Instrumentation & Measurement Magazine 26-1 - 5

fundamentalsmeasurement
of
Richard Davis
The Revised SI Compared to Natural Systems of Units
I
n a useful introduction to " natural units " [1], the author
contrasted natural units to the SI as it existed at the
time of writing (2016). The SI was characterized in [1] as
a system that " relies heavily on precision measurements of
standard prototypes, objects or systems that define a physical
unit. " In particular, the SI indeed relied
on one standard prototype, called the International
Prototype of the Kilogram. The
mass of this unique object defined 1 kg. By
contrast, " natural units, " as the name implies,
are completely defined in terms of
selected constants used to describe the natural
world. Different systems of natural
units are based on different selections of
these constants.
However, the SI underwent sweeping
changes on May 20, 2019 as documented in
[2]. It is now defined through a set of seven
physical constants, two of which feature in common systems
of natural units. Consequently, the SI no longer relies on any
" standard prototypes, objects or systems. " Advances in technology
have made these changes possible.
This article aims to explain through simple examples how
the SI is presently structured. The comparison with the previous
structure will highlight arbitrary choices, mathematical
necessity, and mathematical sufficiency.
Basic Principles of the Historical SI Remain
Dimensions and Base Units
To begin at the beginning, it is useful to organize physical
quantities in a system of base dimensions, whose number and
identity are decided by convention and taken to be mutually
independent. For example, the SI has seven dimensions of
which this article focuses on the first four: time (T), length (L),
mass (M), and electrical current (I). This subset of the seven
dimensions suffices to construct the dimensions of force, energy,
pressure and other mechanical quantities as well as all
electro-magnetic quantities such as voltage, resistance, and inductance.
The dimension of any quantity Q that falls within
this subsystem is given by:
dim Q = Tα Lβ
February 2023
Mγ Iδ
where the exponents α, β, and δ are usually small integers and
γ generally equals 1 or 0. In any SI equation relating a quantity
Q to another term or a sum of other terms, each term must
be the same kind of quantity as Q. Quantities of the same kind
have the same dimension. However, some physical quantities
share the same dimension but are not considered to be quantities
of the same kind. An example of this is energy and torque
[2], which both have the dimension of force times length but
are understood to be different kinds of quantities. (In a widelyused
system of natural units presented below, mass and
energy share the same dimension and are treated in equations
as the same kind of quantity). Physical equations written as
terms that can be added or subtracted because all terms have
the same dimension, are said to be " homogeneous. "
There is a set of SI units that corresponds to the four basic
dimensions. These are the base units: the second (s), the meter
(m), the kilogram (kg), and the ampere (A). All units that can be
constructed from one or more of these base units take the form:

unit s m kg A
p p pp
1 2 34
(1)
The base units are simply those where the exponent is 1 and
all other exponents are zero. All units that are the product of
powers the base units, as described by (1), are said to be " coherent. "
This means that the relations among all such units
automatically mirror the equations of science with no extraneous
numerical factors required. Of course, multiples and
submultiples of coherent units, microseconds and kilometers
for example, are also useful SI units but they are not coherent-
they cannot be constructed from (1) unless a numerical factor,
10−6
or 103
in this example, is introduced.
This article uses coherent units unless otherwise noted. In
the SI, twenty-two combinations of base units corresponding
to certain often-used quantities are given special names such
as the newton, pascal, joule, volt, and ohm.
For an example of dimensional analysis, take the familiar
equation F = m a which tells us that force must have the same
dimension as the product of mass times acceleration:
dim F=T−2 M1
L1
as kg m s−2
I0
The coherent SI unit corresponding to F is usually written
. This product, in whichever order one wishes to
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