1. Physical Quantities and Measurement Techniques

# 1.1 Quantities and Units

## Definitions

Magnitude: The magnitude is an indication of the size of a quantity. Technically, it is the absolute value of a quantity. For example, let’s say we have two velocities: v1 = 10 m/s and v2 = -15 m/s; the magnitude of v1 is 10 and the magnitude of v2 is 15. The magnitude is just a number and the unit is not mentioned while indicating the magnitude of a quantity.
Physical quantity: a physical quantity is any quantity that can be measured. It has two components: magnitude and unit. For example, distance, speed, and acceleration are physical quantities.
Absolute Value: The absolute value of a number tells you the distance between that number and zero on the number line. The absolute value is always positive.The number 10 is 10 units to the right of zero, so its absolute value is 10. However, -15 is 15 units to the left of zero, so its absolute value is 15. The absolute value does not include the sign of the number.
The number line representation is shown below in figure 1.1:
The figure shows two simple equations: |-15| = |15| and |10| =10
The first one says: the absolute value of 10 is 10.
The second one says: the absolute value of -15 is 15.
The two vertical bars on the left-hand side of both the equations represent the symbol that it used to represent the absolute value.

## Key Concepts

Some physical quantities are referred to as base or fundamental quantities.
The base quantities for IGCSE / O-Level physics are: length, mass, time, electric current, thermodynamic temperature, and luminous intensity.
The SI unit of each quantity is given below in Table 1.1
Some quantities are derived quantities, and as the name suggests, such quantities are derived from some combination of the fundamental (or base) quantities.
For example, speed is a derived quantity. It is derived from distance and time. Distance is just a measurement of length so distance is also a base quantity.
Similarly, velocity is another example of a derived quantity. It is derived from displacement (type of length) and time. Other examples of derived quantities include acceleration, force, pressure, etc. Numerous examples of such quantities are given below in table 1.2.
You must ensure that all units are consistent. This means that the units should satisfy the equation that you use. For demonstration, let’s use the following equation:
Let’s suppose that a question provides you with the following units:
Distance in km
Speed in m/s
Time in h
To check if these are consistent units, you need to plug them into the equation shown above and check if the units on both sides of the equation come out to be the same.
Inserting the given units (without any values) into the equation gives you:
You can now clearly see that the unit on the left-hand side of the equation is not the same as the unit on the right-hand side of the equation. This is an example of inconsistent units. They are called inconsistent because they do not satisfy the equation.
Whenever you come across such a scenario, you will be required to perform unit conversions to make sure that all units are consistent.
You have two options for this set of units:
Convert km to m and h to s
Convert m to km and s to h
km represents kilometres
h represents hours
s represents seconds
m represents metres
The original set of units resulted in:
After unit conversions, the same equation will look like this:
Now, the units are consistent. The two ’s’ units on the right-hand side of the equation will cancel out and you will simply be left with the same unit on both sides of the equation. In this case, that will be ‘m’.
Of course, you will have to multiply the distance in km with 1,000 to get the distance in m; similarly, you will have to multiply the time in hours with 3,600 to obtain the time in seconds. However, the values have not been included in the working above because it is only to demonstrate consistent and inconsistent units. The same rules will apply, regardless of the numbers involved.
It is necessary for you to be mindful of units before attempting any problem. All equations require consistent units. If an equation does not have consistent units, it cannot be valid!
Let’s see how we can easily work out an equation using units: you know that the unit of speed is m/s. Notice that the unit ‘m’ is in the numerator and the unit ’s’ is in the denominator. You know that the metre is the unit of length (or distance) and the second is the unit of time. Based on the unit m/s, it should be clear that the distance is being divided by time. Therefore, the unit indicates that speed = distance / time.

