# Algebraic Linguistics

One of the first things they teach you in algebra is that the letters you use to signify variables are arbitrary, and you can use whatever you want[^hjwx8cszbpb]. Like most of the 'first things' students are taught, this is almost entirely a lie: every letter has implicit connotations, and if (for example) you use "n" for a non-integer variable, it'll confuse someone reading your work. More importantly, if you don't know what symbol choices imply, it'll be harder for you to understand what an equation is implicitly communicating, making it even more difficult to grasp the concepts that are actually being laid out.

So I've decided to go through the English alphabet and explicitly explain the connotations of each character as they might be used by a \[unusually-bright-highschooler|reasonably-clever-college-student\]-level mathematician working in pure(-ish) mathematics (with a little statistics/mechanics, as a treat). If you've finished (and still remember) highschool math, you're unlikely to get much out of this: still, I suspect there are a decent number of readers who might benefit from the below being spelled out.

### a

*   The first variable in a list.
*   In mechanics: acceleration.
*   An uppercase A is frequently used as an arbitrary constant you multiply by when k is taken. (usually something like "y = Ae^(kx)")

### b

*   The second variable in a list.
*   Billion.
*   Breadth (of an object).

### c

*   An arbitrary constant you expect to add. (Compare k.)
*   The third variable in a list.

### d

*   Distance.
*   In calculus, a differential operator. ("dy/dx" means "the amount y changes when x changes"; "d/dx" means "the amount whatever-I-write-next changes when x changes".)
*   The fourth variable in a list.
*   Dimensions.
*   Depth (of an object).
*   Diameter (of a circle or sphere).

### e

*   e refers to a specific number, like pi. e=2.71828 to six figures. People like phrasing exponentials in terms of e because it's easy to differentiate: d/dx e^x = e^x.
*   Because it's a number, e is almost never used as the fifth variable in a list (or as a variable in general); a list of five or more variables is typically handled by using the same letter for all of them and distinguishing them with subscripts.
*   In mechanics problems, a capital E sometimes means Energy.

### f

*   A function. ("*f*(x)" doesn't mean "what you get when you multiply f by x"[^icqo5ci50o9], it means "what you get out when you feed x into the function *f*"; if *f*(x) = 3x+1, then *f*(2)=7.)
*   Because it's so often a function, you'll rarely see f used to denote a variable.
*   In mechanics, when capitalized: Force.

### g

*   A second function, when "f" is taken.[^0x5ipn8w6dg]
*   Gravity. A lowercase "g" means the local gravitational constant, which tells you how heavy a given mass is on Earth's surface. An uppercase "G" means the universal gravitational constant, which tells you how much mutual attraction two arbitrary masses have between them.
*   The "Giga" prefix, meaning 1,000,000,000. (If you waited 0.5Gs, you waited half a billion seconds.)

### h

*   Height.
*   A third function, when "f" and "g" are both taken.[^0x5ipn8w6dg]

### i

*   i refers to a specific number, kind of (but not entirely) like e or pi; the "imaginary" number you get when you take square root of -1. It can be conceived of as one unit up from the number line, the same way 1 can be conceived as one unit along it.[^c5v76e0slsi]
*   A part of a vector meaning "along". (If you walked 3 miles East, you could say you moved 3*i* miles.)[^0x5ipn8w6dg] Pairs with x.
*   In programming, the outermost part of an iteration loop. (If you're a waiter collecting orders from five tables, and you're on your third table, i=3.)

### j

*   A part of a vector meaning "forward" or occasionally (if working in two dimensions, one of which is vertical) "up". (If you walked 3 miles East and 5 miles North, you could say you moved 3*i+*5*j* miles[^ag6pem82o3].)[^0x5ipn8w6dg] Pairs with y.
*   In programming, the second-outermost part of an iteration loop. (If you're a waiter collecting orders from five tables, and you're collecting from the fourth customer on your third table, i=3 and j=4.)

### k

*   An arbitrary constant you expect to multiply by. (Compare c.)
*   The "kilo" prefix, meaning 1,000. (If you ate 2kg of something, you ate 2000 grams.)
*   A part of a vector meaning "upward". (If you walked 3 miles East and 5 miles North, then did a little jump, you could say you moved 3*i+*5*j+*0.0002*k* miles.)[^0x5ipn8w6dg] Pairs with z.
*   In programming, the third-outermost part of an iteration loop. (If you're a waiter collecting orders from five tables, and you're writing down the second order from the fourth customer on your third table, i=3, j=4, k=2.)

### l

*   Length.
*   If it's a really fancy capital L, it's an operator, and it means you're doing a Laplace Transform[^sre2vob8hma]; I wish you the best of luck.

### m

*   For the "gradient" part of the equation of a straight line (and *only* the equation of a straight line), this inexplicably steals k's job as an arbitrary constant you expect to multiply by[^1kg1tmhaqq9].
*   Mass.
*   When capitalized: the mass of something big. (An astrophysics problem about a ship orbiting a planet might use m for the mass of the ship and M for the mass of the planet.)
*   Million.
*   The "milli" prefix, meaning 0.001. (If your thimble has 1ml of water in it, it has a thousandth of a liter.)
*   When capitalized: the "Mega" prefix, meaning 1,000,000. (If your pond has 1Ml of water in it, it has a million litres.)

