the most DISLIKED math notation

The rules of exponents make sense. 3^-1=1/3 and x^-1=1/x but f^-1 doesn't mean 1/f
f^-f is one of the most problematic math notations or one of the most disliked math notations. f^-1 actually means the inverse of the function f. We use this notation a lot, especially for trigonometric functions. For example, tan^-1(x) means the inverse tangent, or we can also write it as arctan(x).
BUT!!! What exactly is tan^-2(x) supposed to mean?
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[@waynemv]

How about for inverse functions we just write it with the function name itself inverted? So not arctan() by rather, uɐʇ().

[@blackpenredpen]

Wayne VanWeerthuizen Yooooo how did you type that upside down?????

[@blackpenredpen]

Btw, that reminds me of the omh and hmo : )))) Too bad I am on phone and don’t know how to type those symbols up.

[@kartik4152]

@@blackpenredpen mho*

[@blackpenredpen]

-Chromium- yes

[@trueriver1950]

So given the famous Hamiltonian function H (X, ....) how would we write its inverse?

[@SkateGeneva]

this is why normal people use arctan

[@AugustinSteven]

No. Normal people use brackets.

[@waynemv]

Normally, one would use atan2(,) instead of arctan(), in order to avoid division by zero in its argument.

[@regulus2033]

IDK, for me most common notation is "arctg", "arcsin" e.t.c.

[@kishorekumarsathishkumar1562]

@@AugustinSteven what like (tan(x)) to mean inverse tangent, or tan()(x), or is it tan^() (x)

[@peterkrammer8968]

@@AugustinSteven Hi Steve, I think that the primary problem is not with brackets.. problem is with naming of the function ; different functions (tan() and arctan()) using same name with adding few numeric symbols (^-1) which are in mathematics already defined for another purpose. Power symbol ^ has some previous definition, which is different. Writing tan^-1(x) for meaning inverse tangent function is VERY VERY inappropriate symbolic representation. Much better representation are also argtan(x), invtan(x) , atan(x) .... but I dont see a reason, why dont use name arctan(), which is strictly defined; and chance of misunderdstaning is really minimal. Have a nice day.

[@GGJChannel]

This video could be entitled “the reason for the existence of cotangent”

[@federicodc]

and of secant and cosecant

[@sunilparekh4581]

😂😂

[@angelmendez-rivera351]

Well, historically, the reasons for the existence of the cotangent are completely different, but yes.

[@HugoHabicht12]

Where do you see the cotangent???

[@nou3227]

@@HugoHabicht12 because tan^-2(x)=cot^2(x)

[@fluffymassacre2918]

I hate inconsistent notation more then people expanding (x+y)^2 as x^2+y^2

[@blackpenredpen]

Fluffy Massacre lol

[@Nylspider]

Actually same. At least the x^2+y^2 is understandable

[@sunilparekh4581]

@@Taurus388 he has said that extra 2ab is the energy to make the bracket keep in contact I literally died laughing 😂😂

[@deltacream]

That's quite the butchering.

[@sunilparekh4581]

@@Taurus388 I guess he was out of India🙄🙄

[@kevinmackie4045]

I prefer the arc notation, it clears up the confusion

[@rot6015]

i agree

[@AznJsn82091]

But it's like he mentioned, traditionally it's with the negative1 superscript which was how I was taught and I'm completely fine with it. Personally, writing the arc is a bit much for me.

[@kevinmackie4045]

Jason Lu Yeah I get that, maybe they could shorten the arc part to just a or alpha so it's like atan(x) or αtan(x)

[@angelmendez-rivera351]

Jason Lu Except writing arctan on a computer is more manageable than tan^-1. Also, the arctan notation is inherently consistent and less confusing.

[@angelmendez-rivera351]

My solution is to never actually use the notation tan^2(x) unless I authentically mean tan[tan(x)]

[@xaxuser5033]

"YOU DON'T HAVE THE INVERSE TANGENT ANYMORE" lengend as always

[@blackpenredpen]

XaXuser hahaha thanks!!!

[@chrissekely]

Sounds a bit like Yoda.

[@shayanmoosavi9139]

@@chrissekely more like arnold😂😂😂

[@thecarman3693]

@@chrissekely The inverse tangent you have no more.

[@chrissekely]

@@thecarman3693 🙂

[@thisismycoolnickname]

in Russia we only use the arc notation. We also write "tg" instead of "tan". I guess you could call it our math dialect :D

[@Macieks300]

same in Poland

[@cpazca]

Same in Peru.

[@Liberty5_3000]

And also ctg instead cot or cotan, cosec instead csc and sec intead sc

[@markorezic3131]

Same in croatia, tg and ctg, inverses are arctg and arcctg

[@happydmitry]

Same in Ukraine

[@InTheBeginningTheUniverseWas]

f^2(x) = f(f(x)), but not for sine! sin^2(x) = (sin(x))^2. I don’t like these inconsistencies, it should be all or nothing with maths.

[@angelmendez-rivera351]

I agree. I say that sin^2(x) should be treated as different notation as sin(x)^2. Simple

[@iantaakalla8180]

Especially given sin^-1(x) can mean arcsin(x) or 1/sin(x)

[@wolffang21burgers]

The main problem is the notation tan²(x) = (tan(x))². With usual functional notation f²(x) = f(f(x)), so it would make more sense for tan²(x) = tan(tan(x)), which is consistent with tan^-1(x) = arctan(x). But the prior was introduced because without brackets common expressions of tan x ² etc were ambiguous, probably due to laziness of not wanting to write brackets. With this logic, tan -²(x) should equal arctan(arctan(x)).

