The Big Idea
A bare number tells us an amount, but it may not tell us what the amount describes. The number \(12\) could mean twelve meters, twelve seconds, twelve kilograms, twelve dollars, or twelve objects.
A unit provides a reference scale. A dimension identifies the kind of physical quantity being described. A scalar is a quantity whose value is specified by a number and any necessary unit, without requiring a spatial direction.
These ideas turn arithmetic into quantitative reasoning. They tell us which quantities can be added, how conversion factors work, what a formula is allowed to mean, and whether a calculated answer is physically sensible.
A Quantity Combines a Number and a Unit
A measured or specified physical quantity can be written conceptually as
\[ \text{quantity}=\text{numerical value}\times\text{unit}. \]
For example,
\[ 5\text{ m},\qquad 12\text{ s},\qquad 3.2\text{ kg} \]
represent length, time, and mass. Changing the unit changes the numerical value needed to describe the same quantity:
\[ 5\text{ m}=500\text{ cm}. \]
The number changed from \(5\) to \(500\), but the physical length did not change.
Unit and Dimension Are Different
| Concept | What it identifies | Examples |
|---|---|---|
| Dimension | The physical kind of quantity | Length, time, mass, temperature |
| Unit | A chosen reference scale for that quantity | Meter, foot, second, hour, kilogram |
| Numerical value | How many units are present | \(5\), \(12.7\), \(-3\) |
Meters and feet are different units of the same dimension, length. Seconds and hours are different units of the same dimension, time. A meter and a second cannot be converted into one another because length and time are different dimensions.
The Seven SI Base Quantities
The International System of Units, abbreviated SI, selects seven base quantities and base units. Other SI quantities are built from them.
| Base quantity | Common dimension symbol | SI base unit | Unit symbol |
|---|---|---|---|
| Time | \(\mathsf{T}\) | second | \(\text{s}\) |
| Length | \(\mathsf{L}\) | meter | \(\text{m}\) |
| Mass | \(\mathsf{M}\) | kilogram | \(\text{kg}\) |
| Electric current | \(\mathsf{I}\) | ampere | \(\text{A}\) |
| Thermodynamic temperature | \(\mathsf{\Theta}\) | kelvin | \(\text{K}\) |
| Amount of substance | \(\mathsf{N}\) | mole | \(\text{mol}\) |
| Luminous intensity | \(\mathsf{J}\) | candela | \(\text{cd}\) |
Dimension-symbol conventions vary among books and fields. What matters is that the symbols are declared and used consistently.
Derived Quantities Combine Dimensions
Many useful quantities are defined by multiplying or dividing base quantities. Their dimensions and units follow the same arithmetic.
| Quantity | Relationship | Dimension | Common SI unit |
|---|---|---|---|
| Area | length \(\times\) length | \(\mathsf{L}^2\) | \(\text{m}^2\) |
| Volume | length \(\times\) length \(\times\) length | \(\mathsf{L}^3\) | \(\text{m}^3\) |
| Speed | distance divided by time | \(\mathsf{L}\mathsf{T}^{-1}\) | \(\text{m}/\text{s}\) |
| Acceleration | speed change divided by time | \(\mathsf{L}\mathsf{T}^{-2}\) | \(\text{m}/\text{s}^2\) |
| Density | mass divided by volume | \(\mathsf{M}\mathsf{L}^{-3}\) | \(\text{kg}/\text{m}^3\) |
| Force | mass \(\times\) acceleration | \(\mathsf{M}\mathsf{L}\mathsf{T}^{-2}\) | newton, \(\text{N}\) |
| Energy | force \(\times\) distance | \(\mathsf{M}\mathsf{L}^2\mathsf{T}^{-2}\) | joule, \(\text{J}\) |
Named Units Are Shorthand
Some derived SI units receive names for convenience. A newton is not a separate dimension; it abbreviates a combination of base units:
\[ 1\text{ N}=1\frac{\text{kg}\cdot\text{m}}{\text{s}^2}. \]
A joule is
\[ 1\text{ J}=1\text{ N}\cdot\text{m} =1\frac{\text{kg}\cdot\text{m}^2}{\text{s}^2}. \]
Expanding a named unit into base units can reveal whether an equation or conversion is consistent.
What Is a Scalar?
