2343: Mathematical Symbol Fight

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Mathematical Symbol Fight
Oh no, a musician just burst in through the door confidently twirling a treble clef.
Title text: Oh no, a musician just burst in through the door confidently twirling a treble clef.


This comic imagines which mathematical symbols would be good in a fight if they were made corporeal in two (or three) dimensions.

Generally, objects with longer reach and pointier ends wound up on the right ("more useful") side of the scale, and symbols with less reach and more curves wound towards the left ("less useful") side. A straight line is farthest to the "more dangerous" side; however, the straight line does not appear to be any thicker or thinner, or pointier, than any of the other lines that would make it more "useful" (It should be noted that this chart seems to fall afoul of what Eliezer Yudkowsky (who also wrote HPMoR) calls the intent to kill: that humans tend to define "winning a fight" and "useful" as causing some form of bodily harm on their opponent despite survival and purely defensive strategies being an equally valid goals.).

Below the chart, with the symbols listed in order of usefulness, eight characters wield eight of the symbols. See the table below for the meaning of each symbol.

The comic invokes surreal humor by suggesting that mathematical symbols could be handled as physical objects in the real world. Another component of the humor is the implication that it is useful to prepare to use mathematical symbols in a fight, even though mathematicians, who use mathematical symbols, usually do not conduct their debates violently (though some stories suggest that Hippasus was killed by his fellow Pythagoreans for his proof that irrational numbers exist), and even if they did, they wouldn't use large reproductions of their symbols as weapons.

The title text refers to a treble clef, which is not a mathematical symbol but rather a musical symbol. The note of concern in the text suggests musical symbols may be viewed in such fights as exotic or especially dangerous. See also the last entry in the table below.

Note that Black Hat and Danish are using two of the most useful symbols, and appear to have teamed up against their opponents, resulting in an unfair advantage they would both absolutely exploit.

Table of symbols[edit]

Symbol Meaning Notes on using in a fight
  ℝ The set of real numbers Megan seems to be struggling with a giant version of this symbol because of its weight. A very strong person might find it slightly more useful as a bludgeon.
  θ Theta This symbol is not very sharp, and White Hat is unable to use it in combat. It would not work as a good shield due to the hole in the symbol, but it possibly could be thrown. (The later version of Xena's chakram had a bar in the middle.)
  ∅ Empty set Similarly shaped to theta, but identified as being slightly more useful, presumably because the parts of the bar that extend beyond the circle can be used as thrusting weapons. Alternatively, the round nature of the symbol makes it easier to handle for defense, by holding on to the line and using the edge of the circular part to block.
  ∞ Infinity Could be gripped between the rings and used as a blunt instrument.
  > Greater than Could be used to thrust with the point, but lacks a grip by which to do so. If held by one leg, the angle and balance is wrong to use it as a pick. Could also be used as a boomerang.
  ∝ Direct proportionality (not to be confused with lowercase alpha) Could be brandished by the ring and used to thrust with the prongs, but the two prongs extending at 90-degree angles make thrusting motions awkward.
  ∪ Union Could be used as a boomerang, which is notoriously less practical in real life than in fiction
  ∈/ϵ Member of (set theory) or the "lunate" lowercase epsilon Cueball is holding this in the manner of a Klingon "Bat'leth" from the Star Trek franchise. The Bat'leth is infamous among swordfighters for being rather impractical, and that the Klingon warrior race would have been better suited using swords like humans. It seems that Randall agrees, as the ∈ is quite far on the left of the chart.
  π Pi Could be used as a hammer, but the two handles would make doing so impractical.
  ∀ "For all" Could be held by the crossbar clenched in a fist, and then used as a push dagger, with the legs acting as a guard. Or could be held by the legs.

Compared to the similar Δ (delta) the point is sharper, the grip is closer to the point so it would be easier to control (keep the weapon from twisting). It is unclear why it is listed as much less effective than the delta. Perhaps the guard legs are too long?

