explain xkcd - User contributions [en]

2586: Greek Letters

2022-02-26T10:23:20Z

141.101.77.24: Fixed a minor spelling error - it said "Trition" instead of "Triton" in the last line.

{{comic
| number = 2586
| date = February 25, 2022
| title = Greek Letters
| image = greek_letters.png
| titletext = If you ever see someone using a capital xi in an equation, just observe them quietly to learn as much as you can before they return to their home planet.
}}

==Explanation==
{{incomplete|Created by '''''O R B S''''' PRO®- Missing explanations for some letters. Do NOT delete this tag too soon.}}

Mathematics uses lots of Greek letters, typically using {{w|Greek_letters_used_in_mathematics,_science,_and_engineering|the same letter consistently}} to represent a particular constant or type of variable.
This comic gives a (non-)explanation of what they typically mean.

The letters are:

* '''π (lower-case pi)''' — Typically used to refer to the constant ratio between a circle’s circumference and its diameter (approximately 3.14). This usage of pi commonly applies to equations in introductory geometry classes, which would be considered "simple" by advanced mathematicians. However, pi also shows up seemingly randomly in extremely advanced and complicated equations (that have nothing to do with a circle), as part of the solution to an infinite series or whatnot. (There are also {{w|Pi_(letter)#Lowercase_Pi|several advanced equations}} which use pi to represent variables other than the ratio of the circumference to the diameter.)

* '''Δ (capital delta)''' — Typically used to refer to a change in quantity.

* '''δ (lower-case delta)''' — Also typically used to refer to a change in quantity, but unlike the capital delta, this is only for infinitesimal changes and is used in derivative and integration expressions in mathematics hence the text's reference to "a mathematician's fault".

* '''θ (lower-case theta)''' — Typically used to refer to an angle, and is notably used in the polar coordinate system. The text refers to its close relationship with circles, on which the polar coordinate system is based on.

* '''Φ (lower-case phi)''' — Typically used to refer to another angle other than one referred to by theta. It's used in spherical coordinates, and the text refers to how spheres, or orbs, are important in spherical coordinates.

* '''ϵ (lower-case lunate epsilon)''' — Epsilon is typically used to refer to very small quantities which go to zero in the limit. In this interpretation, the comic suggests that because these quantities are very small, they are unimportant, when in reality the study of quantities that go to zero gives rise to limits and calculus. It is also used for the series of transfinite numbers that are unreachable from ω (see below) using addition, multiplication, and exponentiation. Also used in statistical modelling to denote observational noise.

* '''υ,ν (lower-case upsilon and lower-case nu)''' — If these are being used it implies that the normal u & v characters are already assigned as constants or variables, and thus the math is probably of a higher level. Common in college level physics and engineering equations.

* '''μ (lower-case mu)''' — The SI prefix for "micro" = 10<sup>-6</sup>, representing very small quantities: a micrometer (μm) is tens of times smaller than the width of a human hair, a microgram (μg) is one single fine speck of flour, both of which are barely visible with the bare human eye nor feelable through the skin.

* '''Σ (capital sigma)''' — Typically used as a symbol for summation of a series of numbers.

* '''Π (capital pi)''' — Typically used as a symbol for multiplication of a series of numbers.

* '''ζ (lower-case zeta)''' — Frequently used with number theory, in particular the {{w|Riemann zeta function}}, which is a the focus of a famously unsolved problem in highly advanced mathematics.

* '''β (lower-case beta)''' — This could be a reference to the typical usage of beta to represent coefficients of independent variables in the {{w|Ordinary_least_squares#Linear_model|ordinary least squares regression model}}. Regression can potentially have a large number of independent variables, hence potentially many different betas (differentiated by subscript, or compacted into matrix notation) would be used. Alternatively, the comic might suggest whatever source this equation is from has run out of Latin letters to use as symbols, and is now going through the Greek letters.

* '''α (lower-case alpha)''' — Possibly referring to alpha radiation, which certainly could kill someone. Quite likely refers to angular acceleration, or the acceleration of spinning systems, which are capable of killing people in a number of [https://xkcd.com/123/ interesting ways]...

* '''Ω (capital omega)''' — Omega is the last letter of the Greek alphabet, and thus often seen as momentous (the end, the final word, death). This symbol has been used for a {{w|Omega_function|variety of mathematical functions}}. Also used for the symbol for {{w|ohms}}, a unit for electrical resistance, and for the first uncountable ordinal.

* '''ω (lower-case omega)''' — Lower-case omega is used for the {{w|Transfinite_number|lowest transfinite ordinal number}}, a specific way of referring to a type of infinity in a mathematically robust way. The line about dying here among the transfinite equations may be in reference to the literally infinite scope of the branch of mathematics. It is also used in physics and electrical engineering for angular frequency, equal to 2πf.