A scalar is a quantity specified by a numerical value and any required unit, without needing a spatial direction. Examples include:
- mass: \(8\text{ kg}\),
- elapsed time: \(15\text{ s}\),
- temperature: \(22^\circ\text{C}\),
- energy: \(120\text{ J}\),
- volume: \(3\text{ L}\), and
- speed: \(60\text{ km}/\text{h}\).
A scalar may be positive, negative, or zero when the quantity permits it. Negative temperature, electric charge, profit, or coordinate change can be scalar. “Scalar” does not mean “positive”; it means that no spatial direction is required to specify the quantity.
Scalars and Vectors
| Feature | Scalar | Vector |
|---|---|---|
| Required information | Numerical value and unit; sign where meaningful | Magnitude, direction, and unit |
| Example | Speed \(=20\text{ m}/\text{s}\) | Velocity \(=20\text{ m}/\text{s}\) east |
| Addition | Ordinary signed addition for compatible quantities | Direction-sensitive vector addition |
| Representation | Single numerical value in a chosen unit | Components or a directed geometric object |
Distance is scalar; displacement is vector. Speed is scalar; velocity is vector. A trip can have a total distance of \(10\text{ km}\) while its final displacement is \(0\text{ km}\) if the traveler returns to the starting point.
Scalars Can Scale Other Objects
In algebra, the word scalar also describes an element that multiplies another mathematical object. Multiplying a vector, matrix, function, or geometric figure by a scalar changes its size or intensity according to the setting.
If \(\mathbf{v}\) is a vector, then
\[ 3\mathbf{v} \]
has three times the magnitude of \(\mathbf{v}\) and the same direction. The vector \(-2\mathbf{v}\) has twice the magnitude and the opposite direction.
The allowed scalars depend on the mathematical structure. In elementary real-vector work, the scalars are usually real numbers.
Units Participate in the Algebra
Treating units as algebraic factors is one of the safest problem-solving habits.
Multiplying length by length produces area:
\[ (3\text{ m})(4\text{ m})=12\text{ m}^2. \]
Dividing distance by time produces speed:
\[ \frac{150\text{ km}}{3\text{ h}} =50\frac{\text{km}}{\text{h}}. \]
Dividing a quantity by the same unit cancels that unit:
\[ \frac{12\text{ m}}{3\text{ m}}=4. \]
Addition Requires Compatible Dimensions
Quantities can be added or subtracted only when they represent the same dimension. For example,
\[ 3\text{ m}+40\text{ cm} =3\text{ m}+0.4\text{ m} =3.4\text{ m}. \]
The units need not begin in the same form, but they must be convertible. By contrast,
\[ 3\text{ m}+4\text{ s} \]
has no ordinary physical meaning because length and time are different dimensions.
Multiplication and division are different: they can combine unlike dimensions to create a derived quantity.
Conversion Factors Equal One
If \(1\text{ m}=100\text{ cm}\), then both ratios
\[ \frac{100\text{ cm}}{1\text{ m}} \qquad\text{and}\qquad \frac{1\text{ m}}{100\text{ cm}} \]
equal \(1\). Multiplying by either ratio changes the representation without changing the quantity.
To convert \(2.7\text{ m}\) to centimeters, choose the factor that cancels meters:
\[ 2.7\text{ m} \left(\frac{100\text{ cm}}{1\text{ m}}\right) =270\text{ cm}. \]
The unwanted unit appears once in the numerator and once in the denominator, so it cancels.
Factor-Label Method
The factor-label method, also called unit-factor conversion or dimensional analysis, organizes one or more conversion factors into a cancellation chain.
Convert \(72\text{ km}/\text{h}\) to meters per second:
\[ 72\frac{\text{km}}{\text{h}} \left(\frac{1{,}000\text{ m}}{1\text{ km}}\right) \left(\frac{1\text{ h}}{3{,}600\text{ s}}\right) =20\frac{\text{m}}{\text{s}}. \]
Kilometers and hours cancel, leaving meters per second. The surviving units are an internal check on the setup.
Area and Volume Require Powered Conversion Factors
A length conversion must be squared for area and cubed for volume. Since
\[ 1\text{ m}=100\text{ cm}, \]
it follows that
\[ 1\text{ m}^2=(100\text{ cm})^2 =10{,}000\text{ cm}^2 \]
and
\[ 1\text{ m}^3=(100\text{ cm})^3 =1{,}000{,}000\text{ cm}^3. \]
Multiplying by \(100\) only once is a length conversion, not an area or volume conversion.
