – i.e. identities with 3 players (including the result) – has to live added to mathematics.
As the kids should acquire a chip later, peradventure past times the cease of the high school, this is non genuinely the case. The key operations you lot take away to calculate the full general powers in addition to full general logarithms are functions of i variable only – you lot would take away "diangles" for those, non "triangles". The triangle looks similar an irreducible human relationship with 3 players but the underlying entity is not irreducible. And inwards physics, \(\exp(x)\) in addition to \(\ln(y)\) appears far to a greater extent than oft than the ability \(b^x\) in addition to logarithm \(\ln_b(y)\) with a to a greater extent than full general base of operations \(b\). Also, \(\sqrt{x}\) is the base-two (square) beginning in addition to appears oft plenty in addition to so that it's proficient to receive got a special, simple plenty symbol, instead of a triangle with the number \(2\) written side past times side to i of the vertices that I can't reliably pinpoint fifty-fifty if I think for 5 seconds.
Changes of notation may live costly because the transition may displace lots of problems. But sometimes they may brand sense. However, I am afraid that with the electrical flow ease of power, particularly inwards the Western pedagogy systems, such reforms are beingness pushed past times people who don't think virtually (and don't discuss) the consequences every bit described inwards this spider web log shipping service (just a real specific example) and, inwards many cases, they are completely ignorant of closed to real basic concepts in addition to principles (e.g. the full general logarithm of a complex number etc.) which agency that they can't fifty-fifty think or hash out these of import issues.
That's why a vast bulk of such proposed changes are unavoidably counterproductive. Mathematicians receive got used the same notation for powers in addition to logarithms for a few centuries in addition to if it had closed to obvious plenty in addition to easy-to-fix problems or redundancies, they would receive got already fixed those problems. Most of the people who wishing to brand the fixes are commonly missing something of import in addition to so a bulk of the proposed changes would brand the notation (or to a greater extent than of import things) worse, non better.
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