* '''σ (lower-case sigma)''' — In statistics, commonly refers to the standard deviation of a distribution. Statistics often attempts to use simplified models to explain real-world phenomena.

* '''ξ (lower-case xi)''' — Randall comments that this looks like a strand of curly hair. Xi is used in the {{w|Riemann Xi function}}.

* '''γ (lower-case gamma)''' — Gamma ray is the most powerful classification of electromagnetic radiation AKA "light", and powerful lights are frequently associated with high-tech, futuristic devices and weapons, hence "space noises". Alternatively, this might be a reference to the Lorentz factor, an important variable in special relativity calculations.

* '''ρ (lower-case rho)''' — often used to measure density, such as air density that a wing might be travelling through.

* '''Ξ (capital xi)''' — Resembles the icon of some {{w|Stack Exchange}} [https://stackexchange.com/sites# sites]. This character is also identical to Besh, the second letter of the {{w|Aurebesh Alphabet}} [https://starwars.fandom.com/wiki/Aurebesh].
* '''ψ (lower-case psi)''' — Psi looks exactly like a trident. This is hilarious.{{citation needed}} In quantum mechanics it's used to describe the wave function of a particle, leading to a bad pun. (Psi is also used in mathematics to represent the sum of the inverse of the Fibonacci numbers, the division polynomials, and the supergolden ratio.)

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Header:]
:What Greek letters mean in equations

:[What follows is a list of Greek letters, with explanations next to them.]
:π This math is either very simple or impossible.
:Δ Something has changed.
:δ Something has changed and it's a mathematician's fault.
:θ Circles!
:Φ '''''O R B S'''''
:ϵ Not important, don't worry about it.
:υ,ν Is that a V or a U? Or...oh no, it's one of ''those''.
:μ This math is cool but it's not about anything that you will ever see or touch, so whatever.
:Σ Thank you for purchasing ''Addition Pro''®!
:Π ...and the ''Multiplication''® expansion pack!
:ζ This math will only lead to more math.
:β There are just too many coefficients.
:α Oh boy, now '''''this''''' is math about something real. This is math that could '''''kill''''' someone.
:Ω Oooh, ''some'' mathematician thinks their function is cool and important.
:ω A lot of work went into these equations and you are going to die here among them.
:σ Some poor soul is trying to apply this math to real life and it's not working.
:ξ Either this is terrifying mathematics or there was a hair on the scanned page.
:γ ''Zoom'' pew pew pew [space noises] ''zoooom!''
:ρ Unfortunately, the test vehicle suffered an unexpected wing separation event.
:Ξ Greetings! We hope to learn a great deal by exchanging knowledge with your Earth mathematicians.
:ψ You have entered the domain of King Triton, ruler of the waves.

{{comic discussion}}
[[Category:Math]]
[[Category:Language]]

Talk:2585: Rounding

2022-02-25T16:30:54Z

141.101.77.24:

Wot no {{w|FFF system|furlongs per fortnight}}? [[Special:Contributions/172.70.91.126|172.70.91.126]] 23:14, 23 February 2022 (UTC)

: I, too, was initially surprised that Randall hadn't used the standard joke measure. But, then I realized that F/F is so outrageously large that rounding wouldn't offer much advantage. [[User:MAP|MAP]] ([[User talk:MAP|talk]]) 05:10, 24 February 2022 (UTC)

If we're using the table, can I suggest it be fully filled in, but mark "original (rounded)" value cells one key colour and the chosen conversion in another, so that scanning along (not necessarily adjacent/rightwards) then down (always next row) then along... you see the 'bounce around'. And we also get to appreciate what other fractional values ''could'' have been chosen, prior to rounding... Alternately, some flow-charty layout (perhaps contained within a nominally borderless version of the table?) with arrows leading across the width and filling in-between each down-step. Ideas only. I have others, but those seem the best bet to consider. [[Special:Contributions/172.70.85.113|172.70.85.113]] 01:32, 24 February 2022 (UTC)