[@HaveANceDay]

Just write arctan(x) and be happy

[@Zejoant]

Yakes up too much space on the paper. Annoying to write

[@ianweckhorst3200]

I have a better solution, I believe to fix all of this, and add more use cases to it, tan^2(x) shouldn’t be the same as tan(x)^2, it should instead represent the amount of times you’re applying the tan function, this makes the negative powers more consistent by making tan^-2(x) be arctan(arctan(x)) instead, of course if you want to put exponents between two parentheses, you do still have to expand it, but I think from a functional analysis perspective it makes sense

[@michaelyap939]

there is nothing wrong of (tan⁻¹(x))² .

[@smathlax]

If I designed the notation this is how I would do it: sin^n(x) = sin(sin(...(x)...)) {n times} ∀n∈ℤ+ sin^0(x) = x sin^-1(x) = arcsin(x) sin^-n(x) = arcsin(arcsin(...(x)...)) {n times} ∀n∈ℤ+ sin(x)^p = (sin(x))^p ∀p∈ℝ Of course, this only works if we *always* use brackets whenever we use the trig functions, but I don't see this as a problem as we already do so when we write f(x), and all of the notation I've just described is already in use when we talk about a general function f(x), e.g. f^2(x) = f(f(x)), f^-1(x) is the inverse function of f, f(x)^2 is (f(x))^2.

[@zeeshanmehmood4522]

I think you're notation is pretty cool but, how useful is sin (sin(X)) And what about arcsin(arcsin(X))?

[@smathlax]

Well it's not necessarily about usefulness, but rather about consistency. This is how function composition notation works when we write f(x) so I don't see why it wouldn't work for other functions. This would apply to non-trig as well, e.g. ln^-1(x) would be e^x, and ln^-2(x) would be e^e^x, etc. EDIT: and of course this prevents the confusion that we saw in the video.

[@zeeshanmehmood4522]

Okay, so if f^2(x) is f(f(x)), what is f'2(x)?

[@shayanmoosavi9139]

@@zeeshanmehmood4522 if you mean f'^2(x) it's f'(f'(x)). If not I haven't heard of this notation before. If you mean differentiate it twice it's actually f''(x) or f^(2)(x). (2 should *always* be in a parentheses in order not to be confused with powers). Hope that helped.

[@smathlax]

What @shayan moosavi said. For some reason KZbin didn't notify me when the responses happened and I only saw it now because I got a notification stating "Somebody liked your comment" 😐

[@nico99nrc]

I my opinion writing arctan(x) is much better notation than tan^-1(x)

[@xCwieCHRISx]

yea one time i forgot the ^-1. Since then I use arc... and paranthesise its more save. Most of my failures are that I dont copy the equation parts right rather than do math wrong ^^. I have to do this because my handwriting is messy xD

[@Nylspider]

Yeah the inverse tangent notations is so dumb imo But that begs the question, should the inverse of some arbitrary function f(x) be called arcf(x)? I would be fine with that tbh as long as it's consistent across all of mathematics

[@kelvinyonger8885]

@@Nylspider I mean arc makes sense for trig because angles, not sure about all of math though.

[@Nylspider]

@@kelvinyonger8885 honestly I prefer consistency over names that make sense

[@taekwondotime]

Just use *"inv sin"* for inverse sine, *"inv cos"* for inverse cosine, and *"inv tan"* for inverse tangent. Drop the word "arc" altogether because it makes no sense.

[@litigioussociety4249]

The inverse tangent should be known as Euler's Confusion.

[@Nylspider]

XD

[@adrianhdz138]

XD

[@mysticdragonex815]

YD

[@DontYouDareToCallMePolisz]

XD

[@HeckYeahRyan]

ZD

[@angelmendez-rivera351]

In the grand, vast mathematical literature, (f^n)(x) almost always has referred to iterations of f, not multiplicative powers of f. Multiplicative powers are expressed as [f(x)]^n or more simply f(x)^n, since what you are repeatedly multiplying is the output of f given some input x, not f itself. tan^(-1) is should be the correct notation for inverses. The problem would be solved if people simply stopped writing (tan^2)(x) to mean tan(x)^2, because for any other function, (f^n)(x) and f(x)^n are most definitely not the same. Yes, it may be a tradition, but some traditions objectively should stop existing. This is one of them. However, since it's probable that mathematicians will never stop committing the aberration of writing (tan^2)(x) to mean the "tangent of x " squared, I think it would be best that instead we used arctan as opposed to tan^(-1). As for tan^(-2), it would be best to use it to represent arctan(arctan(x)). Then at least some sort of rule could be established where negative exponents do represent iteration and the positive do not. This is much less ambiguous and much more consistent than simply using every exponent as multiplicative powers except for -1 to mean functional inverse. That is just deluded.

[@xXJ4FARGAMERXx]

tan⁻²(x) should be tan⁻¹(tan⁻¹(x)) which should be inverse tangent of inverse tanget of x but it's actually interpreted like 1/(tan(x))² tan⁻¹(x) should be inverse tangent of x and it is indeed defined that way. tan⁰(x) should be x and it is indeed defined that way. tan¹(x) should be tan(x) which should be tangent of x and it is indeed defined that way. tan²(x) should be tan(tan(x)) which should be tangent of tangent of x but it's actually interpreted like (tan(x))² The thing I hate is the fact that they apply regular function notation on every function except the trig ones and log and ln. WHY?? Is it that hard to write parentheses? How do you know if logxi means log(x)i or log(xi)?

