In this explainer, we will learn how to recognize which units are used to define the values of physical quantities.

It is helpful to begin by clarifying what is meant by a measurement.

A measurement consists of four parts:

- The object that is being measured.
- The quantity of the object.
- The value of the quantity.
- The unit that the value takes.

Let us look at an example.

A measurement is made of the mass of a brick. The mass of the brick is 1.5 kilograms.

Let us consider the four parts of the measurement.

Object being measured | The brick |
---|---|

Quantity of the object | Its mass |

Value of the quantity | 1.5 |

Unit that the value takes | Kilograms |

The meaning of *quantity* is easily confused with the meaning of *value*, as the word “quantity” is often used to mean the amount of something.

When referring to measurements, however, a quantity is not an amount of something; it is something that there can be an amount of.

Let us look at another example.

A measurement is made of the time taken for an ice cube of mass 5 grams to melt. The time taken is 250 seconds.

Let us consider the four parts of the measurement.

Object being measured | The ice cube |
---|---|

Quantity of the object | The time it takes to melt |

Value of the quantity | 250 |

Unit that the value takes | Seconds |

The value and the unit of a measurement have different properties than the object and the quantity of a measurement. The object and the quantity of a measurement are defined in only one way, but the value and the unit of a measurement can be defined in multiple ways.

If we measure something about a brick, we measure that particular brick. Measuring a different object does not directly tell us anything about the brick. If we measure the temperature of the brick, this does not directly tell us anything about the mass of the brick. If we try and define the mass of the brick by referring to a different object or quantity, we are referring to a different measurement than the measurement of the mass of the brick; there is a single unique way of stating the object and the quantity for a given measurement.

“The mass of the brick is 1.5 kilograms” is not a unique way of stating the mass of the brick. This is because “1.5 kilograms” can also be expressed in various ways, such as

- 1 500 grams,
- 0.0015 tonnes,
- 3.30693 pounds.

We can see that if the unit of a measurement changes, the value of the measurement also changes. This can be understood by thinking of the value and the unit of a measurement as being multiplied together.

We can express 1.5 kilograms as where we are using the symbol kg for kilograms.

We know that where we are using the symbol g for grams.

We see then that

We have just shown that a unit can be multiplied by a value to obtain a different unit. We can also divide a unit by a value to obtain a different unit, as shown by the following example:

The same multiplications and divisions can be done with units of other quantities as has just been shown for units of mass:

As well as being multiplied or divided by a value, a unit can be multiplied or divided by another unit.

The simplest case of a unit being multiplied by a unit is a unit being multiplied by itself.

Consider a rectangular object that has sides of lengths 7.5 metres and 1.5 metres, as shown in the following figure.

The area, , of the object is the product of the lengths of the sides. We can express this as

The area should have a unit though, as area is a quantity.

It is not clear from the equation what the unit should be, and this lack of clarity is because the units of the values that were multiplied together were not included in the equation.

If the units of the values are included with the values, we obtain instead

The order in which the terms in an equation are multiplied does not change the result of the equation, so we can express this equation as

We have seen that but we have not defined the result of

If, rather than m, a value was multiplied by itself, for example, the value 4, we would obtain

There is no equivalent result for multiplying m by m, but can also be expressed as

This means that we can express m multiplied by itself as

This unit is the metre-squared, or square metre.

We see then that

Units can be multiplied or divided by other units as well as by themselves.

Let us look at an example involving units that are multiplied or divided by themselves or by other units.

### Example 1: Identifying a Unit of Area

Which of the following is a unit of area?

- Square meter
- Centimeter
- Meter per second squared
- Cubic meter

### Answer

The multiplying of a length by another length is the equation of the area of a rectangle. The equations for the areas of other shapes are given by different multiplications, but the equations still involve multiplications of lengths. Lengths can be measured in metres.

All of the options contain the word “metre,” so it is not immediately obvious that any of the options are incorrect.

A centimetre is a unit of length. An area involves multiplying lengths by other lengths. Multiplying lengths measured in centimetres would involve multiplying centimetres by centimetres. The result of multiplying centimetres by centimetres is not centimetres, so centimetres cannot be a unit of area. In fact, no unit of length can be a unit of area.