Metric Prefixes Encode Powers of Ten
| Prefix | Symbol | Factor | Power of ten |
|---|---|---|---|
| giga | \(\text{G}\) | \(1{,}000{,}000{,}000\) | \(10^9\) |
| mega | \(\text{M}\) | \(1{,}000{,}000\) | \(10^6\) |
| kilo | \(\text{k}\) | \(1{,}000\) | \(10^3\) |
| centi | \(\text{c}\) | \(\frac1{100}\) | \(10^{-2}\) |
| milli | \(\text{m}\) | \(\frac1{1{,}000}\) | \(10^{-3}\) |
| micro | \(\mu\) | \(\frac1{1{,}000{,}000}\) | \(10^{-6}\) |
| nano | \(\text{n}\) | \(\frac1{1{,}000{,}000{,}000}\) | \(10^{-9}\) |
Prefix symbols are case-sensitive. \(\text{M}\) means mega, while \(\text{m}\) means milli or meter depending on its role. Clear spacing and context matter.
Equations Must Be Dimensionally Consistent
Every term added in a valid physical equation must have the same dimension, and the two sides of an equation must describe compatible dimensions.
Consider the distance relation
\[ d=vt. \]
Its dimensions are
\[ \mathsf{L} =(\mathsf{L}\mathsf{T}^{-1})(\mathsf{T}) =\mathsf{L}. \]
The time dimensions cancel, leaving length. This does not prove the formula is correct, but a dimensional mismatch proves that a proposed physical formula cannot be correct as written.
Dimensional Analysis Is Powerful but Limited
Dimension checking can detect missing or extra powers and incompatible additions. It cannot determine every numerical constant or distinguish all quantities that share a dimension.
For example, work and torque both have dimensions
\[ \mathsf{M}\mathsf{L}^2\mathsf{T}^{-2}, \]
but they represent different physical ideas. Likewise, dimensional analysis alone cannot tell us whether a formula needs a factor of \(2\), \(\pi\), or another dimensionless constant.
Dimensionless Quantities
A quantity is dimensionless when its dimensions cancel. Ratios of like quantities are common examples:
\[ \text{slope}=\frac{\text{vertical length}}{\text{horizontal length}}, \qquad \text{strain}=\frac{\text{change in length}}{\text{original length}}. \]
Percentages, probabilities, refractive index, and many coefficients are dimensionless.
Dimensionless does not always mean that notation can be discarded. Radians and steradians are dimensionless derived SI units that communicate the kind of ratio being used. A percentage symbol also communicates a scale factor of \(\frac1{100}\).
Arguments of Mathematical Functions
In physical formulas, the input to functions such as exponentials, logarithms, sine, and cosine must ordinarily be dimensionless.
The expression
\[ e^{-t/\tau} \]
is dimensionally valid when \(t\) and \(\tau\) both have the dimension of time, so their ratio is dimensionless.
Writing \(e^{-t}\) with \(t\) measured in seconds is incomplete unless an implied scale supplies the reciprocal-time factor needed to make the exponent dimensionless.
Mass and Weight Are Not the Same Quantity
Mass is a scalar measure of inertia and is expressed in kilograms. Weight is the gravitational force on that mass:
\[ \mathbf{W}=m\mathbf{g}. \]
Weight is a vector force measured in newtons. Near Earth’s surface, a mass of \(10\text{ kg}\) has a weight magnitude of approximately
\[ (10\text{ kg}) \left(9.8\frac{\text{m}}{\text{s}^2}\right) =98\text{ N}. \]
The mass remains \(10\text{ kg}\) on the Moon, but the weight changes because gravitational acceleration is different.
Temperature Requires Special Care
Kelvin is an absolute temperature scale with a meaningful zero. Celsius and Fahrenheit use shifted zero points, so converting a temperature reading requires both scaling and an offset:
\[ T_{\!F}=\frac95T_{\!C}+32. \]
A temperature difference behaves differently because the offsets cancel:
\[ \Delta T_{\!F}=\frac95\Delta T_{\!C}. \]
Therefore, an increase of \(10^\circ\text{C}\) is an increase of \(18^\circ\text{F}\), even though \(10^\circ\text{C}\) itself equals \(50^\circ\text{F}\).