[@xXJ4FARGAMERXx]

addendum: So I actually typed tan⁰(x) in wolfram alpha and it spat out 1

[@angelmendez-rivera351]

@@xXJ4FARGAMERXx The answer is that people are lazy, and often fail to understand how to use mathematical notation correctly. Most people see no difference between writing (sin^2)(x) and writing sin(x)^2, even though they are notation representing different things.

[@daxramdac7194]

​@angelmendez-rivera351 True, indeed we often are lazy creatures as well as creatures of habit. However, in this case, I don't think it was due to laziness. It was more so that mathematicians in more classical times wrote with a certain style, a certain elegance, and with intention too. They saw the aesthetics of mathematics, not just at a conceptual level, but in its raw expression through notation. My point is that, when they wrote, as best as they could to avoid confusion and write with consistency, seeing that that was also part of the beauty as well as the utility of it all, to make the notation/machinery usable, you can't always have it perfect. But they relied on people reading their writings to have a certain level of mathematical maturity, that they will be able to "get it" without much hassle and move on, so long as it wasn't totally convoluted, bulky, awkward, ugly, etc.

[@angelmendez-rivera351]

@@daxramdac7194 I don't think what you have said addresses anything I have said at all, so as far as I can tell, this definitely still just boils down to laziness.

[@ΜΙΧΑΗΛΚΑΤΤΗΣ]

Arc notation clears all confusion and it actually sounds really cool

[@radadadadee]

I agree, but it's six letters long. I don't like that.

[@ΜΙΧΑΗΛΚΑΤΤΗΣ]

@@radadadadee you can write atan but be sure to specify it

[@xwarrior760]

@@Errenium Russian?

[@BetaKeja]

But how do you handle f^-1(x)?

[@angelmendez-rivera351]

@@BetaKeja Exactly. That is the question. This idea of "arc" notation only creates further inconsistency, as it only makes sense from trigonometric functions.

[@scarbotheblacksheep9520]

Why don't we always just use arctan instead of tan^-1?

[@sahilkumar-qf9tg]

Because we always tend to find shortcut ways

[@jmjmjmjmjmjmjmjmjmjmjm]

I prefer at(x)

[@tipoima]

In Russia we do. But we also use "tg(x)" instead of "tan(x)", so...

[@julien31415]

Because we always use the notation f^-1 for any function

[@Pinjesz]

You also write sqrt (x) as (x^2)^-1 ?

[@BriTheMathGuy]

I’ve personally always disliked radical notation. Radicals just end up confusing students and creating more rules and complications. Always using exponential notation would clear a lot of things up, not to mention make simplification and derivatives easier. I also vote arctanx, the negative exponents in that case just confuse students and are harder to type and write with.

[@vangrails]

What is radical notation?

[@BriTheMathGuy]

vangrails square roots, cube roots, etc.

[@CarlyDayDay]

It's either radicals or fractions. Pick your poison.

[@f.p.5410]

@@CarlyDayDay Or if you want to be even more inconvenient you could use multiple exponentiation: √(3) = 3^(2^(-1))

[@CarlyDayDay]

@@f.p.5410 acceptable

[@MillerIndustriesInc]

tan²(x) should get redefined as tan(tan(x)), then the - exponent could still mean inverse: tan-²(x)= arctan(arctan(x)). Or make a new term for the inverses, like tān(x), sīn(x), cōs(x)

[@justabunga1]

There is no shorthand notation for tan(tan(x)), arctan(arctan(x)), etc. We just leave as this type of notation as the composition notation.

[@scepticusverisimillimenonm8450]

No need to redefine. It's already defined correctly in your Linear Algebra book. f²(x)=f(f(x))

[@cosimobaldi03]

oof

[@zigzagblade]

@@justabunga1 actually, no. The exponentiation of endomaps is well defined and largelly used in group theory.

[@Fox0fNight]

Because I found the notation ^(-1) so confusing, I had decided to use a ~ above the function instead

[@thenewtonium3521]

Another benefit to the arctan notation is that it is a _specific_ inverse (that is, with -π/2 < x < π/2), rather than just a generic inverse of tan. Same goes for arcsin, arccos, arcsec, arccsc and arccot, of course... Fun fact about inverse hyperbolic functions by the way: the reason they use ar-, rather than arc-, is because 'ar' stands for 'area', since the resulting value tells you about the area between a hyperbola and a line from the origin to a point on the line, whereas 'arc' is of course refering to the arc length of a unit circle.

[@lyrimetacurl0]

Funnily enough, if you type arcsin(1.5) in Wolfram Alpha, it gives you an answer (a complex number).

[@GogiRegion]

(+Lyri Metacurl) That’s because that’s mathematically correct.

[@landsgevaer]

I've seen tan^← notation (with an "exponentiated back arrow behind the function name"). To me, that is clearest. I would prefer it over arctan because it works on any function. Disadvantage is that it is uncommon. But, to be fair, the standard f^-1 extends nicely to function composition f^2(x)=f(f(x)), so if anything using it as the square instead of (tan(x))^2 seems like the bad notation.

[@jeeasprint8254]

BUT NO VIEWS WHY???

[@blackpenredpen]

tanuj kumar I have no idea....

[@TreniFS_]

@@blackpenredpen I think it's about the problem of counting from multiple inputs. Tom Scott made a video about this. Every time someone watches a video, it is supposed to add 1 to the views counter; the problem is, if two or more people watch the video at the same time, they see the same views count, thus they add 1 to the same number, instead of adding 1 to the original count and then 1 again to the new count. To solve this problem, they had to work out a system which sometimes lags and takes a while to output the actual count.

