Quantities, units and dimensions

A quantity describes a property; its value includes a unit

A physical quantity is a property that can be described quantitatively, such as a segment’s length, a journey’s duration or a sample’s mass. The object and the quantity are different: one stone has a mass, a volume and several possible measured lengths. Before recording a value, identify which property the question concerns.

If l denotes the length of a particular segment, l = 1.5 m gives a value with two parts: the numerical value 1.5 and the chosen unit metre, symbol m. More generally, let Q denote a quantity value, a its numerical value in a chosen unit u; then Q = a × u. The bare number 1.5 does not identify what was measured or which unit was used. Keep the quantity’s name or symbol with the number and unit when context might otherwise be unclear.

Changing the unit changes the number, while the quantity stays fixed

For a stipulated length of 1.5 m, the exact unit relations 1 m = 100 cm = 1000 mm give 1.5 m = 150 cm = 1500 mm. Metre, centimetre and millimetre express the same kind of quantity with different-sized units. A smaller unit needs a larger numerical count to describe the same length. The segment’s quantity stays fixed while its number-and-unit representation changes.

The unit belongs to the calculation. In 1.5 × m = 150 × cm, the different numbers balance the different unit sizes. Converting a value does not add new physical length or new measurement information. Named SI prefixes specify exact decimal scale factors; here centimetre and millimetre supply enough examples to explain the idea. The next lesson develops systematic unit conversions and dimensional checks.

A dimension describes the quantity’s base-quantity structure

A unit says how a quantity is expressed; a dimension describes its structure in terms of base quantities. A length expressed in metres or centimetres has the same length dimension. In this lesson, L denotes the dimension of length, T of time and M of mass. If d is a length and t a duration, we write [d] = L and [t] = T; these brackets indicate dimensions here, rather than the chosen units.

A physical kind is more specific than its dimension. Length, area and volume have dimensions L, L² and L³, so they are distinguishable dimensionally. Other quantities can share a dimension while playing different physical or contextual roles. A clock reading and a duration both involve time, but reading 09:00 is different from saying that an activity lasted nine hours. A zero length still belongs to the length quantity kind; the number zero does not remove its context.

The SI has seven base quantity–unit pairs

The International System of Units, SI, uses seven base quantities and corresponding base units. Their names and symbols provide a shared reference for expressing measurements. The table gives the current pairs.

Keep the symbols’ case and spelling: m, kg, s, A, K, mol and cd. The base unit of mass is kilogram even though its name already contains a prefix. The seven base quantities organize dimensions and derived units; they are not a list of every quantity that can be studied. This lesson uses the current SI, while the detailed defining constants belong to more specialized study.

Derived quantities combine base quantities and units

A rectangle’s area combines two perpendicular lengths. Its dimension is L², and an SI area unit is the square metre, m². A rectangular box’s volume combines three lengths: its dimension is L³ and its SI unit is the cubic metre, m³. A unit square has sides 1 m by 1 m; a unit cube has edges 1 m by 1 m by 1 m. The powers refer to the repeated length factors, not a repeated count of squares or cubes.

Speed combines distance with duration. If d denotes travelled distance and t the positive elapsed duration, average speed v = d/t. Its dimension is L/T and its SI unit is m/s. These are derived quantities because their dimensions and units are built from base quantities. The same speed can be expressed with different units, such as m/s or km/h, provided each numerical value is interpreted with its stated unit.

Addition needs compatible quantities and a shared unit

To find the total length of two segments joined end to end, add their lengths. A segment of 1.5 m and one of 75 cm give 150 cm + 75 cm = 225 cm, or 2.25 m. The shared unit makes the numerical counts comparable. Adding the bare numbers 1.5 and 75 without their unit relationship would not describe that total length.

A length of 1.5 m cannot be added to a duration of 0.75 s to make one length: their dimensions differ. Compatible dimensions are also only a first check. Even two lengths need a stated question, such as joining segments, finding a perimeter or comparing a difference. If segments overlap, their combined covered length may not equal the sum of their separate lengths. Context determines which operation represents the quantity you seek.

Products and quotients carry their units through the operation

For a rectangle with side lengths 2 m and 0.5 m, area = 2 m × 0.5 m = 1 m². Numerical values multiply, and the units multiply as well. With a third perpendicular edge of 3 m, a rectangular box’s volume is 2 m × 0.5 m × 3 m = 3 m³. Multiplication by a pure scale factor works differently: doubling a 2 m length gives 2 × 2 m = 4 m, still a length.

