Introduction and the International System of Units

Physics is a science of measurement. Before you can study how objects move, how forces act, or how energy flows, you need to be able to measure things reliably. And reliable measurement starts with one simple question: compared to what?

What Is Measurement?

When you measure any physical quantity, what you are really doing is comparing it against a chosen reference standard. That reference standard is called a unit. The outcome of every measurement is a number paired with a unit. Saying “the table is 1.5” means nothing on its own. Saying “the table is 1.5 metres long” tells us that the table’s length is 1.5 times the standard reference length we call a metre.

So every measurement boils down to two parts:

• A numerical value (how many times the standard fits)
• A unit (the standard itself)

Base Units and Derived Units

Although the physical world seems to contain a huge variety of quantities (speed, force, pressure, energy, electric charge, and so on), you do not need a separate, independently defined unit for each one. Most quantities are connected to each other through equations. This means you only need a small set of independently defined units, and everything else can be built from them.

• Base units (also called fundamental units) are the independently defined units for a small set of core quantities. These are chosen by international agreement and do not depend on any other units for their definition.
• Derived units are units for all other quantities, constructed by combining base units through multiplication, division, or powers. For instance, speed is length divided by time, so its unit (metres per second) is derived from the base units of length and time.

A system of units is the complete package: all the base units together with every derived unit that follows from them.

Three Older Systems: CGS, FPS, and MKS

Before the modern international standard was adopted, scientists in different countries used different systems. The three most common ones differed in their choice of base units for length, mass, and time:

System	Length	Mass	Time
CGS	centimetre (cm)	gram (g)	second (s)
FPS (British)	foot (ft)	pound (lb)	second (s)
MKS	metre (m)	kilogram (kg)	second (s)

Having multiple systems in circulation created unnecessary confusion. Scientists working in one country would report results in one set of units, and colleagues elsewhere would need to convert before they could compare or build on that work. The solution was to agree on a single, universal system.

The International System of Units (SI)

The system that the world settled on is called the Systeme Internationale d’Unites (French for “International System of Units”), shortened to SI. It was developed by the Bureau International des Poids et Mesures (BIPM, the International Bureau of Weights and Measures) in 1971. The definitions of the base units were most recently revised by the General Conference on Weights and Measures in November 2018.

The SI is now the accepted standard for scientific, technical, industrial, and commercial work worldwide. One of its biggest practical advantages is that it uses the decimal system: all multiples and sub-multiples are powers of 10. This makes conversions within the system straightforward. Converting kilometres to metres, or milligrams to kilograms, is just a matter of shifting a decimal point.

The Seven SI Base Quantities

The SI defines exactly seven base quantities, each with its own base unit. Every other physical quantity in physics, chemistry, engineering, and beyond is derived from these seven.

Base Quantity	Unit Name	Symbol	Defined Via
Length	metre	m	Speed of light in vacuum, $c = 299\,792\,458$ $m\,s^{-1}$
Mass	kilogram	kg	Planck constant, $h = 6.62607015 \times 10^{-34}$ $J\,s$
Time	second	s	Caesium-133 hyperfine transition frequency, $\Delta\nu_{cs} = 9\,192\,631\,770$ Hz
Electric current	ampere	A	Elementary charge, $e = 1.602176634 \times 10^{-19}$ C
Thermodynamic temperature	kelvin	K	Boltzmann constant, $k = 1.380649 \times 10^{-23}$ $J\,K^{-1}$
Amount of substance	mole	mol	Avogadro constant, $N_A = 6.02214076 \times 10^{23}$ $mol^{-1}$
Luminous intensity	candela	cd	Luminous efficacy $K_{cd} = 683$ $lm\,W^{-1}$ at $540 \times 10^{12}$ Hz

A few important points to notice:

• Every base unit is now defined by fixing the value of a fundamental constant of nature. This is a major shift from older definitions that relied on physical objects (like the old platinum-iridium kilogram prototype kept in Paris). Constants of nature do not wear out, get damaged, or change over time.
• The second anchors everything. The definitions of the metre and the kilogram both reference the second (through $c$ and $h$ respectively), and the second itself is pinned to the caesium-133 atom’s transition frequency. This makes the second the most foundational unit in the SI.
• When you use the mole, you must always specify what entity you are counting. One mole is exactly $6.02214076 \times 10^{23}$ elementary entities, but those entities could be atoms, molecules, ions, electrons, or any other specified group of particles. Saying “one mole” without specifying the entity is incomplete.