### Standard Notation

Another important concept is that of standard notation. Given that quantities tend to get very large or very small, it is important for us to be able to write very large or very small numbers in a format that is easy to comprehend and convey to others. We make use of powers of 10 to accomplish that, and the format we use is called the standard notation.
The general form of the standard notation is:
where A is any number equal to or greater than 1 but less than 10,
n is a positive or negative whole number
Put simply, the general form tells you that to represent a number in standard notation, you need to multiply a number with a power of 10. And, that the number must be between at least 1 and not larger than 10.
It is important to note that A must not be smaller than 1. The minimum value it can have is 1 and the maximum value it can have is 10.
Mathematically, the condition on the value of A is represented as:
For example, the number 100 written in standard form looks like this:
Comparing this to the general form, you should realize that A = 1 (or 1.0) and n = 2. This is the correct way to write 100 in standard notation. However, if you were to write it as , it would not be considered standard notation, even though mathematically, the calculation would still produce a result of 100. It would not be considered standard notation because A would be less than 1!
Similarly, it would also be incorrect to write it as:
Again, comparing this to the general form of the standard notation tells you that A = 10 and n = 1. This is also wrong because A is not less than 10.
To summarize, you have to ensure that A is at least 1 and less than 10.

### Prefixes

The powers of 10 are closely linked to prefixes. A prefix is a number that we attach to a unit to make it more readable and easier to process. For example 1 km = 1,000 m — in this case, the prefix ‘k’, stands for kilo and represents a value of 1,000.
You must be fully familiar with all common prefixes and their associated values. The prefixes (with values) are given below in table 1.3:
It is very important to use the correct symbol — some prefixes are represented by small letters whereas others are represented by capital letters. For example, small ‘m’ means milli whereas capital ‘M’ means mega.
There are many common examples involving prefixes that you encounter in everyday life. For example:
kilometre
centimetre
millimetre
kilobyte
megabyte
gigabyte
Instead of saying 1,000 metres, we simply write 1 km.
Similarly, it would be awkward for us to say 1,000,000,000 (1 billion) bytes. Instead of that, we simply say 1 gigabyte (1 GB). The prefix ‘giga’ takes care of the magnitude because it represents a value 1 billion (109).
As for why we use prefixes or the standard notation, consider two ways of expressing the same information:
1,000,000,000,000 bytes
1 terabyte
The first one is much more difficult to read — in fact, you need a second or two to count the number of zeros, whereas the second one is much easier to read and understand. Both represent exactly the same information but the second format is a lot more meaningful.
Standard notation comes in handy for the same reason — it makes numbers much easier to understand. In this case, option (a) would be written as:
This is significantly easier to read and comprehend, as compared to 1,000,000,000,000 bytes
We have two options while using prefixes: using the prefix name or the associated power of 10.
Using the prefix values, you can easily convert between units. For instance, 1 km = 1000 m because the prefix ‘k’ represents kilo (1000).
Similarly, 1 cm = 0.01 m because the prefix ‘c’ represents centi (10-2). It is very obvious if you simply replace the prefix with its value, as shown below:
1 centimetre = 1 x 10-2 m
1 cm = 0.01 m
If you divide both sides by 0.01, you will get 100 cm = 1 m (or 1 m = 100 cm).

## Common Misconceptions

Misconception 1: The magnitude of a quantity is the number (or value) that precedes (comes before) the unit.
This is not true. The magnitude is the absolute value of the value associated with a quantity, which means that the magnitude of a quantity can only be positive. For example, if v = - 5 m/s, its magnitude would be 5 and not -5.
The magnitude does not include the unit — it is just a number.
Misconception 2: standard notation only requires multiplication with a power of 10
Simply multiplying a number with a power of 10 does not mean that it is in standard notation (or standard form). You must remember that there is a condition on the other number as well.
The general form of the standard notation is:
It is necessary for the number A to be in this range:
The rules for standard notation will not be obeyed if the first number is less than 1 or greater than 10.
For example, is not in standard notation. To convert it into standard notation, you must fix the decimal point of the first number. In standard notation, it would be written as:
Misconception 3: small letters and capital letters are interchangeable while using prefixes
You cannot interchange small and capital letters while using symbols to represent prefixes. The symbols are case-sensitive, which means that it small ‘m’ and capital ‘M’ are not the same. Small letters are called lowercase whereas capital letters are called uppercase.
You must remember when to use lowercase and when to use uppercase. For instance, lowercase m represents milli whereas uppercase M represents mega. Milli stands for 0.001 whereas mega stands for 1,000,000.