  ∂ Partial derivative or boundary operator It's not clear why Randall ranks this symbol as so much less effective than the similarly-shaped 𝜌, but the curl in the "tail" of the ∂ would give it a shorter lever arm when swung and would disrupt the balance.
  + Plus sign Maybe thrown like a shuriken? It could also be used like Kylo Ren's lightsaber.
  Ψ Psi It could be used as a slightly-less-functional trident or pitchfork, with a shorter handle. More particularly, it resembles a sai (which, funnily enough, is close to how "Psi" is pronounced).
  ≠ Not equal to Could be gripped by the crossbar, using the lengths to puncture. Notably, the equal sign (=) is absent from the chart, likely because the lengths are separate and would not be used as a single entity.
  ~ Tilde, meaning "approximately", equivalent, or several transforms of a function. Also used as negation in mathematical logic and, in various programming languages, to symbolize bitwise NOT (amongst many, many other possible uses...). A potentially dangerous throwing weapon. Could also be similar to a wavy bladed dagger (Kris) or a sword (Flame-bladed sword).
  # Cardinality, connected sum (knot theory), or primorial. Blondie uses this symbol, gripping it by two of the prongs on one side. The other three prongs could thus be swung at opponents, but being sharp on all sides would pose a threat to the user as well as the opponent.
  Δ Delta Could be held with one bar clenched in a fist, and then used as a push dagger. It is unclear why listed as much more effective than ∀. Also, delta could be thrown like a star.
  ⇒ Material consequence or logical consequence, meaning "implies" The point is nice, but having two poles (or one board-like thick handle, if the shape is drawn "closed" at the back) would be difficult to wield. A collection of them might make a nice defensive pike line.
  ⋅ Decimal point or multiplication sign Functionally a ball, and could therefore be thrown as a projectile weapon, or scattered on the ground as a trip hazard.
  ζ Zeta Could be sharpened into a scythe or curved saber, depending on the curvature and length of the 'tail' at the bottom.
  ℵ Aleph number The irregular shape of the symbol leads to edges and points on all sides; thus it could be brandished or thrown in the manner of a shuriken.
  Γ Uppercase Greek letter gamma If this letter is formed with serifs, it could be used as an axe or hook, and if it is made sans-serif, it would make a powerful pick or war hammer.
  √ Radical symbol Randall has drawn this sign with a long overbar, which makes it useful like a pole arm. Black Hat has chosen this symbol in his fight against Hairy.
  ρ Italic rho Ponytail is brandishing this symbol against Danish, apparently using it like a club or hammer and striking with the curve.
  ∮ Contour integral This symbol is drawn with a very tight center, giving it an almost identical figure to the regular integral symbol. Presumably, the added bulk for marginal additional use earns it a rating of 'less effective'.
  ∫ Integral Could be used in the manner of a shepherd's crook, or as a billhook. Could be used as a spear thrower, combined with ⋅ or ⇀.
  → Implies; X→Y means that if X is true, then Y is also true Danish is brandishing this symbol against Ponytail, apparently using it like a spear.
  ⊥ Up tack or falsum, indicating a false proposition in logic, perpendicular lines in geometry, or the bottom element in a partial order Used as a war hammer (held by the long stem) so the opponent would be struck with either leg of the top of the T. Held by the short legs (like a wishbone), could be used as a two handed sword.

(If tipped with a knapped stone head, it could become a falsum point.)

  ⇀ Vector notation Could be used as a single-barbed spear or, if held by the other end, similarly to a sabre. The multi-functionality is probably why it is listed as more useful than the 'implies' arrow.
  –– Fraction bar (division) or overline (complex conjugate or mean) or minus sign. Hairy is brandishing this symbol against Black Hat. The single shaft would allow it to be used in the manner of a quarterstaff, , or other stick-fighting weapon. Of the symbols shown in the comic, Randall considers this one the most useful in a fight, presumably because of its greater reach than the vector arrow and its simplicity compared to the radix root symbol. Alternatively, could be handled like a pike, with a reach greater than that of an implication or vector arrow.
  𝄞 Treble clef (title text) Mentioned in the title text; this is not a mathematical symbol, but a musical symbol. The treble clef is a much more complicated symbol than those used in mathematics, hence the musician's "confidence" in his weapon. The curve at the bottom could be used as a hook, the upper curl could be used as a blunt weapon, and the tight curl of the center would serve as a better defensive shield than theta.


