Did You Pay Attention In Math Class?

[sumgai]

What the heck is this:

{^{3}5}

Any takers?





sumgai

[cynical1]

No, I did not pay attention in math class...

HTC1

[JohnH]

Easy!

that is the number 35, as written on a 1940's Underwood travel typewriter that was used during the whole career of a retired war correspondent until he stepped on something that went 'click' 'boom!', and after that he had to slow down a bit.

[reTrEaD]

funny reply:

SPOILER: Click to show

serious reply:

Can you provide a link to where you saw that, so we can understand the context? Or did you just decide to place a number in superscript to the left of another number?

[Yogi B]

Or, a seriously funny reply:

1911012597945477520356404559703964599198081048990094337139512789246520530242615803012059386519739850265586440155794462235359212788673806972288410146915986602087961896757195701839281660338047611225975533626101001482651123413147768252411493094447176965282756285196737514395357542479093219206641883011787169122552421070050709064674382870851449950256586194461543183511379849133691779928127433840431549236855526783596374102105331546031353725325748636909159778690328266459182983815230286936572873691422648131291743762136325730321645282979486862576245362218017673224940567642819360078720713837072355305446356153946401185348493792719514594505508232749221605848912910945189959948686199543147666938013037176163592594479746164220050885079469804487133205133160739134230540198872570038329801246050197013467397175909027389493923817315786996845899794781068042822436093783946335265422815704302832442385515082316490967285712171708123232790481817268327510112746782317410985888683708522000711733492253913322300756147180429007527677793352306200618286012455254243061006894805446584704820650982664319360960388736258510747074340636286976576702699258649953557976318173902550891331223294743930343956161328334072831663498258145226862004307799084688103804187368324800903873596212919633602583120781673673742533322879296907205490595621406888825991244581842379597863476484315673760923625090371511798941424262270220066286486867868710182980872802560693101949280830825044198424796792058908817112327192301455582916746795197430548026404646854002733993860798594465961501752586965811447568510041568687730903712482535343839285397598749458497050038225012489284001826590056251286187629938044407340142347062055785305325034918189589707199305662188512963187501743535960282201038211616048545121039313312256332260766436236688296850208839496142830484739113991669622649948563685234712873294796680884509405893951104650944137909502276545653133018670633521323028460519434381399810561400652595300731790772711065783494174642684720956134647327748584238274899668755052504394218232191357223054066715373374248543645663782045701654593218154053548393614250664498585403307466468541890148134347714650315037954175778622811776585876941680908203125

---

Just to be more confusing, we could even throw some other numbers into the mix:

{}^3 5 = 5 \mathbin{\raisebox{1.5pt}{\fbox{\scriptsize\,4\,}}} 3 = 5 \mathbin{\uparrow^2} 3
Though those alternatives are actually what I'm more familiar with — being generalizations of the concept, thus seeing greater usage, as well as possibly giving a little more of a clue as to what (the hell) it is...

...and are such that (unlike the notation in the OP) I don't forever confuse them with the following, equally WTF inducing, notations:

\begin{aligned}
5^{(3)} &= 210
\\[1ex]
(5)_3 &= 60
\end{aligned}

[sumgai]

{^{3}5}

This is called "tetration", it's a form of writing a very large number that was based on the single digit, 5.

To think of it in a more familiar context, try 5 ^ 5 ^ 5.  That's easy to speak out loud: "raise 5 to the 5th power (which equals 3125), and then raise 5 to that number as a power". It may not look right at first, but your calculator will get it correct.  The answer to that one would be: 298,023,223,876,953,125.

(If your calculator blows up at this, simply type the formula into any search engine, and trust me, you'll get the same answer as above.)

Just for fun, you can do this even higher, like as in tetration is 4th level, pentation is 5th level, hexation is 6th level, and right on up the scale. However, you really need to be an advanced mathematician to need to use numbers on that scale.

Also, I used the notation I learned in Rudy Rucker's book Infinity And The Mind published in, I think, 1982, but he took it from Maurer who made it up in 1901. There have since been many attempts at writing math expressions like this, each "school" having their adherents. To me, it's fascinating, but then again, I'm the twisted one here.

