Unicode mathjs extension proposals

[gwhitney]

This is a companion discussion to #3364 collecting and adding proposals for extensions to the mathjs expression language based on the use of Unicode symbols, now that the standard is long established and it is becoming easier and easier to enter and deal with such characters.

Notation	Meaning	Issue	Comments
√x	sqrt(x)	#767	Concluded this works best as a tightly-binding unary prefix operator.
30°	30 deg	#2672	Seems uncontroversially good, simply a unit alias.
∑[2,3,4]	sum([2,3,4])	#3075	Seems like it would work well as a unary prefix operator.
∏[2,3,4]	prod([2,3,4])	#3075	Ditto.
∑_(i=1)^5 i^2	sum(range(1,5).map(_(i) = i^2)	#3075	This extension to the meaning of sum is what was specifically proposed in #3075, and as it is has a direct AsciiMath notation, it is a proposal that makes a lot of sense.
∏_(i=1)^5 i^2	product(range(1,5).map(_(i) = i^2)	#3075	Ditto.
a·b	a*b		Not clear one needs a synonym for *, but it is the standard mathematical notation.
v × w	cross(v,w)		Unicode really shines here.
v ⟂ w	v * w == 0		Not clear how "necessary" an abbreviation, but good if supporting Unicode in general.
v ⊙ w	dotMultiply(v, w)
A ⊗ B	kron(A, B)
f∘g	_(x) = f(g(x))		Function composition could actually be handy in arguments to map etc.
x ≠ y	unequal(x,y)		Seems straightforwardly good.
a ∧ b ∨ ¬c	a and b or not(c)		Again, not clear we need more synonyms, but if we are supporting Unicode in a wholehearted way, these are the standard symbols.
a ∪ b ∩ c	setUnion(a, setIntersect(b))		Would make working with sets in mathjs much easier.
a ∆ b	setSymDifference(a, b)		Ditto.
a ⊆ b	setIsSubset(a, b)		Ditto.
a ⊇ b	setIsSubset(b, a)		Ditto. (If we are supporting these, we likely should also support their strict forms and negated forms as well.)
a ∖ b	setDifference(a, b)		Ditto, but note this really has to be the Unicode setminus symbol because ascii backslash is just so overloaded already.
x ∈ S	x is an element of S		Not sure if we have a mathjs function for this: just check if x appears as an entry of S.
r ∠ t	r*(cos(t) + i*sin(t))		More of an engineer's notation than a mathematician's one, but clear and compact.
⌊x⌋	floor(x)		One of the best reasons to embrace Unicode :)
⌈x⌉	ceil(x)		Ditto.
a ≤ b	a <= b		A synonym is not needed, but clearly should be supported if mathjs embraces Unicode.
a ≥ b	a >= b		Ditto.
x ≡ y (mod m)	(x-y) % m == 0		Also similar.
a ∣ b	b % a == 0		Along the same lines. Note that this is a different vertical bar than the usual |
f′(x)	derivative(f, x)		Note this is different from any ascii quote character, so unambiguous.
Γ(x)	gamma(x)
𝒫(S)	setPowerset(S)
ζ(z)	zeta(z)
℃			That's a single Unicode character that means degree Celsius. There's one for Fahrenheit as well, not that anyone uses that unit ;-).

Unicode is so huge, it's unlikely this table is exhaustive. So feel free to recommend others or discuss any of these proposals.

[gwhitney]

Oh and of course per #3335, if we were going to go unicode-positive, we should allow standard constants like τ, π, and perhaps γ.

[gwhitney]

A few more possibilities:

Notation	Meaning	Comments
x², x³, etc	x^2, x^3, etc	There are also superscript letters, not sure if we want to support those as exponents
v₁, v₂, etc	v[1], v[2], etc	Would we want to support double-subscripting matrices, e.g. if A = [1,2;3,4], should A₂₁ be 3? I don't see why not. It's very standard and readable.
z̅	conj(z)	I don't think an overbar on a variable is really used for much else besides complex conjugate
x − y, x ∕ y	x - y , x/y	If we are supporting Unicode we need to be sure to interpret the actual Unicode minus symbol and division slash appropriately .
∀x ∈ S x > 7	S.every(_(x) = x>7)	Actually, mathjs does not currently provide access to the every method of arrays, but there is no reason it shouldn't have a similar function that works with sets, arrays, and Matrix objects, whether or not it can be invoked from Unicode in this way. Note that the precedence of this universal quantifier expression would have to be chosen carefully or else you will almost always need parentheses around the condition: ∀x ∈ S (x > 7)
∃x ∈ S x > 7	S.some(_(x) = x>7)	Similar comments.
∅	[]	That's our current representation of the empty set.
½, ⅗, etc	fraction(1,2), fraction(3,5) etc
∛x	cbrt(x)	Should obey exactly the same syntax rules as √
‖v‖	norm(v)	At least with Unicode, the character is unambiguous, but you do still have to decide how to disambiguate when subexpressions close: does ‖a‖b‖c‖ denote norm(a)*b*norm(c), or norm(a*norm(b)*c) ?

[gwhitney]

Also see the very old https://github.com/davidson16807/unicodemath which was (is?) a sort of mathjs clone based on using unicode symbols, with its own version of typed-function for operator overloading and everything! Since it was so long ago, it made the dubious choice of using letters from other alphabets that happen to look like math symbols, which I think is a poor choice, but it could still be a mine for operations that one might want to allow in MaJE (the "MathJs Expressions" language), just with the proper modern symbols as needed.

[josdejong]

Thanks Glen, nice overview! Good to keep these ideas in mind when working on #3420, not all notations needed for the ideas above are supported by the parser right now I think.

[gwhitney]

And now we have #3593 -- we should look for equivalents to standard delimiters in other writing systems supported by Unicode as well.