[A chart is shown with 30 different symbols arranged above a line with arrows in both ends and with 17 ticks between the arrow heads. The symbols are mostly in two rows, but the first two symbols from the left do not have another symbol above them, and towards the right there are a segment with three rows of symbols. Above the symbols there is a heading and a subheading. And beneath those there is a long arrow pointing right with a label above it.]
Mathematical Symbols
By how useful they would be in a fight
More useful
 ℝ     ∅ > ∝ π + Ψ ~ ⇒ ⋅ Γ √ ∮ ∫ ⇀
     θ ∞ ∪ ∈ ∀ ∂ ≠ # Δ ζ ℵ ρ → ⊥ ––
[Below the double arrow line are eight characters that fight each other in groups of two by using some of the symbols shown above them as weapons. They have chosen symbols that are almost above them in the chart.]
[Megan is awkwardly handling a giant "ℝ" holding it up against White Hat who is holding a "θ" (or "∅"?) with both hands, as a shield.]
[Cueball is holding an "∈" in both hands, with its "tines" pointed towards Blondie, who is swatting at him with a "#".]
[Ponytail is leaping at Danish, swinging a "ρ" like an axe, while Danish is leaning back and thrusting a "→" back at her.]
[Black Hat is swinging a long "√" like a polearm at Hairy, who is holding a long "⎯" defensively.]


  • Here is a list of symbols from left to right (by rightmost edge):
    •   ℝ θ ∅ ∞ > ∪ ∝ ∈ ∀ π ∂ + ≠ Ψ # ~ △ ζ ⇒ ⋅ ℵ Γ ρ √ → ∮ ∫ ⊥ ⇀ ––
    • Note: Where two symbols had similar right-most edges, the overlay grid on an ASUS pro-art display was used to decide which one went further right.

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Can I get aleph-null aleph-shaped throwing stars? LunarNapolean (talk) 20:18, 7 August 2020 (UTC)

I'd prefer octothorpe throwing stars. 16:19, 10 August 2020 (UTC)

Apologies to whoever added the "citation needed" that I stepped on. -- brad

That zeta looks conspicuously bad. I wonder if this comic will get a cleaned-up version uploaded. 20:51, 7 August 2020 (UTC)

Megan usually has shoulder-length hair, so the person being attacked by Ponytail is probably not Megan... except in so far as all brunettes in this comic are called 'Megan'. LtPowers (talk) 20:53, 7 August 2020 (UTC)

Is one of them Danish? And one of them Megan? 22:49, 7 August 2020 (UTC)

I think Randall is underestimating the weapon utility of psi. There's a real-world martial arts weapon that looks somewhat like it. 22:04, 7 August 2020 (UTC)

I think he’s also seriously underestimating the value of keeping your fingers attached to your hand. Swords have guards for a reason. I’d pick the contour integral over anything else there.

Considering the title text, a bass clef looks pretty formidable, close to a bat'leth. Nutster (talk) 00:31, 8 August 2020 (UTC)

