The International System of Units

The Need for a Universal System

Before the modern era, scientists in different countries used different systems of units. This created confusion when comparing experimental results across borders. Three older systems were in widespread use:

• CGS system: centimetre, gram, second
• FPS (British) system: foot, pound, second
• MKS system: metre, kilogram, second

The problem is obvious: a measurement of force in the CGS system (dyne) and one in the FPS system (poundal) are not directly comparable without messy conversion factors. What was needed was a single, coherent system that everyone could agree on.

The SI System

The internationally accepted system today is the Système International d'Unités (International System of Units), abbreviated as SI. It was developed by the Bureau International des Poids et Mesures (BIPM) and was originally established in 1971. The most recent major revision came from the General Conference on Weights and Measures in November 2018.

The SI system uses the decimal system, which makes conversions within the system straightforward. For example, converting from metres to kilometres simply means moving the decimal point — no awkward factors like 12 inches to a foot or 3 feet to a yard.

The Seven Base Units

The SI system defines seven base units, each corresponding to a fundamental physical quantity that cannot be expressed in terms of other quantities. These are shown in the table below.

Each of these units is now defined in terms of fixed numerical values of fundamental physical constants. This is a crucial shift from older definitions that relied on physical artefacts (like the standard metre bar or the standard kilogram cylinder). Constants of nature do not change, so definitions based on them are permanent and universally reproducible.

Definition of the Metre

The metre is defined by fixing the numerical value of the speed of light in vacuum, $c$, to be $299\,792\,458$ when expressed in the unit $\text{m s}^{-1}$. The second is itself defined in terms of the caesium frequency $\Delta \nu_{\text{Cs}}$.

Definition of the Kilogram

The kilogram is defined by fixing the numerical value of the Planck constant $h$ to be $6.626\,070\,15 \times 10^{-34}$ when expressed in the unit $\text{J s}$, which is equivalent to $\text{kg m}^2 \text{s}^{-1}$. The metre and the second are themselves defined in terms of $c$ and $\Delta \nu_{\text{Cs}}$.

Definition of the Second

The second is defined by fixing the numerical value of the caesium frequency $\Delta \nu_{\text{Cs}}$, which is the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be $9\,192\,631\,770$ when expressed in the unit Hz, which is equivalent to $\text{s}^{-1}$.

Definition of the Ampere

The ampere is defined by fixing the numerical value of the elementary charge $e$ to be $1.602\,176\,634 \times 10^{-19}$ when expressed in the unit C, which is equivalent to $\text{A s}$. The second is defined in terms of $\Delta \nu_{\text{Cs}}$.

Definition of the Kelvin

The kelvin is defined by fixing the numerical value of the Boltzmann constant $k$ to be $1.380\,649 \times 10^{-23}$ when expressed in the unit $\text{J K}^{-1}$, which is equivalent to $\text{kg m}^2 \text{s}^{-2} \text{K}^{-1}$. The kilogram, metre, and second are defined in terms of $h$, $c$, and $\Delta \nu_{\text{Cs}}$.

Definition of the Mole

The mole is defined by fixing the numerical value of the Avogadro constant $N_A$ to be $6.022\,140\,76 \times 10^{23}$ when expressed in the unit $\text{mol}^{-1}$. This number is called the Avogadro number. One mole contains exactly this many elementary entities.

Definition of the Candela

The candela is defined by fixing the numerical value of the luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12}$ Hz, $K_{\text{cd}}$, to be 683 when expressed in the unit $\text{lm W}^{-1}$, which is equivalent to $\text{cd sr W}^{-1}$ or $\text{cd sr kg}^{-1} \text{m}^{-2} \text{s}^3$. The kilogram, metre, and second are defined in terms of $h$, $c$, and $\Delta \nu_{\text{Cs}}$.

Supplementary Units: Plane Angle and Solid Angle

In addition to the seven base units, the SI defines two supplementary quantities that are dimensionless but have named units.

Plane Angle

A plane angle $d\theta$ is defined as the ratio of the length of an arc $ds$ to the radius $r$ of the circle:

The unit of plane angle is the radian (symbol: rad). Since both $ds$ and $r$ have the dimension of length, the ratio is dimensionless.

Solid Angle

A solid angle $d\Omega$ is defined as the ratio of the intercepted area $dA$ on a spherical surface to the square of the radius $r$:

The unit of solid angle is the steradian (symbol: sr). Again, since $dA$ has dimensions of length squared and $r^2$ has dimensions of length squared, the ratio is dimensionless.

Derived Units

All other physical quantities can be expressed in terms of the seven base units. These are called derived units. For example:

• Speed is length divided by time: $\text{m s}^{-1}$
• Acceleration is speed divided by time: $\text{m s}^{-2}$
• Force is mass times acceleration: $\text{kg m s}^{-2}$, which is given the special name newton (N)
• Energy is force times distance: $\text{kg m}^2 \text{s}^{-2}$, which is given the special name joule (J)
• Power is energy per unit time: $\text{kg m}^2 \text{s}^{-3}$, which is given the special name watt (W)

A comprehensive list of derived units with special names is provided in the appendices of the textbook.

Units Outside the SI

Some units that are not part of the SI system are still retained for general use because of their widespread acceptance in specific contexts. These are listed in Table 1.2 of the textbook.

Table 1.2 — Some units retained for general use (though outside SI):