Instrumentation & Measurement Magazine 26-1 - 8

fundamentalsmeasurement continued
of
electron in a hydrogen atom is a constant of nature given by

v /4π0 and it follows from (5) that r 4π / m e
 e2
  
22
0e

is also a constant of nature. This result for r is still referred to as
the Bohr radius [7]. Its value is about 5.3 × 10−11
m which correctly
predicts that the diameter of a hydrogen atom relative to
one meter is about 10−10
. What about the velocity v? Since the
ratio of any two velocities is the same pure number in all systems
of units, the ratio v/c is a universal constant of dimension
1. In the SI formulation:
ve
cc
 

 1
4π   
2


0 

(6)
The quantity α, which is now known as the fine-structure constant
for reasons that have nothing to do with this derivation, is
approximately 1/137. This is all that dimensional analysis can
confirm, but it is much better than nothing. The experimental
uncertainty of α is currently 1.5 parts in 1010
[7]. Referring to
both (6) and Table 1, note that e, h, and c all have exact numerical
values in the SI. This means that the formerly exact value of
μ0
(= 1/(c2
0 
ε0) ) must now be determined from the experimental
value of the fine-structure constant. The latest result confirms
that
 4π 10 N A

72
[7].
Using the Defining Constants: In Table 1, it can be seen that
each line defines the SI unit given in the last column (Q=
{Q}SI
·[Q]SI). Take the third line, for example. The quantity Q is
the natural constant h. An exact value of {h}SI
the SI unit, [h]SI
is used to define
. Therefore, the third line of Table 1 defines the
joule-second (J s) as:
h/(6.62607015 × 10−34
)= 1 J s = 1 kg m2 s−1
To measure a physical constant whose value is defined to be
exact is a logical impossibility. However, prior to 2019 h was
measured in terms of the artifact definition of the kilogram.
The value of h shown in Table 1 is now exact but the principle
of continuity constrained its exact value to be within the uncertainty
of its pre-2019 experimental value [5].
Are the Historical Base Units Needed to
Define All SI Units?
The short answer is no, but the usual base units are well known
and there is no compelling reason to designate new base units.
It is true that the SI units of the defining constants, four of
which are highlighted in Table 1, do not generally define the
base units directly, but this is nothing new [5]. However, the
path to the definitions of all SI units passes through the defining
constants of Table 1. Again, we take only the first four
defining constants as an example but it is also true that the
complete set of seven defining constants defines any and all
SI units. How this is done was shown rigorously in [8], but
can at least be demonstrated without going into mathematical
8
Fig. 1. The matrix M is shown in yellow. The unit of the constant symbolized
at the top of a column can be written as the product of the base units shown
as labels in the first column, each raised to the power shown in the matrix.
proofs. The main point is: once Table 1 has been established, all
SI units essentially define themselves through a relatively simple
algorithm.
One first sets up a matrix M (Fig. 1) where each column
gives the unit of a different defining constant in terms of the exponents
of the usual base units, as shown in (1).
The matrix M (in yellow) can be inverted using, for example,
the MINVERSE function in Excel. The inverse matrix, M−1
,
is shown in the gray cells of Fig. 2. It is always true that the multiplication
of any matrix M by its inverse matrix, M−1
, results in
the " identity matrix " I, where all the diagonal cells contain 1
and all other cells contain 0. Matrix multiplication is summarized
in [9].
The cells of M−1
contain exponents of the defining constants
used to define the base unit that heads each column.
Fig. 2. The inverse matrix, M−1
is shown in gray. The quantity used to define
the base unit shown at the top of each column can be written in terms of
the product of powers of the defining constants shown as labels in the first
column.
IEEE Instrumentation & Measurement Magazine
February 2023
Instrumentation & Measurement Magazine 26-1

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