[@TreniFS_]

@Abbhinav Bharadwaj That is also true. But no need to be so rude.

[@sunilparekh4581]

@@TreniFS_ whom you are telling 🙄🙄

[@certainlynotthebestpianist5638]

(tan x)^n and no more confusion. Tbh, I prefer this notation even with logarithms

[@angelmendez-rivera351]

Exactly. That or tan(x)^n, consistent with the fact that tan(x) = tan x is just some number y, and powers of numbers are written y^n. This also distinguishes it from (tan^n)(x), which should be reserved to denote n applications of the tangent function, which is consistent with how the notation is used in mathematics for functional iteration and operators, keeping in mind that operators are just a special type of function. Consistent, unambiguous, and concise. Finally someone who understands.

[@cubicardi8011]

tan⁻²(x)=42

[@vivekmathur3514]

Tan ( tan( 42)) = x

[@vivekmathur3514]

Boi.

[@strikerjazz101]

No Vivek

[@ZXC-ey3zu]

@@vivekmathur3514 Get to know interesting people around you. Swipe right to like, and swipe left to forget. Tantan is a fun and easy way to connect you with the people you like.

[@jenga_boy7321]

Ri Soo Keu eat wiener

[@morganmitchell4017]

That's why I just use arctan(x). It sounds better too!

[@MrRenanwill]

Yeah... I use it too. But, I see no problem in notation (tan^-1(x))^-1=1/arctan(x), but problem comes when someone try to do tan^-n( x) or tan^n(x). hahahaha

[@CoasterMagicX2]

I actually hate the whole exponent, log and root notation more than this one. Ones a word, ones a symbol, ones a position, and they are all related. Really awful stuff.

[@infernoxen9998]

Yeah your dp already told so.. 😅

[@groszak1]

I hate the "dx" thing in integrals, if you have an integral of d then it becomes ⌠1 │ ddd ⌡0

[@bob53135]

@@groszak1 Not a problem if you do it right: ∫𝒹·d𝒹

[@groszak1]

@@bob53135 what if both d and 𝒹 are used as variables?

[@bob53135]

@@groszak1Then you've got a confusing and ambiguous way to write variables (should be italic) and notations (should be non italic) and this is why you can't have nice things :)

[@simphiwedlamini.]

I like how you always have something to share with us...shows you passion in maths...love your content man

[@blackpenredpen]

Simphiwe Dlamini thank you!!!!!

[@mariokraus6965]

Arctan x is better than tan^-1 in order to prevent confusion. Longer but better and clearer

[@DancingRain]

Or, taking a page from the programmers: atan(x)

[@mariokraus6965]

That's good as well

[@15schaa]

Wait, do we still have the inverse tangent?

[@redactdead]

When dealing with superscripts after the name of a function (e.g. ln²) I like the idea of thinking about it in terms of function composition, with the _functional power_ representing the number of times the function is composed with itself (i.e. ln²(x) = ln[ ln(x) ] ). And thus negative functional powers represent compositions of the inverse function, allowing us to use power laws (adding powers of the same base) to simplify compositions of different functional powers (e.g. sin⁻¹( sin²(x) ) = sin²⁻¹(x) = sin¹(x) = sin(x), where sin²(x) is a twice composition of sin, i.e. sin²(x) = sin( sin(x) )). However, in practice I always end up writing and interpreting positive superscripts on trig functions as powers, -1 as inverse notion and wholly avoiding any other numbers.

[@mtaur4113]

Composition is equivalent to matrix powers if f is represented by square matrix multiplication, so there's one more reason this is reasonable. But trig functions show up squared and beyond pretty often so it wins out on convenience.

[@mtaur4113]

I have in the past used a big composition circle with "k=1 to n" as you would with a sum or product, once upon a time. It was a past life when I was working with a self-similar set and its finite approximations by unions of polygons which were composed similarity mapping images of the convex hull.

[@benyseus6325]

This really isn’t that big of a deal though because most people just denote the reciprocal of tan(x) as cot(x) meanwhile the inverse of tan(x) because of tradition will remain as tan^-1(x) or arctan(x). It’s really rare that confusion will occur from this.

[@Ligatmarping]

It can still happen for a general function. Like f^2(x), f^(-1)(x), etc..

[@drpeyam]

Totally agree!!!

[@blackpenredpen]

Hold on, did dr. P just comment on my video????

[@JoshuaHillerup]

I vote we say that tan^2x = tan(tan x), and tan^-2x = arctan(arctan x)

[@xaxuser5033]

And this is the tangent

[@subinmdr]

Start a petition

[@angelmendez-rivera351]

I 1000% agree. I have always said this. In fact, in my own writings, I never write (tan^2)(x) unless I legitimately mean to say tan[tan(x)]. I always write tan(x)^2 or [tan(x)]^2 depending on the software.

[@TheReligiousAtheists]

That doesn't make sense. You're just making more parentheses a necessity. Barely ever does one ACTUALLY NEED to use something like tan(tanx) or arctan(arctanx). However, if somewhere (like in a paper) you do need to use something like tan(tan(...(tanx)))...) often enough to need notation for it, you can make up your own notation or use what you just said; tan²x will be tan(tanx) and tan^(-2)[x] will be arctan(arctanx) and use parentheses for the actual squaring. Math notation is flexible and mathematicians don't care about notation as long as it makes sense and conveys what the author means efficiently.

[@srpenguinbr]

@@TheReligiousAtheists I agree, it is very rare to use function iterations like that. You could also use subscript: tan_2(x)=tan(tan(x)).