Yes, but the treble clef is the one in the title text, and that’s nothing like a Klingon bat'leth. I removed the comment from the table. Adam1729 (talk) 02:09, 8 August 2020 (UTC)
If we're talking clefs/klingon weaponry, get on the viola clef. That's bat'leth AF. It's even known as a "K Clef" in some circles. You could do some pretty hefty damage with a viola clef.
Or a viola... 22:05, 9 August 2020 (UTC)
What's shown in the explanation right now is a G-clef, not a treble-clef; I believe this to be an error. (Also the G-clef symbol doesn't display reliably on mobile.) The symbol 'treble-clef' is only mentioned by name, not shown in the comic; therefore, I think we should show a symbol that is actually specifically a treble-clef. A clef note only becomes a treble-clef note if the lines are included; otherwise it could mean any clef note. U+1D11E (𝄞) is specifically a G-clef by name in its Unicode listing. The Unicode emoji U+1F3BC named 'Musical Score' is defined in Unicode as displaying a staff (or stave) with a clef note laid along the lowest of five horizontal lines, thereby forming a treble-clef. Not only is 🎼 (U+1F3BC) a Treble-clef while 𝄞 (U+1D11E) is not, 🎼 displays more universally on mobile than 𝄞 does. Therefore, I'm going to correct the error. Please discuss the issue here if you think 𝄞 should be used instead.
ProphetZarquon (talk) 19:45, 10 August 2020 (UTC)
Yeah, 𝄞 (U+1D11E) just doesn't display on my chromebook. Me[citation needed] 01:58, 20 September 2023 (UTC)
As someone who has studied a lot of music theory, I can say from a purely musical standpoint, there is no difference between a treble clef and G-clef. They are simply two different names for the same thing. While both symbols have their advantages and disadvantages, it really boils down to clarity vs. accessibility. 🎼 may be displayed on more devices, but 𝄞 makes things clearer on the specific symbol by getting rid of the staff. I personally have no preference on which symbol ExplainXKCD should use, but there are my arguments for each side. Trogdor147 (talk) 03:17, 31 October 2023 (UTC)

These “weapons” seem strangely appropriate for xkcd’s stick figures... -cpl

Agreed :)

Are we sure White Hat is holding empty set? There don't appear to be points extending outside the circle in which case I think he's actually holding Theta -jc

Can I use the LaTeX mathwitch? 10:03, 8 August 2020 (UTC)

First time editor here,hope I get the notation right! Question on the pi link to wikipedia: I put in the double link to the main page and the disambiguation because unfortunately https://en.wikipedia.org/wiki/Pi_(disambiguation)#Mathematics doesn't include the mathematical constant definition (though it is listed at the top of the page). Thoughts? Alan g (talk) 10:38, 8 August 2020 (UTC)

I think that’s the is proportional to” symbol rather than just alpha. They are similar but have different Unicode symbols. Thoughts?-- 12:01, 8 August 2020 (UTC)

You're definitely correct. (They don't even look that similar...) It's the direct proportion symbol, not the symbol for Alpha.
ProphetZarquon (talk) 18:03, 8 August 2020 (UTC)
Annnnd somebody has added 'alpha' back in despite alpha not being represented in this comic at all in any way. Even the wikipedia article on Alpha mentions that it is often mistaken for the Proportionality symbol, but the 'fish' looking symbol is not alpha, never means alpha & saying 'Proportionality or Alpha' encourages more confusion. I'm reverting the edit. PLEASE DISCUSS HERE IF YOU THINK ALPHA BELONGS FOR SOME REASON.
ProphetZarquon (talk) 19:45, 10 August 2020 (UTC)

I don't think → is "implies", particularly as we've had ⇒ earlier. → is often used for "maps to", as in f: x → f(x) 19:24, 8 August 2020 (UTC)

→ and ⇒ often mean two different kinds of "implies". The single arrow is for the boolean operator that takes in two truth values and outputs a truth value. The double one is for "things on the left justify/prove things on the right", in somewhat of a metalanguage. Here's an example of two different ways of saying Modus Ponens with the operators: ((p→q) ∧p)→q vs p→q,p⇒q Alan g (talk) 03:52, 9 August 2020 (UTC)

Are we sure that is the multiplication sign (center dot)? The placement makes it seem more a decimal point. 21:51, 8 August 2020 (UTC)

Confusingly, math conventions in some countries use a low dot for multiplication, though it's not as common as it used to be. That's in addition to all the other things that bare dots can represent in math. Personally I can't "see" any particular set of semantics for that symbol, I just see a dot. 22:51, 8 August 2020 (UTC)

We see a greater-than, but no less-than. Where would that appear? I think >≠<, in fact ><<, if wielded properly. Though if thrown, either/both could be a multi-use projectile... 09:08, 9 August 2020 (UTC)

Took me a little bit to figure out what you meant, but I think it hasn't been clarified which side is the point or sharp side, so until that is clarified "<"=">"

Anyone else think the title text is hinting at a pun? "I got scared because his weapon looked like treble" sort of thing?

