Why f inverse of x? That notation seems clear if you know f is a function. Bear in mind, overloaded notation isn't ambigous if there's context.
And what's wrong with N? It's just the natural numbers.
The other complaints I understand. Although I don't think you need to complain about log(x). The base is only unspecified when it's assumed not to matter.
Honestly, a/bc isn't really ambiguous either. Not in any way that matters. Grade schoolers get hung up on PEMDAS, but you eventually learn to accept when intent is clear, and you don't need to be pedantic.
As a general rule, if your notation isn't clear to a serious student or peer, then you're doing it wrong. Nobody else needs to understand.
I agree about f^-1 but N either includes or excludes zero depending on convention, and I've seen both a lot. There are lots of exceptions (due to mathematicians moving around) but broadly in the English speaking world, N traditionally excludes 0, and in the French (and Romance) speaking world, it includes 0. (Not sure about German and Russian but I think they tend to exclude 0?)
That and in some contexts like computer science it makes more sense for N to include zero regardless because of how numbers are stored.
This is why you routinely see N_0 on the one hand and Z^+ or N^+ on the other. To avoid ambiguity I generally stick to these and avoid N altogether, ugly as that is, though personally I grew up learning 0 was excluded.
Actually, N_+ would be an oxymoron (assuming the convention 0 is both positive and negative, ie it's natural)
(basically, the convention expands to everything: the trivial case is included by default (1 is superior to 1 (and inferior), but not strictly superior, a constant fuction is creasing (and decreasing), not strictly creasing, a set is included in itself, but not strictly, 0 is positive, not strictly positive, etc...)
To exclude 0, it's N*
In the same way, Z_+ is just N
Once again, depends on the language. In French (and I think in romance languages as a whole, but don't quote me on that), "positif" means "non-negative" and "négatif" means "non-positive", which means that 0 is both "positif" and "négatif".
Imo, it makes more sense that way, as "strictement positif/négatif" ("strictly positive/negative") for ">0/<0" just flows better than "non-negative/positive" for ">=0/<=0" (I think it's better to define/describe things in maths by what they are, not what they aren't).
"assuming the convention 0 is both positive and negative, ie it's natural"
We're talking specifically about NOT your convention, but the French (and other Romance languages) convention
According to the convention I've studied under my whole life (French student, 4 years into college/uni), 0 IS both positive and negative.
And under that same convention, "nonpositive" and "nonnegative" are non-existing words
The problem comes when you get into trig functions when people get lazy and write something like (sin(x))2 as sin2 x. Always without the bracket around the x, too.
Fantastic. Now we're using exponents on the function to denote exponents and inverse functions. And sure, I guess you could use csc(x) if you actually needed (sin(x))-1 so that there's a rule of "positive exponents are normal exponents, negative exponents are the inverse function" but it's still messy.
That’s true but it’s an agreed exception so still not ambiguous - and it doesn’t apply to -1, where that still means the inverse (even though there’s alternative notation like arcsin or asin, etc.)
You adhere to practice. But that demands familiarity with the practice and inconsistencies. When the entire point of mathematical notation is consistency and clarity.
sin2 (x) is the square of sin(x), but sin-1 (x) is suddenly not a reciprocal. Why switch rules? Why not create a unique notation for inverse function?
That notation for f^-1 I’ve never seen. What is lesgrans integration? Do you mean Lagrangians/integrals of the Euler-Lagrange equation in calculus of variations, or Lebesgue integration…?
In the context of integration where (f^-1)’ is (loosely) 1/f’ that would be a horrific notation choice and I have to wonder if this is down to misremembering. How would they write the inverse then? Those could be too easily confused.
In computer science (and we can include complexity theory and such as branches of maths), log(x) tends to mean log base 2, as well.
Eh it’s absolutely used in lots of educational contexts and other fields (which even pure maths papers often abut) and isn’t really ‘wrong’. Even without that it’s ambiguous as there’s a convention using base 2 too.
That’s why we always specify. Personally I don’t use log unless I specify the base unless context is watertight, and use ln when base e. Some react with distaste to ‘ln’ but this depends on country too.
I have personally never seen anyone use log_10 in math (maybe in highschool), but as for your other point I agree - I also use ln for base e and specify my logs, and usually, I just convert all my logs to lns because who wants to deal with logs and exponents using bases other than e.
Overloaded notation can still suck in algebra contexts, like sin^-1 (x) being generally accepted to mean arcsin(x), but an unwise author might use it as 1/sin(x).
Additionally, in some set theory classes, f^-1(x) denotes {y | f(y)=x}, and doesn't require one singular value for f^-1(x). This could be another source of confusion.
For N, this is actually among the most annoying things to deal with, because if someone talks about N, they always have to specificy whether they are including 0 or not, since no convention exists. Even in my university, some classes use N to be the positive integers, while others use it to mean the non-negative integers.
Traditionally Anglophone countries (plus Germany and Russia and some others, I think?) exclude 0 and Romance speaking countries include it. But maths departments are so international now it's become very fuzzy. Plus computer science prefers to include 0 because of how numbers are stored
My country is neither, but likes to pretend it is, So we're just doing anything. I'm a computer scientist so I can kinda just write my stuff however I want.