[@КонстантинЛукин-с4п]

Arc notation is much easier to understand. And in Russia we use tg and arctg instead of tan and arctan

[@marcioamaral7511]

In Portuguese speaking countries we use tg cotg as well

[@andreffrosa]

@@marcioamaral7511 I'm portuguese and I use tan, cos and sen (not sin) and both tan^-1 and arctan for inverse trigonometric functions

[@justabunga1]

I heard some chemistry, physics, and math teachers use the regular log to mean the base e. Would that confuse the students also? I thought regular log without a base is used to mean the common logarithm (base 10). To me, ln is used for natural log (log base e).

[@ThomasTheThermonuclearBomb]

That's disgusting, ln(x) is the natural log (base e) and log(x) is base 10. Often times though, it really doesn't matter what base is used because of the change of bases property

[@Bhuvan_MS]

In maths, most of the times log=ln In chemistry and physics, log≠ln This is what I learnt in my college actually. The reason this situation arises is because when it comes to maths, we always prefer to use natural logarithm due to the properties of e. Whereas in physics and chemistry, we often use common logarithms as they are easier to calculate and also because we use powers of 10 to plot logarithmic scales such as pH scale etc.. That's why I prefer using ln notation for natural log everywhere. It just avoids all the confusion.

[@Patrickhh69]

No views 473 likes No views=473 Worst math notation, no is 473

[@minecraftrodney]

I think I will rebel and just be using overlined for its inverse, not that bad idea (though better than that disliked notation)

[@karryy01]

Then write 1000 arctan(x) and tan^-1(x) in 2 different papers. Compare how much ink to write them. You will know why inv functions exists

[@PuffPound]

Personally I don’t like the dot notation for derivatives in respect of time

[@MattMcIrvin]

Weighing in on the epic feud of Isaac Newton vs. Gottfried Wilhelm Leibniz?

[@allaincumming6313]

In Classical Mechanics is a tradition and a sort of identity, because it's the Newton notation. But just in Classical Mechanics.

[@David-km2ie]

Iike the D notation of D'Alambert and Euler

[@angelmendez-rivera351]

Using the notation D[y(x)] to refer to "the derivative D applied to the function y" is just objectively the best notation. It is simple and concise to use, without weird symbols that cannot be typed, and is not notationally misleading like the Leibniz notation, and it lends itself to generalizations for multivariable calculus, differential geometry, and linear operator theory much better. It makes the study of ODEs that much more convenient, as well.

[@orvinal2883]

i dislike inverse tangent notation, just use arctan

[@pratyushparmar4595]

hey blackpenredpen pls tell how to integrate 1/(x^4 + x^2 + 1)^1/2

[@dolphinlunggrin6594]

it would have taken less time to put that into wolframalpha and you would have gotten the result instantly. www.wolframalpha.com/input/?i=integrate+1%2F(x%5E4+%2B+x%5E2+%2B+1)%5E1%2F2 and seeing the result, it seems obvious that there is no pretty solution: integral 1/sqrt(x^4 + x^2 + 1) dx = ((-1)^(2/3) sqrt((-1)^(1/3) x^2 + 1) sqrt(1 - (-1)^(2/3) x^2) F(i sinh^(-1)((-1)^(5/6) x) | (-1)^(2/3)))/sqrt(x^4 + x^2 + 1) + constant where F is an elliptic integral of the first kind

[@Apollorion]

How happy would you be if, A: instead of being integrated the expression's antiderivative was derived ? B: instead of blackpenredpen someone else derived it ?

[@groszak1]

1÷(x⁴+x²+1)¹÷2=2÷(x⁴+x²+1)

[@qubix27]

I dislike this notation at all. In particular, in Russia NO ONE writes inverse trig functions like that. And for us sin^-1(x) = (sin x)^-1 ≠ arcsin x, so we don't have any problems.

[@scottrice6969]

this remind me of a test question i had in college. i don't remember the entire question but i remember the concept it was written without parenthesis it was written in a way where you couldn't tell if it was (ln x)^2 or ln (x^2) which is a very important distinction for using the properties of logs it turns out it was supposed to be the former it turned out i was the only one in the class who got the question right just because I interpreted it that way while everyone else interpreted it as the latter.

[@Koisheep]

I'm taking a dynamic systems course and it physically hurt the first time we used exponent notation for composition. Right now thinking sin²(x) USUALLY means sin(x)² and not sin(sin(x)) feels like a triple mental backflip. On the other hand, I think the prefix "arc" is commonplace notation/vocabulary choice in Spain, so I never had major problems with the ⁻¹ I GET the exponent notation comes from group theory because composition groups are not generically abelian but it still feels like there was a better choice for notation and people actively decided to make it the hard way.

[@OnlyTheQuack]

No views but 500 likes... Yeah, well done, you should use the black pen for the easy counting..

[@blackpenredpen]

OnlyTheBest Magic!

[@teavea10]

Next I'd like to see a video on how/why exponents are used the way they are for 2nd, 3rd, 4th ... derivatives.

[@justabunga1]

The subscript for the derivative notation is a Leibniz notation (e.g. d^2y/dx^2).

[@stephenfreel2892]

It probably has to do with how nth derivatives can be written as a summation that involves binomial coefficients. And binomial coefficients are used when raising a sum of two terms to an exponent. So maybe it’s related to that?

[@georgedoran9299]

teavea10 It’s probably when you think of d/dx as an operator and you are talking about how many times you are applying the operator

[@angelmendez-rivera351]

@@georgedoran9299 Yes, that is the correct explanation. This is why the D[y(t)] notation to refer to the derivative D applied to the function y is just better. Less confusing, less misleading, and easier to type and read.

