10 ms·
Much less than, Much greater than symbols
- kelnos 5y agoThis led me to learn: ⋘ U+22D8 VERY MUCH LESS-THAN ⋙ U+22D9 VERY MUCH GREATER-THAN ... which for some reason made me chuckle.
- iratewizard 5y agoThis is getting ridiculous. How long until we have things uncountably more than other things?
- infogulch 5y agoCould you combining-character an aleph-1 symbol on top of one of these?
- thaumasiotes 5y agoIn math, when you care about the details of how small one thing is compared to some other thing, you probably use little-o notation. It's the same thing you're probably familiar with from computer science. So if y is small compared to x, you might say y = o(x), or if it's really small compared to x, you might say y = o(x²), or if it's really really really small, o(x⁶)... whatever you need for your proof. (o(x²) is more impressively small than o(x) because x < 1)
- HPsquared 5y agoNice That reminds me of something about the Greek letters Omega (Ω, ω) and Omicron (Ο, ο). O-mega means "big O" and O-micron means "small O".
- emilsedgh 5y agoIsn't that +∞ and -∞?
- masklinn 5y agoIt seems closer to the aleph numbers.
- jimktrains2 5y agoSome infinities are countable.
- marcosdumay 5y agoYes, but the GP's idea extends to larger infinities.
- deleted 5y ago[deleted]
- isoprophlex 5y agoThis triggers a doge meme in my mind... Wow, such very much less-than!
- afiori 5y agonow I want the also the absurdist VERY MUCH LESS-THAN OR EQUAL
- stavros 5y agoOh God yes
- askvictor 5y agoWould that be an implicit inclusive or (i.e. very much less than, much less than, less than, or equal), or exclusive (very much less than or equal, but not much less than or less than)?
- HPsquared 5y agoExclusive, I reckon.
- drran 5y ago0 ⋘===8
- askvictor 5y agoI want "much equal to", and "very much equal to". Which, in scientific terms, or floating point comparisons, would probably make some sense
- Izkata 5y agoGot me curious so I just did a search for "less than" and found some more interesting ones: ≰ NEITHER LESS-THAN NOR EQUAL TO - https://www.fileformat.info/info/unicode/char/2270/index.htm https://www.fileformat.info/info/unicode/char/2270/index.htm ≨ LESS-THAN BUT NOT EQUAL TO - https://www.fileformat.info/info/unicode/char/2268/index.htm https://www.fileformat.info/info/unicode/char/2268/index.htm ⪒ GREATER-THAN ABOVE LESS-THAN ABOVE DOUBLE-LINE EQUAL - https://www.fileformat.info/info/unicode/char/2a92/index.htm https://www.fileformat.info/info/unicode/char/2a92/index.htm
- thih9 5y agoHow is the ⪒ used?
- this-pony 5y agoI could see people using this symbol when proving for example that x = 0. Say they first show x \geq 0 and then x \leq 0. Then they've shown x ⪒ 0, and therefore x = 0. But yeah, a symbol like '⪒' for this purpose is maybe natural to write on a blackboard, maybe not so much in electronic form. I'm not aware of any other purposes.
- petters 5y agoIf a ≤ b or b ≤ a. That does not always have to be the case. For example, x ≤ y could mean that x is a descendant of y in a tree.
- bruce343434 5y ago>LESS-THAN BUT NOT EQUAL TO what's the point of this? This is already implied with just "<"
- layer8 5y ago"<" is used to represent partial-order relations, and partial orders are often non-strict (i.e. really less-than-or-equal). Then LESS-THAN BUT NOT EQUAL TO can be used to denote that a particular pair is partially-ordered but not equal (i.e. strict partially ordered). In the same vein there’s also SUBSET OF WITH NOT EQUAL TO (U+228A).
- beerandt 5y agoWhen you're trying to convince STEM professionals to go from hand written design documents to computer generated, you'd better be able to reproduce every little symbol that they're accustomed to using, or else it's just an excuse for non-compliance. Something like: (X+1)/X ≈ 1 for X ⋙ 1 We actually use a bit for showing derivation work on engineering design documents, especially for stating requirements for an equation to be valid. Or for equations that need to be simplified to be algebraically solved. A one line equation might have two pages of definitions, assumptions, and constraints. In this case, ⋙ is used to mean not just much greater-than, but big enough to justify whatever the simplification it is that you're making. Eg, 10 could be considered much greater (≫) than 1, but is still a 10% error in the equation above.
