SI units

Pick up a bottle labelled "500 ml" and a runner's watch reading "9.58 s" and you are already trusting a global agreement you never signed. Neither number means anything alone — a bare "500" or a bare "9.58" is just digits — until it is tied to a unit that every physics lab, every sports federation, and every school in the world has promised to measure the same way. That promise is the SI system, the International System of Units, and Class 11 physics opens with it for a very deliberate reason: every formula you will meet for the rest of the year — from $F = ma$ to $v = u + at$ to $PV = nRT$ — is only trustworthy once its units are trustworthy. The idea sounds almost too simple to be "physics": choose a small set of quantities that cannot be defined in terms of anything simpler, fix an exact standard for each, and then build every other unit in science by multiplying and dividing these few. Length, mass, time, electric current, temperature, amount of substance, and luminous intensity — just seven quantities — turn out to be enough to construct newtons, joules, pascals, volts, and every other unit you will ever write. A speed is metres divided by seconds; a force is kilograms times metres divided by seconds squared; nothing more exotic is needed. This article treats SI units as a system worth understanding, not a table worth memorising. You will see why the seven base units were chosen, how modern SI (since the 2019 redefinition) ties every one of them to a fixed constant of nature rather than a metal bar in a vault, how to derive the unit of any quantity you are given a formula for, and why an examiner treats a numerical answer without a unit as an incomplete answer — because in physics, the number and the unit are one inseparable fact.

Key takeaways

A physical quantity is measured by comparison against a fixed, agreed unit; the number alone means nothing without it.
SI recognises seven base (fundamental) quantities — length, mass, time, electric current, temperature, amount of substance, luminous intensity — each with one base unit and symbol.
Every other unit is a derived unit, built by combining base units through the defining formula of that quantity (e.g. $\text{force} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ , named the newton).
Two supplementary units, the radian (plane angle) and steradian (solid angle), are dimensionless and sit outside the seven base units.
Since 2019, all seven SI base units are defined by fixing exact numerical values of fundamental constants of nature (like the speed of light and the caesium-133 transition frequency), not by physical prototype objects.
SI is a coherent, decimal (base-10) system, which is why conversions between its multiples (mm, cm, km) need only powers of ten, unlike the older CGS/FPS systems.
Key idea: every measured or derived quantity in physics is a pure number multiplied by a unit built from the seven SI base units, and no calculation, formula, or exam answer is complete until that unit is correctly attached.

The core insight

SI units express every physical quantity as a number times a unit built from seven fixed base units (metre, kilogram, second, ampere, kelvin, mole, candela), with all other units — like the newton or the pascal — derived from these through the defining formula of the quantity.

Where it fits in the syllabus

This topic opens the CBSE Class 11 chapter "Units and Measurements," and the NCERT/CBSE 2025-26 prescription keeps it close to its classical core: the need for measurement, systems of units, and the SI base and derived units, with the fundamental-versus-derived distinction as the central idea. It deliberately stays short of instrument-level least-count theory and statistical (Gaussian/standard-deviation) treatment of error, which are university-level extensions kept out of the Class 11 board syllabus. Everything that follows in the chapter — significant figures, error propagation, and dimensional analysis — assumes you already know what a "unit" is, why SI is the preferred coherent system, and how a derived unit is built from base units, so this section is the foundation the rest of the chapter is built on.

The physical picture

A physical quantity becomes measurable only when it is compared against a fixed reference of the same kind — that reference is its unit, and the number obtained from the comparison is its magnitude in that unit. Quantities that cannot be expressed in terms of any other physical quantity are called base (or fundamental) quantities; their units are base units. Every other quantity — area, velocity, force, energy, pressure, and so on — is a derived quantity, built from the base quantities through its defining relation, and its unit is a derived unit.

\text{measured value} = (\text{numerical magnitude}) \times (\text{unit})

where: the numerical magnitude is a pure number with no physical meaning by itself, and the unit carries the physical identity of the quantity — change the unit and the numerical magnitude must change to compensate, which is exactly the logic behind unit conversion. This is why "the length is 5" is meaningless while "the length is 5 m" is a complete physical statement. SI recognises exactly seven base quantities:

Figure

A table-style diagram listing the seven SI base quantities length mass time electric current thermodynamic temperature amount of substance and luminous intensity with their unit names metre kilogram second ampere kelvin mole and candela and their symbols m kg s A K mol and cd

A derived unit follows mechanically from the formula that defines the quantity. Speed is distance divided by time, so its SI unit is metres per second, $\text{m/s}$ . Force follows from Newton's second law:

F = ma \quad\Rightarrow\quad [F] = \text{kg}\cdot\text{m}\cdot\text{s}^{-2} = \text{N}

where: $m$ is mass in kilograms and $a$ is acceleration in metres per second squared; the combination $\text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ occurs so often in physics that it is given a compact special name, the newton (N). Several other derived units carry special names for the same reason — the joule (energy), the pascal (pressure), the watt (power), the volt (potential difference) — but every one of them can always be unpacked back into a combination of the seven base units, and doing that unpacking is a standard exam skill. Modern SI (the definitions adopted internationally in 2019) fixes every base unit by assigning an exact numerical value to a fundamental constant of nature, rather than by a physical prototype object. The second is defined through the frequency of a specific radiation from a caesium-133 atom; the metre is then defined using the speed of light and that second; the kilogram is fixed through the Planck constant. This is a deliberate improvement over the old approach — a specific metal bar or cylinder kept in a vault, as the kilogram once was, can be damaged, contaminated, or simply drift in mass over decades, whereas a constant of nature is, by definition, the same everywhere in the universe and at every time, giving the modern SI base units essentially unlimited reproducibility and precision. Two conventions matter directly for how you write physics answers. First, always report a measured or calculated quantity with its correct unit attached — a numerical answer without a unit is treated by examiners as incomplete, even if the digits are right, because the number by itself carries no physical meaning. Second, SI is a coherent, decimal (base-10) system: its multiples and submultiples (millimetre, centimetre, kilometre, and so on) differ only by powers of ten, which is exactly why SI has replaced older systems like CGS (centimetre-gram-second) and FPS (foot-pound-second) in scientific work — conversions inside SI never require awkward factors like 12 or 1760, only shifts of a decimal point.

Derivation ABuilding the SI unit of pressure from base units

The goal of this derivation is to show, step by step, how the SI unit of a derived quantity is constructed purely from its defining formula and the seven SI base units, using pressure as the running example, so that the same method can be applied to any derived quantity — energy, power, density, or momentum — that a board or competitive question might ask you to unpack, without needing to memorise a long list of special unit names by rote.

Step 1. Start from the defining formula. Pressure is force per unit area, $P = \dfrac{F}{A}$ , so before touching units at all, identify that pressure depends on force and area, both of which must themselves be reduced to base units.

Step 2. Reduce force to base units using Newton's second law, $F = ma$ . Mass $m$ is already a base quantity with unit kilogram (kg). Acceleration $a$ is a rate of change of velocity with time, so its unit is metres per second, per second, i.e. $\text{m}\cdot\text{s}^{-2}$ . Multiplying, the unit of force is $\text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ , given the special name newton, $\text{N}$ .

Step 3. Reduce area to base units. Area is length times length, and length is a base quantity with unit metre (m), so the unit of area is $\text{m}^2$ .

Step 4. Divide the unit of force by the unit of area, exactly mirroring the formula $P = F/A$ : $\dfrac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$ . This is the SI unit of pressure expressed purely in base units.

Step 5. Recognise the special name. This particular combination, $\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$ , occurs often enough in physics (and engineering) to be given its own compact name, the pascal, symbol Pa, so $1\ \text{Pa} = 1\ \text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ . Since pressure's unit is entirely built from the base units of mass, length and time — it is not one of the seven base quantities itself — pressure is confirmed to be a derived quantity, and this same five-step method (formula, reduce each factor to base units, combine algebraically as the formula dictates, simplify, name if a special name exists) works for any derived quantity you are asked to unpack.

Worked examples

Board level - unit of pressure from its defining formula

Express the unit of pressure in terms of SI base units, and state whether pressure is a base or a derived quantity.

Step 1. Write the defining formula: $P = \dfrac{F}{A}$ .

Step 2. Find the SI unit of force. From $F = ma$ , unit of $F = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ , the newton (N).

Step 3. Find the SI unit of area: unit of $A = \text{m}^2$ .

Step 4. Combine: $P = \dfrac{F}{A} = \dfrac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$ , which is named the pascal (Pa), so $1\ \text{Pa} = 1\ \text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ .

Answer: the SI unit of pressure is $\text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ , i.e. the pascal, and since it is built from the base units of mass, length and time, pressure is a derived quantity, not a base quantity.

Competitive level - converting a speed to SI units and checking it against a derived-unit formula

A car travels 90 km in 1.5 hours. (a) Express its average speed in SI units (m/s). (b) If the car's engine exerts an average force of 2000 N to maintain this over its 1000 kg mass's motion, verify that the unit of $F/m$ matches the unit of acceleration.

Step 1. Convert distance to metres: $90\ \text{km} = 90 \times 1000\ \text{m} = 9\times10^{4}\ \text{m}$ .

Step 2. Convert time to seconds: $1.5\ \text{h} = 1.5 \times 3600\ \text{s} = 5.4\times10^{3}\ \text{s}$ .

Step 3. Compute average speed: $v = \dfrac{\text{distance}}{\text{time}} = \dfrac{9\times10^{4}\ \text{m}}{5.4\times10^{3}\ \text{s}} \approx 16.67\ \text{m/s}$ .

Step 4. Check the unit of $F/m$ : force is in newtons, $\text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ , and mass is in kilograms, so $\dfrac{F}{m}$ has unit $\dfrac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{kg}} = \text{m}\cdot\text{s}^{-2}$ , exactly the unit of acceleration — confirming $F = ma$ is dimensionally self-consistent regardless of the actual numbers plugged in.

