Principle

Dimensional homogeneity

[X] = M^{a} L^{b} T^{c}

Statement

Every term in a valid physical equation reduces to the same combination of base dimensions.

Symbols and units

$X$	the quantity being checked	—
$M, L, T$	dimensions of mass, length, time	—
$a, b, c$	exponents matched on both sides	—

When it applies

Applies to any physical equation relating measured quantities; does not by itself confirm numerical factors.

Derivation

Assign each base quantity a dimension symbol: mass M, length L, time T, and similarly for current, temperature, amount and luminous intensity where they appear.

Express every quantity in the equation using these symbols, and require the exponents a, b, c … to match on both sides:

[X] = M^{a} L^{b} T^{c}

Worked examples

Checking a pendulum's period formula

For T = 2π√(L/g), with L a length and g an acceleration:

[L / g] = \frac{L}{L T ^{- 2}} = T^{2}

Taking the square root:

[L / g] = T

The right side reduces to a time, matching the period T on the left — the formula passes the check.

Watch out for

Agreement in dimension does not guarantee an equation is correct — a dimensionless factor such as ½ or 2π can be missing or wrong even when the dimensions check out.