### n

*   A variable you expect to be a whole number.
*   When capitalized: a whole number bigger than the whole number you're using n for.
*   When capitalized: the Normal Distribution.[^0x5ipn8w6dg] (*N*(2,4) means "a Normal Distribution with a mean of 2 and a variance of 4".)

### o

*   When capitalized: an operator showing what order a function is. (O(n+4) = n, O(7n^2+5n) = n^2, O(2n^3+3000n) = n^3, etc.)

### p

*   In statistics, a probability.
*   In logic, a proposition.

### q

*   In statistics, another probability.
*   In logic, another proposition.

### r

*   Radius (of a circle or sphere).
*   A ratio.

### s

*   Speed.
*   In the SUVAT equations for constant acceleration, this inexplicably steals d's job and represents distance.[^8qp8fh28ny]

### t

*   Time.

### u

*   In calculus, the conventional choice for a variable you invent to make your life easier. (If y = (x^3+2x+4)^(2/3), and you want to find dy/dx, you can decide that u = x^3+2x+4, and make use of the fact that dy/dx = dy/du * du/dx.)
*   In the SUVAT equations for constant acceleration, this represents the velocity you start at.

### v

*   Velocity.
*   In calculus, the conventional choice for a *second* variable you invent to make your life easier.
*   In the SUVAT equations for constant acceleration, this represents the velocity you finish at.

### w

*   Width (of an object).
*   A variable that gets drafted in when you've already used x, y and z and just need one more dimension/free parameter.

### x

*   The unknown; the variable you use by default (unless you expect it to be a whole number).
*   A measure of something happening horizontally.
*   In Statistics/Machine Learning, x is a predictive(/independent/explanatory) variable, and X is a vector of every predictive variable. If you had a model which used Age, Annual Income and Years of Education to predict something, those would be the three xes which go inside X.

### y

*   The other unknown; the variable you use when you've just used x.
*   A measure of something happening forwardly or upwardly.
*   In Statistics/Machine Learning, y is a predicted(/dependent/response) variable, and Y is a vector of every predicted variable. If you had a model which used some xes to predict Height and Weight, those would be the two ys which go inside Y.

### z

*   A measure of something happening upwardly.
*   The default variable for numbers which might be partly or entirely imaginary (2.4i, 4 + 3i, etc).
*   The last thing.

### Note: Capitalization and the Identical Sign

Something I didn't get a good opportunity to mention during the list proper: capital letters from the start of the alphabet are often used with the triple-equals Identical Sign in a counterintuitive way.

You can say things like "x=3", and this is true for *some* value of x; however, there are statements like "5x=3x+2x" which are tautologically true for *every* value of x. For these, we use ≡, as in "5x≡3x+2x". To put it another way: if "=" means "happens to be equal to"; "≡" means "is literally another way of saying".

Sometimes (usually, using Partial Fractions), it isn't obvious how best to rephrase an equation, so we need to use algebra to figure out what numbers to use in the algebra we're using. This happens like "7y + 6x ≡5y + Ay + 3x + Bx"; here, the identity sign implicitly applies for-and-only-for the lowercase variables, and we 'solve' it by getting the right numbers for the uppercase numbers (in this case, A=2, B=3).

[^hjwx8cszbpb]: "You can draw a little flower if you like!", as I recall one of my teachers saying. 

[^icqo5ci50o9]: If a mathematician is being nice, they might italicize "f" to make it clearer it's not being used as a variable. Most mathematicians are not (this particular kind of) nice, and will expect you to know that f (and sometimes g, and occasionally h) are being used as functions just from context and convention. 

[^0x5ipn8w6dg]: This is another thing which should be consistently italicized due to its non-variable status, but isn't. 

[^c5v76e0slsi]: "But wait!" I hear you cry. "If i means 'along', and j means 'something orthogonal to along', and imaginary numbers are orthogonal to the number line, wouldn't it make more sense to use j as the square root of -1?" During my teenage years there was a movement in some parts of my country's education system to make this exact change for this exact reason, but it didn't catch on; the would-be linguistic revolutionaries eventually abandoned their cause, leaving only confusion and waste in their wake. (Many such cases.) 

[^ag6pem82o3]: Interestingly, you could say this regardless of what route you took to get there. 

[^sre2vob8hma]: If it's a really fancy capital L, it means the world's greatest detective is hunting you down because he doesn't like how you're using your magic notebook. On the bright side, you probably aren't doing a Laplace Transform. 

[^1kg1tmhaqq9]: The only explanation I've heard for why we use "y=mx+c" instead of "y=kx+c" said it was "because the word 'slope' starts with an 'm'". Never let it be said that mathematicians lack a sense of humor, or that they have a good sense of humor[9]. 

[^8qp8fh28ny]: I think the reason is "it's an integration of the distance traveled over time, and s looks like a little integral sign". I think this is a terrible reason and they should be the DUVAT equations. 

[^9gvuj0zoebv]: Okay, I do have a partial justification, which is that an engineer working with slopes is likely to be working with friction as well, and if they're using k for friction coefficients they'll want a different letter for physical gradients. (I still don't think it makes sense for lines you don't intend to ride a sled down.)