Disagree with the current (as of 23:27 US Eastern, 23 February) explanation. According to this site (https://ilovebicycling.com/average-bike-speed/), average downhill bike speed is over 45 mph. Since Cueball doesn't specify "on flat terrain", he should have no problem going 45 without exploiting imprecise conversions. [[User:Nitpicking|Nitpicking]] ([[User talk:Nitpicking|talk]]) 04:30, 24 February 2022 (UTC)
:Huh? This does not say average downhill speed is > 45, it says "fastest". Also why would Cueball need to do this bizarre rounding if he can actually go 45mph? This is an exaggeration because he can only go a typical speed of 17mph.[[Special:Contributions/172.69.33.145|172.69.33.145]] 04:52, 24 February 2022 (UTC)
::Fastest for average cyclist. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 05:05, 24 February 2022 (UTC)
:As a cyclist of several decades experience, who has indeed attained such speeds on rare (reckless) occasions, I think that "fastest downhill speed for an average rider" is overstated. Maybe it is what average people are capable of on a well-surfaced, steep, straight, non-undulating road with sufficient vision (forward and of anything potentially moving into the road from the side) or at least confidence that you're not dealing with traffic/pedestrians/other unaware cyclists. Oh, and sufficient stopping distance for whatever brakes you have.
:Maybe everybody can do it ''once'', but a good bike-ride should be one you can walk away from at the end.
:(Also, that cycling-centric site might have a different idea of 'average' cyclist. The average person on a bike here can't even put their feet on the pedals correctly. If we're talking club-/competitive-cyclists (but still sub-pro) then I'd much more readily agree, but there are far more people these days who can't even ride on the roadway, it seems.)
:That bike, as drawn, looks like it'll be Okish (if kept well maintained) but not exactly set up as functional downhill racer, nor probably is the rider. I really think the machine probably could be ridden at 20+mph on the flat for as long as the rider can stand to, but the characterisation makes me not confident they're able to maintain that kind of average speed for a [https://www.cyclingtimetrials.org.uk/race-results/22059#anchor long ride], and I think they'd overbake a downhill speed-run too, or (sensibly) be more cautious. [[Special:Contributions/172.70.85.143|172.70.85.143]] 05:14, 24 February 2022 (UTC)
:: Yep - the speeds on that site are for road bikes. Cueball looks to be riding a hybrid (flat bars), which would tend to put him in a more upright position, creating a higher frontal area and air resistance, and so slowing his progress. That would have even more of an effect at higher speeds. [[Special:Contributions/162.158.159.43|162.158.159.43]] 11:14, 24 February 2022 (UTC)

Arguably, once you're up to numbers around 45, you're as likely, if not more so, to be rounding to the nearest 5 than the nearest unit (depending on context). So Cueball's initial statement could be taken as suggesting that he can ride at around 42.5 - 47.5mph (rather than 44.5 - 45.5mph). And if he could actually ride at over 45mph then he presumably wouldn't need to add the 'if you round' qualifier, so it could further be taken as just suggesting that he can exceed 42.5mph. [[Special:Contributions/162.158.159.43|162.158.159.43]] 11:22, 24 February 2022 (UTC)

Note I find it kind of disappointing that the insane "KPH" unit is used in the comic. Nobody uses that in places where speed is actually measured in km/h.
: yes, but we are talking about a US based comic, one of only 3 countries (Myanmar, Liberia, USA) that don't use the metric system for measurement...oh, except for money, but that isn't really metric, it is money ;o) [[Special:Contributions/108.162.250.190|108.162.250.190]] 00:50, 25 February 2022 (UTC)
:: Erm, I think you'll find the UK uses miles as well. And we're just putting ourselves through a massive political and economic upheaval so that we can have our old imperial weights and measures back (at least, I think that was the point of it all).[[Special:Contributions/141.101.77.24|141.101.77.24]] 16:30, 25 February 2022 (UTC)

Ironically, by the same standards it only takes one conversion to say that he can't move at all on a bike. he goes 0 parsecs, lightyears or AU (for example) per year, decade or century (for example).

Can we remove the rounding errors in the "exact" values in the tables? For instance, the final value should be "45.0000" not "45.0001". In fact, all three values ending with 0001 are rounding errors. (These were probably a result of converting to metric and back, using low precision conversion factors.) [[User:Divad27182|Divad27182]] ([[User talk:Divad27182|talk]]) 15:49, 24 February 2022 (UTC)

Whoever decided to display that information in that table deserves an award. Gg. [[Special:Contributions/172.70.126.65|172.70.126.65]] 16:38, 24 February 2022 (UTC)

It's nice how the rounding of exact half-integers only ever has to deal with odd-numbers-and-a-half, so Cueball can't be charged with violating the "round to even" rule, nor with violating the "round away from zero" rule. [[Special:Contributions/172.70.131.122|172.70.131.122]] 18:06, 24 February 2022 (UTC)