[@xXJ4FARGAMERXx]

@@angelmendez-rivera351 the problem is when the thing your differentiating is not y(t), just y.

[@cmck362]

My guess is that tan^(-1) is explicitly defined as arctan in math programs because it's a common notation, but other negative numbers are just done how the program usually handles exponents.

[@rohithbanda9127]

5:16 turn on captions.

[@billeyzambie]

XD

[@Cpt.Zenobia]

Hi, Just a question, What is the inverse of this function: u = (e^(−2) * (2^x)) / x! || so, x = ???? I'm having a problem with the factorial part. some on SE suggested that the inverse is n = e exp(W(1/e * log(n!/√2π))) − 1/2 where (W) is the Lambert W-function. BTW this is the CDF function of the Poisson distribution. thanks dude. =)

[@chinesecabbagefarmer]

I didn't get a notification for this video 🤔

[@blackpenredpen]

I haven't gotten a view yet, but there are like over 200 likes...

[@chinesecabbagefarmer]

@@blackpenredpenI noticed the same thing! Your ratio of views to likes is very good .. if it exists.

[@AugustinSteven]

I don't like the notation.

[@blackpenredpen]

Down the Rabbit Hole lol

[@linearealgebra4006]

I dislike this notatation 😠

[@b4ttlemast0r]

Inverse functions aren't only used for functions like trig where we have a specific name for the inverse though. I think it's fine if you use it consistently like f^-1(x) for inverse function and f(x)^-1 for 1/f(x), just never put the power between the function and the argument when you actually mean f(x)^n, and consistently put parentheses when you mean f(x^n).

[@ZoroarkLover98]

I loved this video! But I felt shivers down my spine with all these notations. Math is supposed to be a confusion-free land, when there is no ambiguity.

[@Wyldina]

hahahahaha, maths confusion free.. Yeah, sure :D

[@ZoroarkLover98]

@@Wyldina Let me live in my ideal world hehe

[@jannesl9128]

I just always write down arctan, cause everything else is just confusion xd

[@saswatsarangi6669]

6:06 i thought it would be "tan inverse of tan inverse X"

[@justabunga1]

That’s like saying arctan(arctan(x)), so no shorthand notation for this.

[@angelmendez-rivera351]

@@justabunga1 Except that is the correct shorthand notation.

[@erebo1423]

I dislike the notation but like the video xD

[@jmd448]

Never liked the -1 notation. It breaks the symmetry of the notation. I always use arc notation.

[@moistness482]

i will call this the devil's notation

[@DjSamvy]

I'm team atan(x), mostly because it's often the one used in programming languages :)

[@Gawkie]

/r /TeamArctan yeah #'s are old now

[@matthewleitch1]

My preference is to use atan, acos, asin for inverse functions. I also always use brackets around the argument to the trig function and the brackets I use are square brackets. e.g. tan[45]. I then put powers in the usual sense on the outside of that final square bracket. e.g. tan[45]^2 = (tan[45])^2. The square brackets are an idea from Stephen Wolfram and it appealed to me more perhaps because I've done a lot of computer programming. Now, having read misotanni's comment, I will be willing to use the function notation, e.g. tan^2[45] = tan[tan[45]] but it's flirting with danger as sin^-2[30] would then mean asin[asin[30]] and I suspect few people would interpret it that way.

[@poisonoushallucinations3168]

Personally I prefer the inverse trigonometric function notations since it’s quite similar to how you would write the inverse of a function f(x) being f^-1(x). Of course, it was confusing when it was first taught, (much like logarithms and exponentials), but I think using having “arc” placed in front of every trigonometric function is more out of place than having the inverse of the function. (Yes, by this logic, I also dislike using cosines, cosecants, and cotangents) Another gripe I have with the arc functions would be how they’re seemed to obtain to measure the length of something (arc) instead of the angle of something. While the angle in radians corresponds to the length of the arc formed by that angle on a unit circle, I feel the differentiation between the arc length and the angle has to be made more apparent Of course, if we could come up with a better notation for these things or agree on one single way to express something, math would be a lot simpler and easier to understand.

[@Pete-Prolly]

I love how I aced Trig, thought I was taught to graph anything, & saw this in Calc1: ƒ(x) = x³ cos(x) 🤷🏻‍♂️ At 1st I thought I had the wrong textbook, we didn't have to graph it but still...the horror. (Yes, "Apocalypse Now" reference)

[@MarkMcDaniel]

They should've just kept the reverse trig operators as being the arc-functions.

[@vangrails]

I prefer arctan, arcsin...I mean what does ln^-1x mean? Does that mean e^x? I don't think so.

[@Otomega1]

Lol, true

[@burk314]

Except, that's exactly what ln^(-1)(x) does mean mathematically. No one actually writes it that way, but it would be correct.

[@Otomega1]

solve this equation and find v with your logic of ln^(-1)(x) = e ^ x: u ^ (- 1) = v In fact you can not, just because you can not mix language of mathematics and mathematics, that would make no sense, a function can not be added, multiplied or put to a power. it's an abuse of language, in my opinion.