Worked Example: Density
A sample has mass \(540\text{ g}\) and volume \(200\text{ cm}^3\). Density is mass per volume:
\[ \rho=\frac{m}{V} =\frac{540\text{ g}}{200\text{ cm}^3} =2.7\frac{\text{g}}{\text{cm}^3}. \]
The unit tells us the quantity is not merely \(2.7\); it is \(2.7\) grams for every cubic centimeter.
Since
\[ 1\frac{\text{g}}{\text{cm}^3} =1{,}000\frac{\text{kg}}{\text{m}^3}, \]
the density is also \(2{,}700\text{ kg}/\text{m}^3\).
Units as an Error-Checking System
Suppose a travel calculation divides distance by speed:
\[ \frac{180\text{ km}}{60\text{ km}/\text{h}} =180\text{ km} \left(\frac{\text{h}}{60\text{ km}}\right) =3\text{ h}. \]
Kilometers cancel and hours remain, which matches the requested travel time.
If the units at the end were \(\text{km}^2/\text{h}\), the setup would be wrong even if the arithmetic looked tidy. Always check both the numerical value and the resulting unit.
A Practical Quantity Workflow
- Name the quantity. Is it length, time, mass, area, speed, energy, or something else?
- Write every value with its unit.
- Identify whether the quantity is scalar or vector.
- Convert compatible inputs to convenient units.
- Arrange conversion factors so unwanted units cancel.
- Carry units through multiplication, division, and powers.
- Check that added terms share a dimension.
- Check dimensional consistency before trusting a formula.
- Report the answer with its surviving unit and appropriate precision.
- Interpret what the quantity means in context.
Common Confusions
Are meters and feet different dimensions?
No. They are different units of the same dimension, length.
Can \(3\text{ m}\) be added to \(4\text{ cm}\)?
Yes, after converting to a common unit. The result is \(3.04\text{ m}\) or \(304\text{ cm}\).
Can \(3\text{ m}\) be added to \(4\text{ s}\)?
No. Length and time are incompatible dimensions for ordinary addition.
Is every scalar dimensionless?
No. Mass, time, temperature, and energy are scalars with dimensions and units. Some scalars, such as a probability, are dimensionless.
Does a negative scalar have a direction?
A negative sign may indicate side, sense, deficit, or orientation within a chosen convention, but it does not by itself make the quantity a spatial vector.
Why is \(1\text{ m}^2\) not \(100\text{ cm}^2\)?
Both dimensions of the area must be converted: \(1\text{ m}^2=(100\text{ cm})^2=10{,}000\text{ cm}^2\).
Does dimensional consistency prove a formula?
No. It is a necessary check for a physical formula, but different quantities and formulas can share the same dimensions.
Are kilograms units of weight?
Kilograms are units of mass. Weight is a force and is measured in newtons in SI.
Working Summary
A unit supplies a reference scale, a dimension identifies the kind of quantity, and a scalar supplies magnitude without spatial direction.
Compatible quantities may be added after conversion to common units. Multiplication and division combine dimensions to create derived quantities. Conversion factors equal \(1\), so they change the representation without changing the underlying quantity.
Carry units through every calculation. They help organize the algebra, reveal the meaning of the result, and expose many setup errors before those errors become final answers.
Check Your Understanding
What is the difference between a unit and a dimension?
A dimension names the kind of quantity, such as length. A unit supplies a reference scale for it, such as meters or feet.
Convert \(3.5\text{ km}\) to meters.
\(3.5\text{ km}\left(\frac{1{,}000\text{ m}}{1\text{ km}}\right)=3{,}500\text{ m}\).
What is the dimension of speed?
Length divided by time: \(\mathsf{L}\mathsf{T}^{-1}\).
Is temperature a scalar or vector?
Temperature is a scalar. A temperature field may vary by position, but the temperature at each point is still a scalar value.
How many square centimeters are in \(2\text{ m}^2\)?
Since \(1\text{ m}^2=10{,}000\text{ cm}^2\), \(2\text{ m}^2=20{,}000\text{ cm}^2\).
Is \(v=d/t^2\) dimensionally valid as a formula for speed?
No. \(d/t^2\) has dimension \(\mathsf{L}\mathsf{T}^{-2}\), which is acceleration, not speed.
Convert \(90\text{ km}/\text{h}\) to meters per second.
\(90\frac{\text{km}}{\text{h}}\left(\frac{1{,}000\text{ m}}{1\text{ km}}\right)\left(\frac{1\text{ h}}{3{,}600\text{ s}}\right)=25\text{ m}/\text{s}\).