It looks like Randall picked a starting speed (within a reasonable bike-riding range) to maximize his gain. Groups of starting speeds round to the same final speeds, and some groups have a higher maximum speed earlier in the rounding chain:
::{| class="wikitable"
! Start Speed
(mph)
! Max Speed
(rounded to mph)
! Final Speed
(mph)
|-
|1
|1
|0
|-
|2 to 9
|9
|8
|-
|10
|10
|8
|-
|11 to 16
|16
|15
|-
|17 to 45
|45
|45
|-
|46 to 54
|54
|53
|}[[Special:Contributions/172.70.131.122|172.70.131.122]] 21:24, 24 February 2022 (UTC)
:Are you assuming the exact same chain of conversions, just with different input values? Surely if he'd chosen to start at (say) 16, he'd have chosen whatever ''other'' chain of conversions would have sent him towards some decent high-value. Might have differed only by the initial conversions before it found itself landing on the same late-path, or could be completely different (to get to a different end) as the biased random-walk of choices hit a different useful stride pattern. [[Special:Contributions/141.101.99.20|141.101.99.20]] 22:39, 24 February 2022 (UTC)
::Yes, I put different starting speeds into the same conversion chain. Perhaps I should have said "He chose a reasonable starting speed and chain of conversions to maximize the gain." I was initially surprised that starting at 16mph ends at 15mph, then decided to plot it. The grouping of ending speeds also surprised me, but in hindsight that's to be expected with multiple round offs. [[Special:Contributions/162.158.75.17|162.158.75.17]] 23:02, 24 February 2022 (UTC)
:::Not surprising at all. Given any random (not selectively chosen) conversion-then-rounding function, you'd expect about half the time you get a lowered (absolute) value rather than a raised one, for the input number somewhere in the range 1 to infinity. For any pair of measures of unequal scales but sharing zero. (Possibly also viable in non-equal and dislocated scales, like °C and °F, but that's just a hunch that I've not emperically checked, and not applicable here anyway.)
:::The chain chosen was conspicuously optimal to get the starting value 17 to always rise. Possibly by the maximum possible amount, on each chosen step, from amongst all those considered conversions, but I haven't checked this. It even has a viable unit_A=>unit_B for one rounding rise then unit_B=>unit_A for yet another rounding rise, because it happily works like that at the respective points of each scale.
:::But when you start from a different value, you lose the initial upwards-bias in the same 'meshing' and on each subsequent Randall-chosen one. It's pretty much a random sequence, as far as the value that it wasn't designed for is concerned. Logic dictates that it will downplay the value about as often as it will up-play it, for most scenarios. Except maybe at resonant multiple/divisors of the original (which will still chaotically drift, as rounding up .6 for a value would mean rounding down .3 for value/2 or down from .2 for value*2, setting you up for the next function in the adopted sequence to fail), but then 17 is prime so you'd have to start with 34 for that to (sometimes) work.
:::And, assuming the sequence is chosen for maximising upwards, you've got the function at each stage that is selected precisely because ''for that exact state-value'' it is specifically upward-trending, so when you try that in a different context reversion-to-the-mean suggests you're perhaps more likely to hit one of the downward-trends in the relationship.
:::My theory is that for any given starting value, some convert-then-round (from a sufficiently diverse choice of options) will always maximise the resulting magnitude. And that result will always have its own maximal conversion. Although those two operations may be less maximising in combination than a submaximal first operation (maybe, in some cases, a slight ''reduction''?) that 'lands' on a better number for a differing secondary maximiser step to act upon. So a full search-path needs to consider an N-step look-ahead method rooted in a breadth-first trial of each step-1, etc, to optimise the maximiser-optimiser process. But I haven't the time to test it right now. Maybe later! [[Special:Contributions/172.70.162.77|172.70.162.77]] 00:53, 25 February 2022 (UTC)

A note about the propulsion system in the mouseover text: This system is not entirely novel and was first proposed by Douglas Adams who suggested using the notebooks of waiters in bistros to achieve the desired precision loss. He suggested it should be possible to achieve speeds of round ∞kph (∞mph) [[Special:Contributions/162.158.202.247|162.158.202.247]]
:The books don't mention those details in their description of "bistromathics", and I don't recall them having been added to the radio adaptations. [[User:BunsenH|BunsenH]] ([[User talk:BunsenH|talk]]) 23:15, 24 February 2022 (UTC)
:The Improbability Drive (in the Hitchiker's Guide) also seems somewhat related.
:What relation can that have? I'm looking at {{link|https://hitchhikers.fandom.com/wiki/Bistromathics|this link}}. [[User:GcGYSF(asterisk)P(vertical line)e|GcGYSF(asterisk)P(vertical line)e]] ([[User talk:GcGYSF(asterisk)P(vertical line)e|talk]]) 03:32, 25 February 2022 (UTC)
:The various things that {{w|Hex (Discworld)|Discworld's "Hex"}} can do (including occasionally providing magical teportation) can rely upon it trying lots of 'impossible' things very quickly "before the universe notices". [[Special:Contributions/162.158.159.125|162.158.159.125]] 14:19, 25 February 2022 (UTC)
:My favorite "impossible" thing mentioned in the Hitchhiker's Guide is be able to fly by "learning how to throw yourself at the ground and miss". I have done this successfully while dreaming, but have never accomplished it while wide awake. But it is surely worth trying. [[Special:Contributions/108.162.219.49|108.162.219.49]] 15:13, 25 February 2022 (UTC)