For 120 m travelled in 30 s, average speed is 120 m ÷ 30 s = 4 m/s. Dividing compatible quantities of the same kind can instead give a ratio with unit one: 2 m ÷ 50 cm = 200 cm ÷ 50 cm = 4. Here the denominator must be nonzero and the quantities must be expressed compatibly before the numerical quotient is taken. Keep the kind and interpretation of the result beside its unit.

Unit one still allows a named quantity and meaningful context

Some quantities have dimension one, often called dimensionless. A ratio of comparable lengths or a share of a whole has all base dimensions cancelled, so its coherent SI unit is one, symbol 1, often omitted in writing. For example, a length ratio of 4 says that the first length is four times the second. A percentage expresses a ratio per hundred: 25% = 1/4. The lack of a remaining base dimension does not erase what the comparison concerns.

A plane angle is also treated as a quantity with unit one in the SI, with the named unit radian, symbol rad, retained to identify its angle meaning. In a circle, an angle’s measure in radians uses the arc length divided by the radius, with that geometric angle context. Dimensional cancellation alone cannot tell you that an arbitrary length ratio is an angle. An angle of 1 rad and a share of 25% both have dimension one, but their different kinds do not automatically permit a meaningful sum.

Matching dimensions is necessary, but does not prove the formula

A proposed speed formula should have dimension L/T. Distance divided by duration does; distance multiplied by duration has dimension L × T and fails that check. For an area formula, the result should have dimension L². Yet both width × height and 2 × width × height have that dimension, although only the first gives the area of one rectangle. A wrong dimensionless coefficient can survive a dimensional check.

Meaning also depends on the quantities’ roles. Adding a duration of 30 minutes to a clock reading of 09:00 gives 09:30 on the same time basis. Adding two clock readings by itself does not establish the elapsed duration between them; that needs a suitable difference and a common time reference. A formula can therefore use time units consistently while choosing the wrong time quantity. Check the dimensions, then the physical or mathematical reason for the relationship.

Change the displayed unit while keeping one length

Begin with segment A of 3/2 m and centimetres in the explorer. Its numerical value is 150 in that unit, while the equivalence table also gives 1.5 m and 1500 mm. Switch between metre, centimetre and millimetre displays. The length’s position on the fixed 0-to-3-metre axis stays the same, while the numerical labels change with the chosen unit. Explain which quantity stayed fixed and which part of its expression changed.

Choose the end-to-end joining context with segment B of 3/4 m. Their sum is 9/4 m = 2.25 m = 225 cm, and it remains a length. Change A’s stipulated length, including zero, and inspect the corresponding total. Then select the attempted addition of a 3/4 s duration. The time quantity keeps its own kind and dimension; it cannot be converted into or drawn as an added length. These are exact model quantities, allowing you to study unit meaning separately from an actual measurement’s uncertainty.

An exact unit relation does not make a measurement exact

The defined relation 1 m = 100 cm is exact. A stipulated mathematical length of 1.5 m is therefore exactly 150 cm. An actual measured segment may instead have limitations from the instrument, method or conditions. Converting the unit preserves that measurement information; it does not remove uncertainty or establish that the reported value is the exact physical length.

For example, a reported length of 1.50 m with stated uncertainty 0.02 m becomes 150 cm with stated uncertainty 2 cm. Both the value and its stated uncertainty use the same exact unit factor. Extra written digits alone do not establish greater accuracy, and an uncertainty statement needs its own method and interpretation. Keep exact model relationships distinct from measured values and rounded reports. Detailed conversions and dimensional checks are the next lesson’s focus.

State the quantity, its unit and what the result means

Before calculating, identify the quantity kind and select a useful unit. Separate the numerical value from the complete number-and-unit value, and define any quantity symbols locally. For addition, check compatible quantities, a common unit and an appropriate combining question. For multiplication or division, carry the units and dimensions through the operation; a ratio may have unit one while still needing its named context.

After calculating, ask whether the result’s dimension matches the requested kind, whether its numerical magnitude is reasonable, and whether the relationship itself is justified. A length, an area and a duration are different answers even if their bare numbers happen to agree. Keep source measurement limitations when reporting an actual measurement, and use exact equivalences only where they are justified. The next lesson develops unit conversions and dimensional checks into a systematic procedure.

Worked examples

Try it yourself

1. A stipulated segment is 1.5 m long. Identify the quantity kind, numerical value and unit. In 150 cm, which of those changes, and what is the dimension?