Where is the opening parenthesis? 21:43, 9 August 2020 (UTC)

Here! (: 09:33, 10 August 2020 (UTC)

If I could implement a line integral as a machine rather than a symbol, I'd use that to encircle my enemy. WIN! Cellocgw (talk) 13:15, 10 August 2020 (UTC)

i'm sure that the radix symbol could be used in a manner similar to sly cooper's crook 13:32, 10 August 2020 (UTC)

The partial derivative could be used as a grappling hook, which is possibly why it's ranked where it is. I think Delta is higher than the A shape, because there isn't a chance of damage from push-back - both ends of the latter are pointy. 18:20, 10 August 2020 (UTC)

I'm kind of shocked that no one has mentioned that the WhoWouldWin subreddit had a couple of threads which likely inspired this comic: https://www.reddit.com/r/whowouldwin/comments/epm3tf/which_number_would_make_the_best_weapon/ and https://www.reddit.com/r/whowouldwin/comments/eq1ezq/which_letter_in_the_alphabet_would_make_the_best/. Of course, xkcd took it one step farther, but I wouldn't be surprised if those threads got the ball rolling. 19:31, 10 August 2020 (UTC)

I think you're right! It's a shame Reddit threads lock in less than a year... I hope someone created a new thread based on this comic, linking the only old threads for reference.
ProphetZarquon (talk) 21:22, 10 August 2020 (UTC)

The treble clef at the bottom should probably be standardized, not that colorful clip art thing. Dogman15 (talk) 09:33, 12 August 2020 (UTC)

Something about this seems like Animation vs. Math by Alan Becker... AzurusTheFirst (talk) 02:54, 22 February 2024 (UTC)

Would anyone else like to join me in creating an anti-Gamma interest group?[edit]

Just asking. 21:47, 9 August 2020 (UTC)

I hate the U+0393 "Γ" symbol for Gamma as rendered in sans-serif, as it too closely resembles the Extended ASCII code 218 line-drawing character "┌". The lowercase "ɣ" is more distinctive, but it just doesn't look like a voiced-velar fricative "gh" sound to me. The IPA symbol "ɤ" for a close-mid back unrounded vowel actually looks more like a "g" sound to me for some reason, but that's taken & I don't think this situation is worth fighting for it. When shown with serifs as Randall drew it, the uppercase "Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \Gamma " Gamma symbol ([1]) is plenty distinctive enough for me to identify visually. The sans-serif version bears very little resemblance to the symbol displayed in this comic. I would go so far as to say that displaying this symbol without its serifs is a bad practice. How do we go about getting some serifs shown on that symbol for more devices\fonts?
ProphetZarquon (talk) 21:22, 10 August 2020 (UTC)
Russian Г is basically a "beck-de-corbin"; but, again, you have to make sure you use Russian "Cyrillic" uppercase symbol Г, not, for.ex., lowercase "г". RELEASE!!! 21:06, 14 August 2020 (UTC)

What about... digits?[edit]

I thing, Ein, Dos, Troix, Ne, Fem, Lu... E, Eight, Kyuu, Ten would do better.

1 is literally a spear 2 is quite a matcatcher-like hook. 3, unlike ballets, also has hook-like uses 4 is not just a spear. Its an "alspiece": a Swiss sword-made-of-broken-lance. 5 is a sickle. 6 is a... padlock. Attached to a chain, its being able to do damage 7 is a scythe 8 is... a chain. Combine 6 and 8 to fight in bludgeon fashion 9 is... I know, it's almost like 6. But in some fonts, 9 has serifs 6 doesn't get (9-corner 9, opposed to 6-corner 6) -- 9 would be a hammer. (talk) 21:18, 14 August 2020 (please sign your comments with ~~~~)

treble clef is not showing. 00:30, 10 April 2024 (UTC)Dev