Besides being the first and last letters of the alphabet, why do we like abc and xyz so much?
In algebra, both abc and xyz are used, and occasionally pqr as well. Why those specifically? And for vectors, one of the notations is ij. Why is that? And in programming, it is a convention to use loops with ijk as the index. Why not something like cde (for count) or tuv(tally)? Why index? And in coordinates, we use xyz. Why is that? Why not abc? What about ijk, pqr, etc.? Also, why is y and z flipped in some 3D programs, and why is in some 2D programs, up is negative y rather than positive?
Finally, why do these kind of triplet letter combinations always use the Latin alphabet? Why aren’t there any conventions that use Cyrillic, Greek, Arabic, or the Latin letters with cool hats? Is it because English became the dominant language? That can’t be the only reason, as we still use some Greek symbols for some things like pi, phi, alpha(for angles as well as the particle), beta, gamma (rays), etc. The Greeks figured out things like coordinates, algebra, etc., so why Latin letters? Is it because of the Romans?
The following are not-properly-researched theories I made up
I think x came from “unknown”. Perhaps at one point people used a cross (x) to represent something unknown. The symbol may have extended to mean “variable”. Then math advanced and people begin making graphs with variables and the logical next variable is y.
abc are used when these variables run out, because they are intuitive to pick.
ijk in programming loops came from “index”. The reason we use “index” and not “count” or “tally” is because in early computer days, someone decided to use “index”. For example, your browser still automatically fetches “index.html” if a path doesn’t return a valid response.
ijk in vectors may have come from quaternions, which in turn came from complex/imaginary numbers. Since i represents “imaginary”, that could have been carried over to vectors.
The xyz orientation in 3D program is purely a choice by the developer. It is which ever orientation they are comfortable with developing.
y is up because of it’s commonly used for graphs, but y is down for computer graphics. The top-left corner of a screen is (0, 0) and as the position moves downwards, y increases.
We do use Greek letters in other coordinate systems! For example, a spherical coordinate system often uses (r, θ, φ). I think these symbols may have already had connections with other aspects of math (like θ is commonly used for angles). They are not used generally to avoid confusion.
Everything you said fits what I remember. Although I remember abc being more for constants where xyz were variable. Maybe I’m remembering wrong.
i,j,k for basis vectors is an interesting one. Historically, Hamilton invented his quaternions before any notion of “vector” existed (as an algebraic object; I believe the geometric notion is older). (So, what, did people just write out everything componentwise? Yes, yes they did. For example, that’s how things like Maxwell’s equations were originally presented.) The reason he chose i,j,k for the unit quaternions is because i was already in use for complex numbers, and i was in use for complex numbers probably to stand for “imaginary”.
The notion of “vector” was invented specifically as a “de-algebraicization” of quaternions. People did not like working with quaternions because they thought it was weird, particurlary because they required 4 numbers but space only required 3, so the likes of Gibbs and Heaviside gutted them and gave us modern 3D vector calculus. The reason we work with the dot product and cross product in 3D is specifically because, given pure imaginary quaternions v, w the product (vw) has real part (-v.w) and imaginary part (v x w).
Also, your last paragraph is somewhat misinformed. Sequences of Greek letters are used all the time, and Hebrew letters are also used in set theory to denote cardinalities (though I can only think of aleph and beth, no sequences of such letters). It is also well-known that some people like to use Japanese よ (yo) for the Yoneda embedding in category theory. But beyond Latin and Greek, there is quite a dearth.
A lot of it comes down to convention and convention is often set by those who did it first or whose work dominated a field. The whole mathematical notation system we use today is just a convention and is not the only one that exists, but is the one the world has decided to standardise to…
Rene Descartes is usually regarded at he originator of the current system. He used abc for constants and xyz for unknown variables amongst other conventions.
Sequential letter sets are easy to use as they are easily recognised, and convenient as a result, plus are generally accepted to have non specific or less specific meaning. For example:
a2+b2=c2
That formula is a much simpler concept to get round using sequential leffer than:
V2 + G2 = z2
Under the common system, when you don’t use sequential letters it also implies much more specific meaning to the individual letters, and that can introduce ambiguity and confusion.
When writing a proof there can be many many statements made and you’d quickly run out of letters if you didn’t have a convention for accepting abc are variables and can be reused.
We also do use symbols from other alphabet sets, and alpha/beta/gamma is commonly used trio. But in mathematical notation there are a huge range of defined constants and symbols now that many have been ascribed specific uses. Pi for example. So you risk bringing in ambiguity of meaning by moving away from the accepted conventions of current maths by using other sets.
Even e has specific meaning and can be ambiguous if you need to stretch to 5 variables. When working with e it’s not uncommon to use a different string of lwtters in the latin alphabet to avoid confusion if you need to use variables
And we don’t stop at 3; abcd etc is used.
abc and xyz make sense because they lie on opposite ends of the alphabet, so if you need more variables for either sets, there’s a lot more room to maneuver. If you’re looking for the history of these variables in the context of Cartesian coordinates, I’d start with looking at Descartes’ work. This whole system originated from him, so if he used xyz, then that must be where it came from.
I’ve seen p and q used in various contexts. For example, they could be probability distributions (e.g. KL(p|q)), they could be two points. In these scenarios, we just use p because it’s the first letter of whatever they represent, and q comes after while looking similar so it suggests that they’re the same type of mathematical object.
For indexing, it’s what we commonly use just because we call it an index. You’re not counting or tallying things. It’s a reference to a location in memory. But if you are counting, then c makes perfect sense and I’ve definitely used it in that context. I’ve also used t for indexing if that index represents time. But if there’s no other meaning associated with it, then it’s just an index, hence i.
p and q often also represent arbitrary rational numbers (Q), like m and n often denote natural numbers (N).
They don’t always use the latin alphabet. In University I hated my prof using the same letter over and over again in different writing systems. x, chi, Gothic x, x with hat, x with dash, x as a vector etc. etx.
This was crazy hard for me as I internally verbalize when I read formulae, so I had to “invent” different pronunciations for evey different version of x. Because (for example) one is the vector, but the lowercase latin version is just the length of the vector.
Along with the fact that people use slightly different conventions and then conventions in math are different in the anglosphere vs here - I frequently couldn’t understand a paper or script without having an idea how things worked in the first place. A didactical nightmare.
In programming things are a lot easier, because there’s much more common convention for the field not being a few hundred years old.
Aaah, maybe that’s even the simple answer to your overall post: Conventions, even if they are arbitrary, make things easier to understand.
I see i, j - I think counting loop variable. Does it matter it is i? No. Do I think “index”? Nope.
There are code bases where some clever persons used (for example) g or p for loop variables, they are just a tiny bit harder to read - until I get used to THAT convention.
When I write code, I always try to mirror, what’s already there, to make it easier for the next guy - even if I don’t like the style.
I believe the IJK convention comes from an early programming language where those variables defaulted to a decimal type so thry were sane choices for loop counters.
Actually, the use of i,j,k as counters is older than programming. It’s more like the other way round. They implemented making variables starting with i, j, k implicitly integer by default, as i, j, k were commonly used for indexing.
X as the unknown https://youtu.be/Bq_ZgV4OURI
Index in programming - someone did that, everyone copied, became a standard
Coordinate systems - just comes to personal preferences of the creator of the engine, no “correct” way to do them AFAIK
Letters from other languages - I don’t know how it is in countries with different alphabets, but you delay teaching school kids another alphabet just for math
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