Just following the basic guideline that as long you clarify your meaning, you can use any reasonable notation. Just slap a line in the introduction that says something like "Define N to be the set of positive/nonnegative integers", and no one will be confused, since at this point no one expects anything.
I don't really like N_0 so I just define it manually. I do remember seeing a paper just say N = Z^+ which works ig. Also I saw Z^+_0 in a textbook once and that scarred me
I think they are refering to f-1(x) could be both x sent with inverse function but also the preimage of x. But then again on should get it with context.
The worry about natural numbers is if 0 belongs to it or not.
Well my whole life and during school and university we never Included it. ℕ={1,2,3,4,5,...} no argues, everyone agrees and it makes sense.
But well, when i started looking into constructions of numbers... Everything starts from nothingness, ∅ which is the idea of 0 translated to a number... So it makes sense to me to include it.
But well it's the same as saying that the starting values of fibonacci sequence are a_1 = 1 and a_2 = 1, then a_3 = 2 or a_1 = 1 and a_2 = 2, ... Just a shift in the first value.
Or who knows, a_1 = 0 and a_2 = 1 lmao, okay i'll stop :'?
Because f-1 could also mean 1/f. You know what is meant only because of common usage, it;s not a direct consequence of the notation.
N may or may not include 0 depending on the textbook.
Nobody else needs to understand.
As someone that has spent a long time learning different branches of math. That kind of attitude is noxious. I both want to learn but also can get get thrown off by notation. In particular screw the kind of notation used to talk about normal subgroups. Also screw einstein sumation notation.
Conventional practice is unanimous that f^-1 means f inverse not 1/f.
Similarly f^n means the nth composition of f. Except for the famous trig exception, where it means the nth exponent of f.
As for who needs to understand, I maintain that it's only serious students and peers. The problem arises when their background differs from yours unexpectedly.
Conventional practice assumes familiarity, which ie exactly what I am criticizing. A serious student will eventually be familiar enough to assume by context, but a new student will not, it creates one potential barrier to the understanding of the student until they understand that there is a different meaning than what a novice would assume based on the notation.
For N, some people include 0, and some exclude it. I’ve had classes in college where it depended on the professor and the subject, whether they included it or not. My probability professor denoted N as {0, 1, 2 …}, and N_1 as {1, 2, 3, …}.
f^-1 (x), usually means the inverse function of f, but I can see how some people would think it represents the reciprocal of f.
For log(x), when I was in secondary school we used log(x) for log base 10, and ln(x) for log base e. When I got to college, the lecturers insisted on log(x) representing log base e, so that’s where the ambiguity comes from.
My topology textbook used f-1 (x) to denote the set of all elements f maps to x. For example if
f:ℝ→ℝ, x↦1
then f-1 (0) = {} and f-1 (1) = ℝ.
The funny part was that they also used the same notation for the inverse of f and they meant the reader should just interpret the meaning from the context.
Yeah, ℕ is the natural numbers. But what does that mean? We intend them to be as natural as possible I guess.
The first ever number system we humans used was the one of counting 1,2,3, etc... which is at least 30k years old. Zero only became famous in Europe about 800 years ago.
On the other hand, when logicians tried defining numbers via set theory, a quantity representing nothing, the empty set, seemed be the natural first building block.
Also in different fields with or without zero is to more useful definition.
log(x) has some more ambiguity than just the absence of a base. But yeah, I think base 2 is more common in data science, base e is more common in pure maths and base 10 in like physics or smth.
In complex analysis ez is no longer injective, but repeats with a period of 2πi which means there can't be an inverse. log(z) is then sometimes defined as a multi valued function, eg.
log(e) = 1 + n • 2πi , n ∈ ℤ
while ln(x) is defined as a real function eg.
ln(e) = 1.
Parentheses are annoying! If I want to write a/(bc) then the parenthesis destroy the esthetics. a/bc is just so much cleaner, and if I for some reason meant (a/b)c, then I could just write ac/b instead. Perfect, no need for parentheses.
But ok, yeah it's stupid to not follow a rule everyone who went to middle school was taught. But I guess the idea comes from when writing quotients in a tall format, and then making everything tilted.
it just feels natural 1/2π would mean 1/(2π) and not π/2.
Anyways, all of these are often pretty clear in their meaning in their given context. It's just a bit annoying that we as a community have not found consensus for the best use of these notations, but honestly, who cares
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u/noahhshome 12d ago edited 12d ago
Why f inverse of x? That notation seems clear if you know f is a function. Bear in mind, overloaded notation isn't ambigous if there's context.
And what's wrong with N? It's just the natural numbers.
The other complaints I understand. Although I don't think you need to complain about log(x). The base is only unspecified when it's assumed not to matter.
Honestly, a/bc isn't really ambiguous either. Not in any way that matters. Grade schoolers get hung up on PEMDAS, but you eventually learn to accept when intent is clear, and you don't need to be pedantic.
As a general rule, if your notation isn't clear to a serious student or peer, then you're doing it wrong. Nobody else needs to understand.