Answer: the car's average speed is about $16.67\ \text{m/s}$ , and the unit check confirms $F/m$ correctly reduces to $\text{m}\cdot\text{s}^{-2}$ , the SI unit of acceleration. Timing note: this pattern — convert every quantity to SI first, then compute — should take under a minute for the arithmetic in part (a). Part (b)'s unit check is a near-instant sanity habit competitive exams reward: before trusting a numerical answer, glance at whether the units on both sides of a formula actually match, since a mismatch instantly flags an arithmetic or conceptual slip without needing to redo the whole calculation.

Common traps

Writing a numerical answer without its unit — in board exams this loses marks even when the number itself is correct, because the unit is treated as part of the answer, not decoration.
Confusing base units (exactly seven exist, by definition) with derived units (there are infinitely many, each built from combinations of the base seven).
Mixing unit systems mid-calculation — for example using centimetres for length and hours for time in the same formula without converting — instead of first converting every quantity to one consistent system, almost always SI, before calculating anything.
Treating the radian and steradian as if they were an eighth and ninth base unit; they are supplementary, dimensionless units for plane and solid angle respectively, kept conceptually separate from the seven base quantities.
Assuming a "special name" unit like the newton or pascal is somehow a base unit because it has its own symbol — every special-named unit is still fully derived from the base seven, as the derivation above shows.

Common trapA number without a unit is not a physics answer

The single most common way marks are lost in this topic is treating the unit as an optional label tacked onto a number rather than as part of the physical quantity itself. Physically, " $16.67$ " and " $16.67\ \text{m/s}$ " are not the same kind of object at all — the first is a pure number with no physical meaning, the second is a complete statement about how fast something moves. This matters beyond exam marking too: every formula in this chapter, and every formula for the rest of the year, is only valid when every quantity plugged into it carries its correct, consistent unit; silently dropping units, or mixing units from different systems in the same formula, is the single fastest way to get a numerically wrong answer while believing the arithmetic was right.

You should be able to…

List all seven SI base quantities with their correct unit names and symbols from memory.
Explain the difference between a fundamental (base) unit and a derived unit, with at least two examples of each.
Derive the SI unit of any given derived quantity — pressure, energy, density, momentum — starting only from its defining formula.
Convert a quantity given in non-SI units (km/h, hours, grams) correctly into consistent SI units before using it in a formula.
State why SI is preferred as a coherent, internationally standardised, decimal system over older systems like CGS or FPS.
Explain, in one or two sentences, why modern SI base units are defined through fixed constants of nature rather than physical prototype objects.

Ask the tutor: Still unsure why some derived units get special one-word names like newton or pascal while others (like m/s) don't — ask the AI tutor to walk through a few more derived-unit examples from your NCERT textbook.

Why do we need seven base units — why not just have one unit for everything?

Because the physical quantities that matter in nature — how far, how heavy, how long, how much electric charge flows, how hot, how much substance, how bright — are not reducible to one another; you cannot express "temperature" purely in terms of "length" no matter how you try. Physicists found that exactly seven independent quantities were enough, when combined through formulas, to build the unit of literally every other physical quantity ever measured. Fewer than seven would leave some quantities impossible to express; choosing more would just be redundant, since some "extra" quantity would turn out to be expressible in terms of the existing seven.

Are the radian and steradian base units or not, and will this be asked directly?

They are neither pure base units nor ordinary derived units — they are called supplementary units, and they are dimensionless, meaning they have no combination of metre, kilogram, second, and so on in their definition (a radian is literally arc length divided by radius, so the metres cancel). CBSE occasionally asks you to identify them correctly as supplementary and dimensionless in a one-mark or assertion-reason question, so know the label, but you will not be asked to derive anything complicated with them at this stage.

Why did SI redefine the kilogram using the Planck constant instead of just keeping the metal cylinder?

The old kilogram was defined as the mass of one specific platinum-iridium cylinder kept in a vault near Paris, and comparisons over decades showed its mass could drift by tiny amounts relative to copies of itself — an unacceptable flaw for the standard that all other mass measurements ultimately trace back to. Fixing an exact numerical value for the Planck constant instead means the kilogram is now defined through an unchanging fact about nature, reproducible in a properly equipped lab anywhere on Earth (or off it), rather than through one irreplaceable physical object that could be damaged, lost, or simply age.

If I get a formula wrong on a numerical, can checking the units still save me marks?

Not directly for the final numerical answer, but it is an extremely valuable self-check habit: if you compute a quantity you know should be a velocity, and your unit-tracking gives you $\text{kg}\cdot\text{m}$ instead of $\text{m/s}$ , that is an immediate, reliable signal that a formula or algebraic step was applied incorrectly, letting you catch and fix the error before submitting. This same instinct — tracking units alongside numbers through every step of a calculation — is exactly the seed of the dimensional-analysis techniques the next parts of this chapter build on.