fundamentalsmeasurement continued of electron in a hydrogen atom is a constant of nature given by v /4π0 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

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