Of the remaining options, two involve the word “square” or “squared” and one involves the word “cubic.”

Squaring a value, or a unit, means multiplying it by itself. This is consistent with multiplying a length by a length. Cubing a length means multiplying the square of the length by the length, which results in a volume rather than an area. We can, therefore, eliminate cubic metre as a unit of area.

We are left to choose between the options “square metre” and “metre per second squared.”

The word “per” indicates division, so “metre per second squared” means “metre divided by second squared.” A metre is a unit of length. We must decide whether dividing a length by a second squared is equivalent to multiplying a length by a length.

A second is a unit of time rather than length, and a second squared is a unit of a time multiplied by a time, which is unrelated to length. We can, therefore, eliminate metre per second squared.

We have seen that squaring a value or a unit means multiplying it by itself. A square metre is, therefore, a metre multiplied by a metre. A metre is a unit of length, so a metre multiplied by a metre is a unit of area.

Let us now look at another such example.

### Example 2: Identifying the Symbol for a Compound Unit

Which of the following is an appropriate symbol for the unit of a quantity found by dividing a temperature by a distance?

- K/m
- km
- K⋅m
- K/m
^{2} - mK

### Answer

The question asks which symbol would be used for a unit of a quantity found by a temperature divided by a distance. It is worth considering what such a quantity could represent.

One example of a temperature divided by a distance could be if one end of a long metal rod was heated and at some instant the temperature was recorded at different points along the length of the rod. Measurements would be made of temperature and of distance. These measurements could be plotted in a graph as shown in the following figure.

The gradient of this graph would equal change in the temperature per change in distance, which is temperature divided by distance.

Considering the options, we see that they all involve the symbols m and K or k. This is because there are quantities of temperature and distance that have units with these symbols, as follows:

- Distance is a quantity that can be measured in metres, which has the symbol m.
- Temperature is a quantity that can be measured in kelvins, which has the symbol K.

The option “km” has only lowercase symbols, but the correct symbol for kelvin is “K,” which is uppercase. The symbol “km” is actually the symbol for the unit kilometre, which equals 1 000 metres. This then is a unit of distance, not a unit of temperature divided by a distance, so it is not correct.

The option “mK” includes the symbol for kelvin but is actually the symbol for the unit millikelvin, which is kelvins. This then is a unit of temperature, not a unit of temperature divided by a distance, so it is not correct.

The symbol “K⋅m” corresponds to a quantity of a temperature multiplied by a unit. We know that we want the symbol for a unit of temperature divided by a distance. There will probably be a division symbol in the unit as is the case with
“K/m” and with “K/m^{2}.” We see then that “K⋅m” is not correct.

We know that “m^{2}” is the symbol for metre-squared or
square metre,
which is a unit of area. The option “K/m^{2}” corresponds to a
temperature divided by an area, so it is not correct.

We know that “m” is the symbol for metre, which is a unit of distance. The option “K/m” corresponds to a temperature divided by a distance, so it is correct.

Let us now look at an example involving the units of quantities plotted on a graph.

### Example 3: Identifying the Units Corresponding to Quantities Determined Using a Graph

The graph shows a plot of a change in distance over time.

What is the unit of the gradient of the line?

What is the unit of the area under the line?

### Answer

Two quantities are plotted against each other on the graph.

The quantity plotted on the vertical axis is distance and has the unit metre, with the symbol m.

The quantity plotted on the horizontal axis is time and has the unit second, with the symbol s.

The gradient of a graph is the change in the quantity plotted on the vertical axis divided by the change in the quantity plotted on the horizontal axis.

The gradient of the graph corresponds to the quantity that is the result of dividing a change of distance by a change of time. This quantity is called “speed.” The unit of the quantity is the unit metre divided by the unit second, or metres per second.

Written in symbol form, this unit is m divided by s, m/s.

The area under the line of a graph is the change in the quantity plotted on the vertical axis multiplied by the change in the quantity plotted on the horizontal axis.