   Show the answer and reasoning

   The kind is length, the numerical value is 1.5 and the unit is metre. In 150 cm the numerical value is 150 and the unit centimetre; the same length and dimension L remain.

   The chosen unit determines the numerical count, because 1 m = 100 cm. The equality 1.5 m = 150 cm relates complete quantity values, not the bare numbers 1.5 and 150.

2. Name the SI base unit and symbol for length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity.

   Show the answer and reasoning

   Metre m; kilogram kg; second s; ampere A; kelvin K; mole mol; candela cd, respectively.

   These are the seven current SI base quantity–unit pairs. Unit symbols preserve their case. The base mass unit is kilogram, and thermodynamic temperature uses kelvin. Other quantities can have units derived from these base units or unit one.

3. Give the SI units and dimensions for area, volume and speed. Compute 2 m × 3 m and the ratio of 2 m to 50 cm. Explain why the first result keeps a unit while the second has unit one.

   Show the answer and reasoning

   Area: m² and L²; volume: m³ and L³; speed: m/s and L/T. The product is 6 m²; the length ratio is 4 with unit one.

   Products multiply length units, so two length factors give area dimension. For the ratio, convert 2 m to 200 cm and divide 200 cm by 50 cm; equal unit factors cancel. The dimension-one result still compares the named lengths.

4. Join lengths 1.5 m and 3/4 m end to end, then express the total in centimetres. Could 1.5 m instead be added to 3/4 s as a length? Does adding two clock readings by itself give elapsed duration?

   Show the answer and reasoning

   The joined length is 9/4 m = 2.25 m = 225 cm. A duration cannot be added as a length. Adding two clock readings by itself does not establish elapsed duration.

   The joining question supplies a meaningful sum of two lengths with a common unit. Length and time dimensions differ in the attempted second operation. For clock readings, a suitable difference on a common time basis supplies duration; having time units alone does not choose the correct operation.

5. What dimension do a length ratio, a percentage share and a plane angle in radians have in the SI? Does that make their kinds interchangeable? Find 25% of a stipulated 8 m length and state the result’s unit.

   Show the answer and reasoning

   They have dimension one, but their kinds and contexts remain distinct. Twenty-five percent of 8 m is 2 m, still a length.

   Ratios can cancel base dimensions, and angle uses unit one by SI convention with radian identifying its angle meaning. That does not turn every ratio into an angle. The factor 25% = 1/4 is dimensionless; multiplying it by a length preserves the length unit.

6. For a rectangle, compare area = width × height with area = 2 × width × height. Do dimensions alone detect the second formula’s error? For sides 2 m and 0.5 m, calculate both results. Would distance × duration have a speed dimension?

   Show the answer and reasoning

   Both area expressions have dimension L², so dimensions alone miss the wrong coefficient. They give 1 m² and 2 m², respectively; the rectangle’s area is 1 m². Distance × duration has dimension L × T, not speed’s L/T.

   The pure number 2 does not change a product’s dimension, although it doubles its value. Geometry supplies the rectangle-area rule. The wrong speed expression changes the dimension itself, so that error is visible to a dimensional check.

7. A measured length is reported as 1.50 m with stated uncertainty 0.02 m. Express both in centimetres. Does the exact relation 1 m = 100 cm establish a perfectly known physical length or improve its measurement accuracy?

   Show the answer and reasoning

   The report is 150 cm with stated uncertainty 2 cm. Exact unit conversion does not establish a perfectly known length or improve the measurement by itself.

   Multiply both the reported value and its stated uncertainty by the same exact unit factor 100. This changes their representation while preserving the source information. Measurement uncertainty concerns the result and method; an exact definition of the unit relation does not remove it.

What to carry forward

• A quantity value is a numerical value multiplied by a unit. Changing the unit can change the number while preserving the same quantity.
• Physical kind, dimension and unit are distinct. The seven SI base quantity–unit pairs organize dimensions and derived units, including kilogram for mass and kelvin for thermodynamic temperature.
• Add compatible quantities in common units with a meaningful context. Products and quotients carry units and dimensions into area, volume, speed or other result kinds.
• Quantities with unit one retain their meaning; radians identify angle context and percentages identify per-hundred ratios. Matching dimensions is necessary but does not prove a formula or interpretation.
• Exact unit relations preserve a representation’s equivalence; they do not remove measurement uncertainty. Define symbols, retain units and check the requested quantity’s meaning.