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 IEEE Instrumentation & Measurement Magazine 1094-6969/23/$25.00©2023IEEE 5

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