Dimensions

The Relationship Between Quantity and Dimension

Base Quantity	Typical Symbol for Quantity	Symbol for Dimension
time	\(t\)	\(T\)
length	\(l\), \(x\), \(r\), etc.	\(L\)
mass	\(m\)	\(M\)
electric current	\(I\), \(i\)	\(I\)
thermodynamic temperature	\(T\)	\(Θ\)
amount of substance	\(n\)	\(N\)
luminous intensity	\(I_v\)	\(J\)

SI Brochure:The International System of Units (SI) (2019). Bureau International des Poids et Mesures. Ninth Edition. Table 3 (Base Quantities and Dimensions Used in the S). pg. 136.

In general the dimensional product forms the dimension of any quantity \(Q\), as (The International System of Units (SI) (2019), pg. 136)

\[\dim\ Q = T^α L^β M^γ I^δ Θ^ε N^ζ J^η\]

where the exponents \(α\), \(β\), \(γ\), \(δ\), \(ε\), \(ζ\) and \(η\), are often small integers. These exponents are the dimensional exponents and can be positive, negative, or zero. Exponents which are zero are not usually used in the simplified form of the quantity dimension.

Relating Quantity Values to Units

Any quantity \(Q\) in SI relates to the numerical value \(\left\{ Q \right\}\), and the unit \(\left[ Q \right]\) by

\[Q = \left\{ Q \right\}\left[ Q \right]\]

The SI derived unit \(\left\{\text{Q}\right\}\) of \(Q\) is then obtained by replacing the dimensions of the SI base quantities. Therefore the dimension of \(Q\), with the symbols for the corresponding traditional base units, is

\[\left\{\text{Q}\right\} = \left\{\text{m}^α\right\} \cdot \left\{\text{kg}^β\right\} \cdot \left\{\text{s}^γ\right\} \cdot \left\{\text{A}^δ\right\} \cdot \left\{\text{K}^ε\right\} \cdot \left\{\text{mol}^ζ\right\} \cdot \left\{\text{cd}^η\right\}\]

The exponents \(α \ldots η\), are dimensional exponents. Dimensions which are not required in the definition of a specific unit have a zero value: other exponents give the number of dimensions required.

Where all dimension exponents are zero, as for ratios of quantities, the defined result of the dimension equation is

\[\left\{ Q \right\} = 1\] \[\dim\ \text{Q} = 1\]

For example, the quantity value of velocity \(\nu\) is physically defined by the relationship

\[\nu = \frac{l}{t}\]

where \(l\) is the distance a particle in uniform motion with velocity \(\nu\) travels in the time \(t\). The dimension of \(l\) is \(L\), and thus the exponent \(\left\{ \text{m} ^ \alpha \right\}\) is \(1\). Likewise the dimension of \(t\) is \(T\), and hence the exponent \(\left\{\text{s}^γ\right\}\) is \(-1\). All other exponents are zero, and so the dimensional equation for \(\left\{ \nu \right\}\) is

\[\left\{ \nu \right\} = \left\{\text{m}^1\right\} \cdot \left\{\text{kg}^0\right\} \cdot \left\{\text{s}^{-1}\right\} \cdot \left\{\text{A}^0\right\} \cdot \left\{\text{K}^0\right\} \cdot \left\{\text{mol}^0\right\} \cdot \left\{\text{cd}^0\right\}\] \[\left\{ \nu \right\} = \left\{\text{m}^1\right\} \cdot \left\{\text{s}^{-1}\right\}\]

Hence the derived SI unit for \(\nu\) is \(\text{m}^1\ \text{s}^{-1}\).