Dimensional analysis

Physical equations must balance not only numbers but kinds of quantity. The solver tracks SI base dimensions as an exponent vector.

Base dimensions

[M] mass          [L] length       [T] time
[I] current       [Θ] temperature  [N] amount
[J] luminous intensity

Example — force is mass × acceleration:

[F] = [M][L][T]⁻²    →    N = kg·m/s²

Algebra of dimensions

• Multiply quantities → add dimension vectors
• Divide → subtract vectors
• Power x^n → scale the vector by n
• Add / subtract only if dimensions match exactly

Consistency checks

For LHS = RHS, both sides must share the same dimension vector. Matching dimensions is necessary but not sufficient for physical truth — coefficients and constitutive laws still matter.

Unit conversion

convert: 100 km/h -> m/s

Both sides must share dimensions. The numeric factor is the ratio of SI scales. Prefixes (k, M, m, μ/u, n, …) and many imperial / engineering units are supported.

Variables

let m = 2.5 kg
let a = 9.81 m/s^2
F = m * a

Bind quantities once, then reuse them. let may reuse names that are also SI symbols (e.g. let m = 2 kg). An unbound identifier that is not a known unit is an error.

Limits

• Not a full computer algebra system
• Absolute temperatures (°C, °F) involve offsets — prefer kelvin differences for thermodynamics
• Arguments of log, exp, sin must be dimensionless in rigorous work