First reference:  Wikipedia - Tetration


After that, you're on your own!







sumgai

[Yogi B]

Jan 29, 2024 22:50:16 GMT -5 sumgai said:

To think of it in a more familiar context, try 5 ^ 5 ^ 5.  That's easy to speak out loud: "raise 5 to the 5 power (which equals 3125), and then raise 5 to that number". It may not look right at first, but your calculator will get it correct.  The answer to that one would be: 298,023,223,876,953,125.

That number's only 5 ^ 25 — i.e. 3125 ^ 5 = (5 ^ 5) ^ 5, the result if exponentiation were left associative — rather than (as your words correctly say) 5 ^ 3125 = 5 ^ (5 ^ 5), which is the monstrously large number I posted.

---

After finding not a single instance of the box notation I used in my reply on the tetration article you linked (nor the hyperoperation article) and, googling "box notation" resulting in the hyperoperation page but on the Chinese language version of Wikipedia, I went digging through the edit history. According to this May 2020 edit, box notation is "unambiguously non-notable". Whereas, apparently, the ASCII friendlier notation 5 [4] 3 predates box notation, and now seemly outlives it.

By its foremost location in the table, there is an implication that Knuth's up-arrow notation is the most popular (despite its offset of 2) — with which I do agree.

I think my most recent encounter with tetration was written using Knuth's notation, and was as a means to calculate a limited range of values on the principle branch of the Lambert W function (useful for solving the Shockley diode equation):

W_0(x) = x \left( e^{-x} \uparrow\uparrow \infty \right) \text{,\quad for $-e^{-1} \le x \le e$\quad(i.e. where $-1 \le W(x) \le 1$)}

[sumgai]

Yogi is correct. Both my OS-provided calculators and every online calculator I found have all gotten the order of operations wrong.

When evaluating tetration (and higher order eponentiation), you work from the right to the left. Thus, 5 ^ 5 ^ 5, as a matter of tetration, should be rewritten not as 3125 ^ 5 (as now obviously done by the aforementioned calculators), but as 5 ^ 3125. That results in Yogi's ridiculously large number. In fact, the only website that got it correct that I could find after a very diligent search was Microsoft's MathSolver. Everyone else either used the incorrect method (left to right) or just went off on a tangent about some unrelated topic having nebulous ties to my query.

So strike my answer above, for now obvious reasons.   Sorry 'bout that. 


Why did I post this?  Because I was recently reminded of something that I used to do in solving equations for graphical functions, back before the advent of graphing calculators.  In electronics and electrical theory, one can go the other way around and invert the tetration by using logarithms.  An example would be how to calculate the dynamic resistance of PN junction, as pretty much invented by Shockley.  (The actual formula, not the invention of the PN junction itself.  That honor is shared between Shockley, Bardeen and Brattain.)

It's getting pretty 'old' trying to write complex equations using Katex, so rather than go down that road (the formatting for such is way out of bounds for something that should be simple), let me just send you over to the best article I could find for an explanation:

Shockley's Diode Equation (which takes you to physics.stackexchange.com)

HTH





sumgai

p.s. Yogi, these formulas shouldn't be center-justified, they should be almost completely to the left. The only way I could find to make this happen was with a table, inside of a blockquote. That's cumbersome, even though it makes for a presentation style that's easier on the eyes. Can we change the 'title=math' button to more-or-less left-justify equations? Please?

[Yogi B]

Jan 30, 2024 13:30:08 GMT -5 sumgai said:

In electronics and electrical theory, one can go the other way around and invert the tetration by using logarithms.  An example would be how to calculate the dynamic resistance of PN junction, as pretty much invented by Shockley.