[@johnschdtpedersen977]

@@Otomega1 burk314 is correct. The notation is ambiguos and that is why it is hated. Exponential functions were historically refered to as antilogarithms but now they have their own name. You ask for the solution of two equations. The latter is purely algebraic and the other contains the inverse function, so they are not the same unless you choose to interpret ln^(-1)(x) as the reciprocal of ln(x) which it could ALSO mean and that is why it is ambiguous. I do not understand why anyone would want to use the inverse of a function when that inverse actually has a name i.e. tan^(-1)=arctan etc. If that function was a person, it would be rude to refer to that person as the inverse of a person ;)

[@justabunga1]

That’s another way to write as e^x. We don’t see that type of notation like this of what you said in your comment.

[@Temari_Virus]

Unpopular opinion: instead of using arctan, tan^2 x should equal tan(tan x) instead of (tan x)^2 I think this notation would be more consistent as we have things like d^2(x) = d(d(x)) and not (dx) ^2. After all, we're writing the 2 in front of tan, and not the whoele expression tan x, so doesn't it make more sense to do tan twice instead of the whole expression twice? Edit: I should also touch on tan^-2 x. Since we're writing in superscript, it would make sense for the power rules to apply. So I suggest that tan^-2 x = (tan^2)^-1 x = (tan tan)^-1 x = tan^-1 tan^-1 x = (tan^-1)^2 x = tan^-1 tan^-1 x It works out mathematically, it's consistent, and it just makes so much more sense

[@ccbgaming6994]

I agree

[@Tzizenorec]

Pretty sure that isn't unpopular at all.

[@tambuwalmathsclass]

If tan⁻¹(x) is not = (tan(x))⁻¹ as we all know• So why is tan²(x) = (tan(x))² ?

[@NotBroihon]

Because whoever came up with tan⁻¹(x) := arctan(x) was a dumb idiot.

[@mevnesldau8408]

In Ukraine we always use the arc notation

[@zackmercurys]

I don't /shrug

[@GMPranav]

I think paranthesis are still kinda underused. People are lazy to put paranthesis so they invented weird notation. As a programmer eho deals with 6 billion paranthesis in one line I think it should be normalised.

[@Alolyn]

YOU _DONT_ HAVE THE INVERSE TANGENT ANYMORE

[@Γιώργος-ε6τ]

When I guessed the tan^-2(x), I thought it would be something like this: if tan^-1(x) is the inverse of tan(x) then tan^-2(x) might be tan(tan(y))=x and then solving for y. However I am glad the real meaning was simpler.

[@Peter_Schluss-Mit-Lustig]

I always thougt applying ^-1 to the function means applying the function -1 times (because tan^-1(tan^1(x))= tan^0(x)=x this always made sense to me personally)

[@WindsorMason]

Yeah, this is the functional composition notation f^{n}(x) means applying f n times :) it's the collision between that and the exponential shorthand where cos^{2}(x) means (cos(x))^2 that causes the unfortunate trouble :/

[@Peter_Schluss-Mit-Lustig]

@@WindsorMason fun fact: i knew the function composition notation before the exponential shorthand form caued me to screw up a math question

[@MrCmon113]

Precisely.

[@justabunga1]

tan^0(x)=1. tan^-1(tan^1(x))=x or another way of saying it as arctan(tan(x))=x.

[@mambosenkoatruaywnn1429]

Hi , here is a strange equation which I did not manage to solve, maybe you will understand and succeed, I would like it if you could show us... F=rot4(degree19)^119+256f(A,B)div(m) Number 4 from the rotor, and number 19 from the gradient are written down as antisymmetric indices, but here I can't write like that , everything else is as you can see ... Thank you .

[@chikyushimin]

Arc notation is better

[@mambosenkoatruaywnn1429]

Sorry, there was a mistake in the editor,,,, the correct equation is like this F=curl4(gradient19)^119+256f(A,B)div(m)

[@MathNerd1729]

Really? Okay, who was the rebel that disliked?

[@mimzim7141]

tan^(-1+0)(x) tan^(-1*1)(x) gives cot(x) tan^(cos180)(x) gives something completely different

[@gourabghosh5574]

I am a follower of you. After I pass class 12, I will make a math channel REDPENBLACKPEN.

[@Jinx-iw6zb]

Very original name

[@sab1862]

Is REDPENBLACKPEN the inverse channel of BLACKPENREDPEN? :D

[@vincentpace]

@@sab1862 So would (BLACKPENREDPEN)^-1 be equivalent to REDPENBLACKPEN or 1/BLACKPENREDPEN?

[@estebson]

People that write tan¯¹(x) for arctan(x) (and equivalents for the other trigonometrics) are the same people that write log(x) for ln(x). The second one is a pet peeve of mine; I thought we had already established that log(x) was for base 10 logarithm, don't create unnecessary confusion by also using it for base e!

[@KelfranGt]

lim is my least favorite, because you gotta write it down millions of times ;-;

[@wurttmapper2200]

Just write the function, and the x->a Saves a lot

[@waynemv]

I dislike using superscript notation for inverses. One because it causes confusion with exponents, as you discussed. But also because with inverses the number -1 is just used as a placeholder symbol and not as an actual meaningful numeric value! In that case, inverse maybe ought to just be it's own non-numeric symbol. Or maybe just append "inv" to the function name. But, I also dislike how exponents and powers both use superscript notation, while logarithms use function notation. Yes, that does provide a compact way to write down polynomials. But I would have preferred a unified notation for all three, given they are all the same relationship, just with the free variable in a different position. Others have already suggested a triangle notation instead. Were it all up to me, we'd only use superscripts in mathematics for the indices of tensors.

[@Polaris_Babylon]

I use: tan^-1(x) as arctan(x) And (tan(x))^-1 as cot(x)

[@henrytang2203]

For inverse function notation in general, maybe we should change it to "f" arrow instead.