- DominikPeters 5y agoTo me, a << b also connotes that the inequality a < b is "robust". For example, even if we mismeasured the values of a and b quite severely, we'd likely still have a < b. Or perhaps later on in my mathematical argument I will consider values that are a bit larger than a, say 5*a. If I claimed before that a << b, it means the reader can be assured that I won't do anything to a that'll make it bigger than b.
- marcosdumay 5y agoAlso, a <<< b connotes that for all practical purposes a + b = b.
- simonblack 5y agoThat depends on the context or the language you are using. In some computer languages those signs can mean entirely different things. In the bash scripting language '>>' means 'append to the named file' as in: echo "what?" >> my_file In C, '>>' means 'shift the value right by one bit' while '<<' means 'shift the value left'. However, in C++, '<<' can also mean 'write to standard output'. No confusion at all possible, ya think?
- anandoza 5y agoYep, this blog post is in the context of using these symbols in mathematics, where this is the only usage of << and >> that I've seen. I agree that programming languages, it can mean various things, just like < and > mean different things in C++ vs bash vs drawing arrows.
- booleandilemma 5y agoI mean, are we gonna talk about ‘=‘, ‘==‘, and ‘===‘?
- yjftsjthsd-h 5y agoDon't forget the languages that use := for assignment! And golang, which I vaguely think can use both = and := to do slightly different assignment (assignment vs initialization?).
- Jtsummers 5y agoGolang uses := to represent declaration and simultaneous assignment to at least one new variable. = only works if the LHS variables have already been declared. If you use :=, at least one of the variables on the LHS has to be new. So the following snippet would be an error: a := 3 ... a := 4 This is valid: a := 3 ... a = 4 As is this: a := 3 ... a, b := 4, 2 And this is an error again: a := 3 b := 4 a, b := 1, 2 Since neither a nor b are new variables.
- 5y ago
- zabzonk 5y agoI think every sentence in this post is more or less wrong. Also, we do not want to introduce non-ascii glyphs into programming languages - APL is about the only time this has been tried and (merits of the language aside) it was hardly a success.
- kelnos 5y agoScala also used non-ASCII arrows in some places, though you could always use a two-ASCII-char version instead. I remember some IDEs had auto-replace rules to give you the "fancy" version, but I think that syntax is deprecated now.
- nerdponx 5y agoAm I missing something, like another page of text? This looks like a short exposition of the symbols and some examples of their use in math notation.
- anandoza 5y agoIt seems pretty correct to me (excluding a few things I'm not familiar enough with to evaluate). Is it possible that you misinterpreted this as a post about << and >> in code rather than their usage in mathematics?
- zabzonk 5y agoPossibly so, but this is a site that tends to concentrate on programming rather than maths. Anyway, I stand by my opinions about introducing extra glyphs into programming languages.
- Jtsummers 5y agoJulia is a rather popular (though not the most popular) language which makes extensive use of Unicode symbols in the language. I had some fun with it at the start of this year building out a package for astrodynamics where, thanks to being able to use math symbols, the code was a near exact translation of the formulas in the textbooks I was using. Fortress is a much less popular language (perhaps, to be accurate, it should be described as a failed experiment) which was intended to permit the same thing. Offering an ASCII, Unicode, and prettified presentation of code. There are quite a few other languages which support Unicode characters, though most of those don't aim for the same symbol heavy presentation of mathematical code. However, the submitted article is not about programming languages so your post and this response are both off-topic.
- SamReidHughes 5y agoA funny thing is, I've seen Congressman Thomas Massie (EE undergrad, MechE masters) casually use this notation on his Twitter account.
- divbzero 5y agoThe issues we face are complex and technical enough that I would like our leaders to be well-educated, preferably better educated than me or the public in general. It reminds me of the time ‘orthogonal’ came up in oral arguments and it was such a novel word for the Supreme Court that a justice paused to ask and remark on the word.
- alexchamberlain 5y agoI feel like orthogonal is a fairly basic logical concept, which makes that a little concerning.
- wruza 5y agoI once made a bet that most people know what a prime number is, being sure that at least for my closest environment of 5-7 highly educated people it is undoubtedly true. A bet was lost after a quick asking round - I’ve got only a couple of confident answers. But what hit me hard was not that the specific term was forgotten, but that most of them were not even familiar with the concept. Another thing I figured out too late is that after someone (not me) explains a complex topic to a group, everyone seems to agree by silence and ready to get to work on it. But my habit of asking if they understand the terms A and B and how it all works made me discover that essentially nobody gets it. People just don’t ask questions and delay their research for some reason. A thought that maybe they do that all the time in every situation or conversation is pretty discomforting.