I've known of both tetration & the Lambert W function for over a decade, but it was only last autumn that I came across the specific link between the two. Someone posted a spreadsheet based calculator for simple single BJT stages (bias voltage, input/output impedance, etc.), but was assuming a constant (650mV) voltage across the base-emitter junction. That prompted me for the first time to look at how to actually implement the function: you can't just "from scipy.special import lambertw" inside a spreadsheet. Ultimately I sidestepped needing to use the function directly and instead settled on doing a handful of iterations of Halley's method applied to the equation of bias current and B-E current. Picking an initial guess from a pair of values calculated using two upper bounds of the Lambert W function (using whichever is lower for the given set of inputs).

p.s. Yogi, these formulas shouldn't be center-justified, they should be almost completely to the left. The only way I could find to make this happen was with a table, inside of a blockquote. That's cumbersome, even though it makes for a presentation style that's easier on the eyes. Can we change the 'title=math' button to more-or-less left-justify equations? Please?

Maybe? There's a few issues at play here.

KaTeX does have a global "fleqn" (flush left equations) option that aims to do what you wish, however it isn't perfect. Firstly, in the old version that we're currently using, flush is applied a little too zealously with no margin added, but that's the easiest fix.

(In fact it was fixed in the very next release, but that's also when IE 9 & 10 support was dropped so I paused updating at the time. There's a small chance I was forgetting IE 11 existed so was equating that to totally dropping IE, which at the time still had a market share comparable to Edge (around 2%). Three and a half years later and IE in total is < 0.5%, with IE9 & 10 < 0.05%, so now that's less of a concern.)

Second is a stylistic concern, centre alignment is the default in (Ka/La)TeX for 'displayed' equations and seems more prevalent than left alignment in the math papers I've read and on sites such as math/physics stackexchange, but granted they're formatted to use a narrower page width than what we have here.

A further issue that's demonstrated in this example is that although the entire (gather) environment may be aligned to left (click the cog menu to verify fleqn is selected), the content within is still centre aligned — which is not how it should be. This can be manually worked around by using an align environment instead, or maybe with some clever CSS but that could be tricky to do correctly.

Finally is an issue that doesn't concern the equations themselves, but with the possibility of numbering tags (as was shown in the previous example).To be honest no-one's yet made use of this feature, but if they did: there'd already be a large distance between the equation and its label even with centre alignment, and enabling fleqn would make it even worse. (You can maximise the editor on the KaTeX site, from the same menu, to get a feel for how this looks with a wider page width.)

Above the fleqn checkbox, you may have noticed another option "leqno" (left equation numbers) that would solve the above problem, however enabling both that and fleqn causes the tag & equation to overlap (try it to see). There's a CSS tweak I could apply, but it's not a proper fix and longer labels will still overlap.

---

P.S. going back to what you said:

Can we change the 'title=math' button to more-or-less left-justify equations?

note that title="math" is for 'inline' use and is (or should be) left aligned (or at least inherit the alignment of its parent element). The default centre alignment is only applied to title="displaymath".

[JohnH]

I've noticed a puzzling side issue with this thread. What should the equation really look like?


On a new Windows laptop running Edge, it looks like this (screenshot):





On a Samsung S22 running Android, it looks like this:






(ignore size difference, just a due to different screenshot parameters between phone and laptop)

[reTrEaD]

Jan 31, 2024 16:42:48 GMT -5 JohnH said:

I've noticed a puzzling side issue with this thread. What should the equation really look like?

The intent is that it looks like the first screenshot you posted:

Jan 31, 2024 16:42:48 GMT -5 JohnH said:

On a new Windows laptop running Edge, it looks like this (screenshot):

However, because plugins don't function in the mobile version of proboards, what you're seeing on your mobile device looks like the second screenshot.

Jan 31, 2024 16:42:48 GMT -5 JohnH said:

On a Samsung S22 running Android, it looks like this:

This is the portion of the post that was intended to be processed by the plugin and result in the 3 being displayed in superscript to the left of the 5:

{^{3}5}

[kitwn]

My problem with maths at school was that nobody ever explained what use it might be. I wasn't even as bright then as I am now.

Only when I wangled my way into a job with the BBC engineering department as a trainee at the age of 24 did I discover that trigonometry explained everything about how radio works. If somebody had mentioned that a decade earlier I might have paid more attention.

And now I've reminded myself of that lovely exchange in The Hitch-hikers Guide To The Galaxy: "I should have listened to my mother when I was younger." "Why, what did she say?" " I don't know, I wasn't listening!"

Kit

PS  The answer to your question is 42, but you already knew that.