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[1]. 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”[2], 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.[3]
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.[3]
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.[4]
A part of a vector meaning “along”. (If you walked 3 miles East, you could say you moved 3i miles.)[3] 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 3i+5j miles[5].)[3] 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 3i+5j+0.0002k miles.)[3] 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[6]; 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[7].
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.[3] (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.[8]
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).
- ^
“You can draw a little flower if you like!”, as I recall one of my teachers saying.
- ^
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.
- ^
This is another thing which should be consistently italicized due to its non-variable status, but isn’t.
- ^
“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.)
- ^
Interestingly, you could say this regardless of what route you took to get there.
- ^
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.
- ^
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].
- ^
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.
- ^
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.)
Obligatory xkcd.
m—often used together with n to denote the height and width of a matrix
Related: In the equation y=ax+b, the values of all four variables are unknown, but x and y seem to be more unknown (more variable?) than a and b. It’s not clear what the difference is exactly.
Traditionally, the y variable is another way of defining the output of a function. y=f(x)=ax+b, where a and b are numerical constants. You may not know the constants, but they are specific numbers. The x is a variable you can change to get different values of y=f(x). y=ax+b has two unknown constants, a parameter, and an output.
Concrete example: Say I am looking at a hill with a straight line sloped side. For every x feet I walk forward I move ax feet up. The a has a single numerical value defined by topography. The b has a single numerical value defined by the altitude at the base of the hill where x=0. The x can be anywhere from zero (a point on the perimeter of the base of the hill) to however far horizontally it is from there to the top of the hill. As x varies, y goes from b to b + the height/prominence of the hill.
Note: y isn’t always intended to mean a function output. That depends on context. For example, say instead you have z=ax+by+c. This defines a plane that crosses the z axis at c, has slope a in the x direction, and has slope b in the y direction. It has two parameters (x and y) and three constants that generate the output, z=f(x,y)=ax+by+c
Parameters are also called the arguments of a function
Different concrete example from computer science: Say I want to write a program that multiplies a number by 4. I write code to take input from the user. I call the input x. I return x*4. y=4x. Or, instead, I could add a line in the code that defines a constant, a=4, and then I return ax. Why? Maybe I want to make it easy to see where to change a if I update the code in the future (I put a=4 at the top so I don’t have to read to hunt for the place where I use the number 4). Maybe I’m writing code for a more complicated function that uses a multiple times and I want to be able to make updates while only changing it manually in one place, for readability and lower risk of making mistakes. Basically the constant a is defined in the code, while the argument x is defined by the user.
This is a quote from George Orwell’s unpublished manuscript The Theory and Practice of Algebraic Collections. He eventually split it into two separate novels which did see print. The 2+2=5 stuff went into 1984 and the “some are more equal than others” went into Animal Farm.
If you can let letters mean whatever you want then there’s nothing to stop you from doing the same with numerals. Let 5≡S(S(S(S(0)))).
Well that’s because it’s meant to be quantifying over linear equations.x and y are not meant to be replaced but a and b are.
That is the usage in electrical engineering (since i is current) and from there, the syntax in Python.
z: The default for complex numbers.
I can’t believe I forgot that one; edited; ty!
Also:
m: used for a second whole number when n is already taken.
p: used for primes
q: used for a second prime.
In general, it feels like the alphabet can be partitioned into “sections” where you can use other letters in the same section for additional variables that will play similar roles. Something like:
[a,b,c,d]; [f,g,h]; [i,j,k]; [m,n]; [p,q]; [r,s,t]; [u,v,w]; [x,y,z]
Sometimes these can be combined: [m,n,p,q]; [p,q,r,s,t]; [r,s,t,u,v,w]; [u,v,w,x,y,z]
Yep, and when you run out of letters in a section you use the core letter from the section with a subscript.
See also Latin, Greek and other letters used in math, science and engineering on Wikipedia.
Never saw “SUVAT” before. Might be a term known specifically in the US?
I’m confused by what you say about italics. Mathematical variables are almost always italicized, so how would italicizing something help to clarify that it isn’t a variable?
If I recall correctly, in contexts where variables are italicized by default, non-variables are roman instead.
Yes, that sounds much more normal to me.
Though in the particular case here, something else seems off: when you write f(x) you would normally italicize both the “f” and the “x”, as you can see in the rendering in this very paragraph. I can’t think of any situation in actual mathematical writing where you would italicize one and not the other in order to make some distinction between function-names and variable names.
For that matter, I’m not wild about making a distinction between “variables” and “functions”. If you write f(x) and also sin(x) then it would be normal for “f” and “x” to be italicized and not “sin”. I was going to say that the reason is that f and x are in fact both variables, and it just happens that one of them takes values that are functions, whereas sin is a fixed function and you’ll never see anything like “let sin = 3″ or “let sin = cos”—but actually that isn’t quite right either, because named mathematical constants like e are usually italicized. I think the actual distinction is that single-letter names-of-things get italicized and multiple-letter ones usually don’t.
I think it’s quite common to use i as index variable (for example, in a sum)
(edit: whoops, I see several people have mentioned this)
You’re right. I’ll delete that aside.
i is often used as an index in math, similar to how it is used as an index in for loops.
Usually, k is not just an arbitrary real number, but an integer, like in cos[(2k+1)π]=−1.
For arbitrary constants to multiply by I think λ (lambda, greek letter) is used.
I agree that k strongly connotes integer, but I wouldn’t expect to see λ used like that if not as a Lagrange multiplier.
In this context, “i” often stands for “index”, as in an offset into an array.
c—a constant for the speed of light, e.g. E=mc2
And h is Planck’s constant. I think abstractapplic is limiting this to classical mechanics.
I would add that bold capital letters are often used in linear algebra to represent matrices.
j—can also be the unit of the imaginary axis.
In the entire field of Electrical Engineering, there is a consensus to use “j” instead of “i” for the imaginary axis, because “i” in all its shapes and forms (upper case, lower, with subscript that’s either upper, lower) is used to refer to electrical current and all the types signal components it can be decomposed into.
Not sure how relevant this is for others, but in the day to day life of an electrical engineer it is much more common to work with complex currents, voltages, powers, rather than real ones. Every now and then I have to be reminded that “i” was used originally :P