[@heliocentric1756]

Also ln^-1(x) on Wolfram Alpha is exp(x)

[@blackpenredpen]

Heliocentric lollll

[@terapode]

I have a particular trigonometric notation. Firstly, I live in Brazil and in portuguese sine is written "seno" so sin(x) is written sen(x); cos(x) is cos(x) and tan(x) is tan(x), although I particularly write tg(x). The inverse functions are arcsen(x), arccos(x) and arctan(x) but I write as asen(x), acos(x) and atg(x). C´mon... much easier write atg(x) than arctan(x), isn´t it? It makes expressions much shorter.

[@asusmctablet9180]

Personally I really hate and despise how economists write things like cⁱₜ (using superscripts as indices along with subscripts, in this case "consumption c of person i at time t"). It literally causes mental block in me, I just can't read it. I learned math back in the 80s and if you used superscripts for anything except exponents or Einstein notation, you would literally get beaten over your knuckles with an iron rod. But nowadays economists think they are math gods because they took second year calc and maybe passed real analysis, therefore they get to make their own math. Drives me nuts.

[@saysikerightnow3914]

tan^(-1)(x) doesn't mean 1/tan(x) because cot(x) already means that, so it's not actually that confusing.

[@animishprateek6172]

NO VIEWS , 324 LIKES. KZbin DRUNK AGAIN

[@melonenlord2723]

^-1 means inverse of something. Is it after a function, it's the inverse function, is it after a number, it is the inverse number. With bracket around, it meant inverse of the whole thing, without inverse of the last thing. Inverse is only -1, not -2, so -2 makes no sense with inverse funktion, so it gets interpretated as something to the power of it. So for me it is pretty clear, that tan x² = tan(x²) and tan x^(-1) = tan(x^(-1)).

[@destroyercs5720]

i thought that means applying inverse tangent function to times.

[@angelmendez-rivera351]

That is what it should mean, but this gets obscured by the fact that people write (tan^2)(x) to refer to tan(x)^2 when they should not.

[@Lucroq]

The function x^-1 is its own inverse. So basically (x^-1)^-1 = x^-1. Can you think of a more confusing mathematical notation?

[@brucefrizzell4221]

I thought tan ^-2 (x) would be arctanh(x) . :-) I always use arctan (x) instead of tan^-1(x) . Loved the jazz exit.

[@3ckitani]

Before I really learned about functions, I thought that f²(x) means f(f(x)). For example sin²x is sin(sin x).

[@justabunga1]

There’s no shorthand notation for this e.g. tan(tan(x)) is still tan(tan(x)).

[@JuvStudios]

Trigonometric functions are like that but not for all functions.

[@dudurododoizi8547]

maybe tan^-2 (x) can be interpreated like arctan (arctan (x))

[@angelmendez-rivera351]

Dudurododo izi That IS objectively how it should be interpreted since (f^-1)(x) has always meant the inverse function of f and (f^2)(x) has always meant f[f(x)] in the literature. Don't know who came up with the terrible idea of (tan^2)(x) = [tan(x)]^2 and (sin^2)(x) = [sin(x)]^2, but it is a bad tradition either way.

[@valeriobertoncello1809]

But the inverse funcion of arctan is tan so I think it should mean tan(arctan(x))

[@dudurododoizi8547]

@@valeriobertoncello1809 obviously tan(arctan(x)) =x

[@valeriobertoncello1809]

@@dudurododoizi8547 exactly

[@justabunga1]

Not really but that actually means the same as (cot(x))^2. arctan(arctan(x)) has no shorthand notation for that or for repeated composition of functions.

[@Mical2001]

It's weird to me that people don't *always* use parentheses when dealing with trig functions. When you have a function 'f', you say that y = f(x), not that y = f x !! So why, then, if trig functions are functions, too, does everyone not just use parentheses‽

[@artey6671]

tan^2(x) sometimes has another meaning, namely composition: tan(tan(x)). I guess composition can be regarded as a multiplication when you don't have one. For example, consider linear maps f, g and h between two fixed vector spaces. Then we have fg+fh = f(g+h), where the "multiplication" is now composition, meaning the set of said maps is a ring, which is quite nice. That said, it can surely cause confusion, but I guess tan^2(x), tan(x^2) and tan(x)^2 represent all different meanings this kind of term can have.

[@electricengine8407]

This kind of stuff happens in the JavaScript programming language, and it's use is widely discouraged

[@priyanksisodia5889]

Hi sir, blackpenredpen

[@sonicpawnsyou]

I hereby present to you an unambiguous and fully generalized notation to invert functions: (x)uɐʇ Thank me later

[@younesabid5481]

I used to write arctan(x) but after watching your videos and seeing when tan(x) and tan-1(x) cancel each other out I mean OH MY GOD IT'S SO SATISFYING !!!!!!

[@Peter_Schluss-Mit-Lustig]

tan^-1(tan^1(x))=tan^0(x)=x (implying tan^1(x)=tan(x) and tan^0(x)=x

[@angelmendez-rivera351]

1815-1898 Iron Man tan^-1(tan^-1(x)) = x is most certainly not true.

[@Peter_Schluss-Mit-Lustig]

@@angelmendez-rivera351 oops fixed it

[@blackpenredpen]

Hahaha thanks

[@MrEdrftgyuji]

tan^-1(tan^2(x)) != tan(x)

[@_wetmath_]

why i dont use arctan is because im too lazy and prefer to write -1 instead of arc because it takes slightly less time. very bad reason i admit, if i had 12 weeks to finish my math paper i would write arctan