oneunit

Domain Specific Language (DSL) for Système international d'unités (SI) arithmetic

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SI units

Common Derived Units and Their 7D Vectors

We use the standard order of the 7 SI base units as coordinates:

$$\bigl[L,\; M,\; T,\; I,\; \Theta,\; N,\; J\bigr]$$

where

• $$(L)$$ = length (meter, m)
• $$(M)$$ = mass (kilogram, kg)
• $$(T)$$ = time (second, s)
• $$(I)$$ = electric current (ampere, A)
• $$(\Theta)$$ = thermodynamic temperature (kelvin, K)
• $$(N)$$ = amount of substance (mole, mol)
• $$(J)$$ = luminous intensity (candela, cd)

In the vectors below, any unmentioned base units have exponent 0 (e.g., temperature, amount of substance, or luminous intensity). For brevity, we typically focus on mechanical/electrical dimensions:

Mechanics & Thermodynamics

Unit	Symbol	Dimensions in Base Units	7D Vector
Frequency	Hz	$$(T^{-1})$$	$$((0,0,-1,0,0,0,0))$$
Force	N	$$(M^1 L^1 T^{-2})$$	$$((1,1,-2,0,0,0,0))$$
Pressure	Pa	$$(\tfrac{F}{A} = M^1 L^{-1} T^{-2})$$	$$((-1,1,-2,0,0,0,0))$$
Energy/Work	J	$$(M^1 L^2 T^{-2})$$	$$((2,1,-2,0,0,0,0))$$
Power	W	$$(M^1 L^2 T^{-3})$$	$$((2,1,-3,0,0,0,0))$$
Velocity	m/s	$$(L^1 T^{-1})$$	$$((1,0,-1,0,0,0,0))$$
Acceleration	m/s²	$$(L^1 T^{-2})$$	$$((1,0,-2,0,0,0,0))$$
Action	J·s	$$(M^1 L^2 T^{-1})$$	$$((2,1,-1,0,0,0,0))$$
Dynamic Viscosity	Pa·s	$$(M^1 L^{-1} T^{-1})$$	$$((-1,1,-1,0,0,0,0))$$
Kinematic Viscosity	m²/s	$$(L^2 T^{-1})$$	$$((2,0,-1,0,0,0,0))$$

Electricity & Magnetism

Unit	Symbol	Dimensions in Base Units	7D Vector
Charge	C	$$(T^1 I^1)$$ (ampere × second)	$$((0,0,1,1,0,0,0))$$
Electric Potential	V	$$(M^1 L^2 T^{-3} I^{-1})$$ (joule per coulomb)	$$((2,1,-3,-1,0,0,0))$$
Capacitance	F	$$(M^{-1} L^{-2} T^4 I^2)$$	$$((-2,-1,4,2,0,0,0))$$
Resistance	Ω	$$(M^1 L^2 T^{-3} I^{-2})$$	$$((2,1,-3,-2,0,0,0))$$
Conductance	S	$$(\Omega^{-1} = M^{-1} L^{-2} T^3 I^2)$$	$$((-2,-1,3,2,0,0,0))$$
Magnetic Flux	Wb	$$(M^1 L^2 T^{-2} I^{-1})$$	$$((2,1,-2,-1,0,0,0))$$
Magnetic Flux Density	T (tesla)	$$(\frac{\text{weber}}{\text{m}^2} = M^1 T^{-2} I^{-1})$$	$$((0,1,-2,-1,0,0,0))$$
Inductance	H	$$(M^1 L^2 T^{-2} I^{-2})$$	$$((2,1,-2,-2,0,0,0))$$

Additional Examples (Radiation, Etc.)

Unit	Symbol	Dimensions (subset)	7D Vector (partial)
Becquerel	Bq	$$(T^{-1}) (1/s)$$	$$((0,0,-1,0,0,0,0))$$
Gray	Gy	$$( \frac{\text{J}}{\text{kg}} = L^2 T^{-2})$$	$$((2,0,-2,0,0,0,0))$$
Sievert	Sv	same as Gray ($$(L^2 T^{-2})$$)	$$((2,0,-2,0,0,0,0))$$
Katal	kat	$$(\tfrac{\text{mol}}{\text{s}} = N^1 T^{-1})$$	$$((0,0,-1,0,0,1,0))$$