The area under the line of the graph corresponds to the quantity that is the result of multiplying a change of distance by a change of time. This quantity does not have a name. The unit of the quantity is the unit metre multiplied by the unit second, or metre-seconds.

Written in symbol form, this unit is m multiplied by s, m⋅s.

Units that result from multiplying units by themselves are equal to the square of the unit. For example, the unit of area is the unit of distance squared.

The unit of area multiplied by length equals the unit of length cubed. We can express this as

This unit is called the metre-cubed or cubic metre. The quantity for which this is the unit is called volume.

Distance, area, and volume are related as shown in the following figure.

Let us now look at an example involving units of distance, area, and volume.

### Example 4: Identifying an Appropriate Unit for Area Divided by Volume

Which of the following is an appropriate symbol for the unit of a quantity found by dividing an area by a volume?

- m
^{−1} - m
^{−2} - m
- m
^{2} - m
^{3}

### Answer

The unit for area is square metre, which has the symbol
m^{2}. This can be expressed as

The unit for volume is cubic metre, which has the symbol
m^{3}. This can be expressed as

The question asks what unit symbol is obtained from the following calculation:

This calculation is equivalent to the following calculation:

By eliminating factors of m that appear in both the numerator and the denominator of the expression, we find that

This can be expressed as

In words, this unit would be called “ per metre.”

A unit can be determined for a quantity even when such a quantity does not correspond to anything that can be directly measured. Expressing such a unit is not incorrect; it just may not have obvious uses.

Let us now look at an example involving multiplying the values of the same quantity expressed in different units.

### Example 5: Identifying an Appropriate Compound Unit for Two Measurements of the Same Quantity with Different Unit Prefixes

Which of the following is an appropriate symbol for the unit of a quantity found by multiplying a length in millimetres by a length in centimetres?

- cm
^{2} - m
^{3} - mm
^{3} - cm
^{−1} - m

### Answer

Millimeters and centimetres are both units of the same quantity, distance.

We know that and

This means that when we convert between a value in centimetres and an equivalent value in millimetres, the value changes by a factor of 10.

For this question, however, no values are stated. The question only asks what unit would be appropriate to use.

An appropriate unit would be one that is the result of multiplying a distance by a distance.

The SI base unit of distance is the metre (m). Multiplying a metre by itself gives us

Such a unit must be some multiple of square metres (m^{2}). The only option that corresponds to this is cm^{2}, so this must be the answer.

It is of interest to consider how conversion factors affect values of the same quantity that are expressed in different units. For example, if a length equal to some number of millimetres is multiplied by a length equal to the same number of centimetres, what unit should be used for the result?

The question can be understood in either of two ways:

- What is the appropriate unit of a quantity found by multiplying a length of millimetres by a length in millimetres of ?
- What is the appropriate unit of a quantity found by multiplying a length of centimetres by a length in centimetres that has the value of ?

We have a choice, then, of giving an answer where the unit is or

The factors of and 10 in these answers are not actually part of the unit obtained; they are what a value that is expressed in that unit must be multiplied by.

This is easier to see if we consider how many cm^{2} are equal to
1 m^{2}.

We know that

If we square both sides of this equation, we see that

This is equivalent to

This is equivalent to

The factor of 10 000 is not part of the unit square metre or the unit square centimetre; it is a factor that a value must be either multiplied or divided by when converting between units of square metres and units of square centimetres.

We see then that the result of multiplying a length in millimetres by a length in
centimetres can be expressed in mm^{2} or
cm^{2}, and the value of the result must be multiplied by the conversion factor between these units. The result
could just as well be expressed in m^{2} as long as the appropriate conversion factor was applied to the value of the result.

Let us now summarize what has been learned in this explainer.

### Key Points

- A measurement consists of the object measured, the quantity of the object, the value of the quantity, and the unit of the value.
- The value and unit of a measurement can be expressed in multiple ways.
- For a measurement that has two possible values, and , with the associated units and , it must be the case that For example,
- Units can be multiplied by and divided by either themselves or other units, even if the resulting unit does not correspond to a directly measurable quantity.
- A unit multiplied by itself equals that unit raised to a power equal to the number of times the unit is multiplied by itself. For example,