Supplementary Units: Radian and Steradian

Besides the seven base units, the SI also defines units for two geometric quantities: plane angle and solid angle. These are sometimes called supplementary units, though both are actually dimensionless.

Plane Angle: The Radian

Fig 1.1(a): Plane angle $d\theta$

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

$d\theta = \frac{ds}{r}$

Its unit is the radian (symbol: rad). Since both $ds$ and $r$ are lengths, their ratio is a pure number with no dimensions. One full revolution sweeps through $2\pi$ radians.

Solid Angle: The Steradian

Fig 1.1(b): Solid angle $d\Omega$

The solid angle $d\Omega$ is the three-dimensional version of a plane angle. Imagine a cone with its tip at a point O. The solid angle is defined as the ratio of the area $dA$ intercepted on a spherical surface (centred at O) to the square of the sphere’s radius $r$:

$d\Omega = \frac{dA}{r^2}$

Its unit is the steradian (symbol: sr). Again, both $dA$ (area) and $r^2$ (length squared) have dimensions of length squared, so the ratio is dimensionless.

Derived Units

From the seven base units, you can construct the units for every other physical quantity by combining base units through multiplication, division, and raising to powers. These are the derived units. For example:

• Speed = length / time, so its unit is $m/s$ or $m\,s^{-1}$
• Acceleration = speed / time, so its unit is $m/s^2$ or $m\,s^{-2}$
• Force = mass $\times$ acceleration, so its unit is $kg \cdot m\,s^{-2}$, given the special name newton (N)
• Energy = force $\times$ distance, so its unit is $kg \cdot m^2\,s^{-2}$, given the special name joule (J)

Some derived units are used so frequently that they have been given their own names and symbols (like newton, joule, watt, pascal, hertz, coulomb, volt, and others). But under the hood, every one of them can be broken down into the seven base units.

Non-SI Units Retained for General Use

Although the SI is the international standard, a handful of older units remain in common use because they are deeply embedded in everyday life, industry, or specific scientific fields. The table below lists the most important ones:

Unit	Symbol	SI Equivalent
minute	min	60 s
hour	h	3600 s
day	d	86,400 s
year	y	$3.156 \times 10^7$ s
degree (angle)	$°$	$(\pi/180)$ rad
litre	L	$10^{-3}$ $m^3$
tonne	t	$10^3$ kg
carat	c	200 mg
bar	bar	$10^5$ Pa
curie	Ci	$3.7 \times 10^{10}$ $s^{-1}$
roentgen	R	$2.58 \times 10^{-4}$ C/kg
quintal	q	100 kg
barn	b	$10^{-28}$ $m^2$
are	a	$10^2$ $m^2$
hectare	ha	$10^4$ $m^2$
standard atmosphere	atm	101,325 Pa

These units are “outside SI” but officially recognised for use alongside it. When performing calculations, always convert to SI units first, carry out the computation, and then convert back if needed.

SI Prefixes

The SI system uses a set of standard prefixes to denote multiples and sub-multiples of units. Each prefix represents a specific power of 10. For instance, “kilo-” means $10^3$ (a thousand times), “milli-” means $10^{-3}$ (a thousandth), and “nano-” means $10^{-9}$ (a billionth). These prefixes attach to any SI unit: a kilometre is 1000 metres, a milligram is 0.001 grams, and a nanosecond is $10^{-9}$ seconds.

This decimal structure is what makes the SI so practical for calculations. Converting between scales never involves awkward factors (like 12 inches in a foot or 5280 feet in a mile). It is always just a shift of the decimal point.