- ptr 5y agoI'm coming to the same conclusion; it blows my mind how people can just run off and confidently implement something vaguely related to what was talked about. People seem very sure about what's supposed to be built, its context, and how. No questions asked. (am I the stupid one for not understanding?) Turns out they just made something up in their head that didn't make sense. As a leader, how does one work with such people? It feels condescending quizzing your team members to verify understanding and I can't spend my time specifying work down to the most minute details.
- jrochkind1 5y ago≪ / U+226A / MUCH LESS-THAN ≫ / U+226B / MUCH GREATER-THAN not to be confused with: « / U+00AB / LEFT-POINTING DOUBLE ANGLE QUOTATION MARK » / U+00BB / RIGHT-POINTING DOUBLE ANGLE QUOTATION MARK Switching one for the other is a frequent tactic of phishing attacks. Kidding.
- madrox 5y agoMaybe not phishing attacks, but certainly text-to-speech attacks
- jrochkind1 5y agowhat is the goal of that attack?
- madrox 5y agoMostly tongue-in-cheek. Twitch donations get text read aloud on streams at times: https://www.twitch.tv/a_seagull/clip/BlightedPreciousEchidnaTBTacoRight https://www.twitch.tv/a_seagull/clip/BlightedPreciousEchidna...
- Vanit 5y agoHuh, it's funny but I have seen >> to the nth degree used in casual internet slang to indicate a similar concept. Almost certainly just used intuitively for emphasis, but it's interesting that it's also a real notation.
- kelnos 5y agoYes, I remember on IRC in the 90s people would use ">" to mean "better than", and ">>" to mean "much better than" (like "chocolate >> vanilla"). I still see this usage occasionally on other platforms (matrix, slack), but not all that often anymore.
- ghoshbishakh 5y agoIsn't this notation very common and widely used in school Mathematics?
- thaumasiotes 5y agoWell, sort of. The notation is common in the sense that if you're taking a math class where the concept is relevant, that's what's going to be used. But it's not so common to take a class like that.
- taneq 5y agoI’d imagine both are the result of the same emphasis-by-repetition.
- travisgriggs 5y agoI saw these and thought “meh, nested generic” I’m ruined.
- taneq 5y agoI always think RPN object. :P
- thih9 5y agoThere are more puzzling mathematical symbols in “Supplemental Mathematical Operators” unicode block: https://en.m.wikipedia.org/wiki/Supplemental_Mathematical_Operators https://en.m.wikipedia.org/wiki/Supplemental_Mathematical_Op... . I’m especially fascinated by ⪋ ⪌, ⪑ ⪒, and ⪓ ⪔. I wish that wikipedia page came with notes on the intended use (or a link to the relevant page) for every symbol.
- petters 5y agoFor partial orderings, we can have neither a ≤ b nor b ≤ a. Perhaps it's that?
- joppy 5y agoI’ve seen this used in the context of statements like “x is less than, equal to, or greater than y if and only if w is less than, equal to, or greater than z”. Which might be written “x ⪋ y if and only if w ⪋ z”.
- wodenokoto 5y agoWhen is x not less than, equal to, nor greater than y ? Is it like, x is complex and y isn't?
- joppy 5y agoThe original text is meant to mean a triple of statements: x < y if and only if w < z x = y if and only if w = z x > y if and only if w > z This can be tiresome (and obviously confusing) if you need to keep using it in many places, hence some authors define the odd symbol to start with, and then use that in many places.
- gilrain 5y ago“If and only if” also has a shorthand in “iff”.
- SAI_Peregrinus 5y ago
- azepoi 5y agoif you want to know where the expression for sqrt(a+b) come from it is a taylor series expansion of the first order (linear approximation) of f(x)=sqrt(a+x) arount 0. f'(x)= 1/(2*sqrt(a+x)). For x close to 0 you get f(x)≈f(0)+x*f'(0)
- williamdclt 5y agoI remember using this in maths lessons. IIRC, it was used to analyze how functions behave, so `999*x + y + 99999` where `y ≫ x` behaves basically the same as just `y`. Something like that, maybe it was for big-O analysis or something.
- samvega_ 5y agoThat's a bad wording for mathematics. It usually means "sufficiently greater than", and only makes sense as part of a sentence or wider context where what it is sufficient for is clear. "Much greater than" is misleading because it could be confused with a valid statement by itself.