What makes x so special in equations?


From a purely mathematical perspective, there is absolutely nothing special about choosing the letter x as your label for a variable. Labels are used in mathematics to represent numbers that are not yet known or can change (variables), a collection of numbers (functions and vectors), and numbers that are known but are too complicated to write out explicitly every time (constants). You can choose to label the unknown thing however you want and still end up with the same answer. Labels need to be used in order to keep track of the mathematical objects. Consider a simple example: I walk into a classroom with three identical cardboard boxes, each containing some unknown item. The items in each box are different. I give the boxes to the students in the room and ask them to try to figure out what each box contains without opening them. The students start weighing the boxes, shaking them, smelling them and so forth. They find that one box contains something heavy. But a few minutes later, the boxes have been handed around and they can’t remember if the one that contains something magnetic was also the one that contains something heavy because the boxes all look the same. What do they need? labels! With a pencil, the students mark one box “A”, another box “B”, and the last box “C”. Now they can keep track of which properties belong to which box. It doesn’t matter which box they decide to call “A”. In fact, from a mathematical perspective, it doesn’t matter what they call each box. They could have labeled the boxes “1”, “2”, and “3” or “red”, “green”, “blue”, or even “Freddy”, “Sally”, and “Joe”, and the labels would still have served their purpose of keeping the boxes differentiated until their contents can be known.

While there is total mathematical freedom in choosing label names, there is still some human advantage to wisely choosing the names. For instance, what if the students labeled the boxes “Michael Jordan”, “Micheal Jackson,” and “the moon”. Observations such as “Micheal Jordan is heavy but Micheal Jackson is light”, “the moon sounds like it contains powder” , and “Michael Jordan seems more magnetic than the moon” are confusing. The problem is that these words already have meanings on their own. In contrast, letters of the alphabet are vague enough entities that they can be used as labels without creating confusion. The best labels for the boxes are probably “A”, “B”, and “C”. The same is true in mathematics. The equation “red = blue2” is a perfectly valid mathematical equation if “red” simply labels the area of a square and “blue” labels the length of the square. But to humans, this equation looks confusing because these words have meanings beyond how they are being used as labels. The best labels are the ones that have as little meaning as possible on their own. Good labels for variables in mathematics are therefore the letters of the alphabet. Even better are the letters that get used the least in everyday English: x, y, and z. I believe these letters are used so often as variable names in mathematics because they are used so little in conversational English.

To further reduce confusion, certain traditions have arisen with regards to assigning labels. Following these traditions makes the equations easier to read, but does not make their mathematical content any different. People who use non-traditional labels may still get the same answers in the end, but they will confuse a lot of people along the way (perhaps including themselves). Below are the traditions for mathematical labels. I suggest you follow these whenever doing mathematics. In general, letters from the beginning of the alphabets are used for constants, letters from the middle of the alphabet are used for functions, and letters from the end of the alphabet are used for variables.

Labeling traditions to follow in mathematics:

  • Variable distances: x, y, z, r, ρ
  • Constant distances: a, b, c, d, h, w, L, R, x0, y0, z0
  • Variable angles: θ, φ
  • Constant angles: α, β, γ
  • Variable points in time: t
  • Constant points in time: T, τ, t0
  • Functions: f, g, h, u, v, w
  • Indices: i, j, k
  • Integers: m, n, N
  • Special constants: π = 3.14… and e = 2.71…
  • Vectors: A, B, C, D, E, F, G, H, x, y, z
  • Physical properties: use the first letter of the word (see below)

Labels to avoid in mathematics:

  • the letter o is too easily confused with the number 0
  • the Greek letters ι, κ, ο, ν, and χ are too easily confused with the letters i, k, o, u, and x

What if you need to keep track of many time variables? There is only one traditional label for time: t. The solution is to use primes or subscript letters. For example, one reference frame follows time t, while another follows time t ‘, and still another follows time t “. Or the time on earth can be tracked with the label tE and the time on the moon can be tracked with the label tM. In general, multiple variables that are very similar should be handled in this way using primes or subscript letters. On the other hand, multiple constants should be differentiated by subscript numbers. For instance, use t0, t1, t2, t3… to keep track of multiple points in time. If you are curious, here are the traditional labels for various physical properties.

Traditional labels for physical properties:

  • a : acceleration
  • b : beat frequency
  • c : speed of light in vacuum, specific heat capacity, viscous damping coefficient
  • d : diameter, distance
  • e : electron charge, eccentricity
  • f : frequency
  • g : acceleration due to earth’s gravity
  • h : height, Plank’s constant
  • k : wavenumber, spring constant, Boltzman’s constant
  • l : length
  • m : mass, magnetic dipole moment
  • n : index of refraction, number density
  • p : momentum, electric dipole moment, pressure
  • q : electric charge, velocity
  • r : radius, distance
  • s : displacement
  • t : time, thickness
  • u : energy density
  • v : velocity
  • w : width, weight
  • x : position in dimension 1
  • y : position in dimension 2
  • z : position in dimension 3
  • A : area, magnetic potential, amplitude
  • B : total magnetic field
  • C : capacitance, heat capacity
  • D : electric displacement field
  • E : total electric field, energy
  • F : force
  • G : Newton’s gravitational constant, Gibbs free energy
  • H : auxiliary magnetic field, Hamiltonian, enthalpy
  • I : moment of inertia, electrical current, irradiance, impulse, action
  • J : electrical current density, total angular momentum
  • K : kinetic energy
  • L : length, angular momentum, Lagrangian, self inductance, luminosity
  • M : magnetization, mutual inductance, magnification
  • N : number of objects
  • P : electric polarization, power, probability, momentum-energy four-vector
  • Q : total electrical charge, heat
  • R : electrical resistance, radius, curvature
  • S : spin, entropy
  • T : torque, time, period, temperature, kinetic energy
  • U : potential energy, velocity four-vector
  • V : volume, potential difference (voltage)
  • W : work
  • X : space-time four-vector
  • Z : electrical impedance
  • α : angular acceleration, spatial decay rate
  • β : normalized velocity
  • γ : Lorentz factor, sheer strain, heat capacity ratio, gamma ray
  • δ : small displacement, skin depth
  • ε : electrical permittivity, strain
  • θ : angular displacement
  • κ : transverse wavenumber
  • λ : wavelength, line density, temporal decay rate
  • μ : magnetic permeability, reduced mass, chemical potential, coefficient of friction
  • ν : frequency
  • ρ : electrical resistivity, volume density
  • σ : electrical conductivity, surface density
  • τ : torque
  • ψ : quantum wavefunction
  • ω : angular frequency
  • Φ : electrical potential
  • Λ : Cosmological constant
  • Ψ : quantum wavefunction
  • Ω : precession angular speed


Leave an answer