## Key Connections

The magnitude and unit of a quantity will apply to topics throughout the syllabus.
The absolute value is relevant to a few different areas of the syllabus. For instance, the absolute value of the instantaneous velocity of an object always gives the instantaneous speed. However, the absolute value of the average velocity does not necessarily give the magnitude of the average speed.
The point above can be demonstrated through circular motion: let’s say that an object begins from point A, travels along a circle, and then ends its journey at point A. In such a case, the average speed would be non-zero because the object covers some distance in a specific time interval. However, its displacement would be zero because it starts and ends at the same point, which means that its average velocity would also be zero. So, its average speed would not be equal to its average velocity.
The prefixes are universal, which means that they can be used with any unit. Prefixes are not linked to any particular type of quantity. However, it is not required at all in some cases — for instance, you do not need any prefix with an angular measurement because the values are not too large or too small. You will see prefixes in a wide range of situations and be required to use them with any quantity. For example, you can use the prefix kilo (k) with length, energy, force, etc.

## Past Exam Questions

You are required to indicate the symbol for “millions” of watts.
You simply need to know the value that a million represents — it is 106. The prefix used to represent a value of 106 is mega and its symbol is M.
Therefore, the answer is MW. The correct choice is D.
Understand why the other options are wrong!
A: lowercase m represents milli — a value of 10-3 and lowercase w does not represent watts. Therefore, both symbols are incorrect.
B: as explained above, lowercase m does not represent a million. Uppercase W is the symbol of watts, so that is correct. This choice shows the wrong prefix but the correct symbol for the unit of watts.
C: uppercase M represents the correct value (million) but lowercase w does not represent watts. This choice shows the correct prefix but not the correct symbol for watts.
You need to identify the choice that shows the correct unit for the given quantity.
The latent heat represents the amount of energy needed to change the state of a substance. The keyword is energy — in other words, the latent heat is just energy and the unit of energy is the Joule (J).
Therefore, the correct choice is B.
Understand why the other options are wrong!
A: the electromotive force is not a force! It is the amount of energy needed to drive a unit of charge around a complete circuit. So, its correct unit is J/C and not N.
C: pressure is defined as the force per unit area, as shown by the equation pressure = force / area. Using the equation, you can easily work out the unit of pressure. You need to divide the unit of force by the unit of area: that results in the unit N/m2. The unit shown in choice C is the unit of density.
D: weight is the gravitational force acting on an object. It is just a type of force, so its unit should simply be N. The unit shown in choice D is the unit of mass.
You need to identify what lowercase m and uppercase W represent
Lowercase m represents milli (10-3) and uppercase W represents watts. Therefore, mW represents milliwatt. The value of milli is 10-3.
Therefore, the correct choice is C.
Understand why the other options are wrong!
A: lowercase m does not represent mega. Additionally, mega represents 106, but its value is indicated as 10-3. So, there are two errors in this choice.
B: the indicated name and value are matching — mega does represent 106, as shown in this choice. However, the required prefix is milli (lowercase m) and not mega. The name and value do not represent mW, as asked in the question.
D: lowercase m is correctly identified as milli but its value is incorrect. Milli does not represent 106.
This problem relates to a concept known as estimation. You are required to estimate the mass of an adult person.
In estimation problems, you are “not” required to remember exact values. Instead, you only need to know the “approximate” values of quantities. In some cases, like this one, you just need to know the correct order of magnitude (the correct power of ten) — which means that you need to know whether the value is in the tens, hundreds, or thousands, etc.
The mass of a normal adult usually varies between 40 kg to 100 kg, depending on the gender and body type.
Therefore, the correct answer is B.
Understand why the other options are wrong!
A: the given value is way too low — the mass of a normal adult cannot be 7.5 kg, of course.
C: the given value of 750 kg is certainly too high — an adult does not have a mass of 750 kg, even in case of obesity.
D: similar to option C, the given value is way too high. It is not possible for an adult to have a mass of 7500 kg. In fact, that is even higher than the mass of a car.
For this question, you need to understand unit conversion.
Recall that 1 cm = 10 mm.
The question wants you to identify which one is given to “one tenth of a millimetre”. One-tenth of a millimetre means 0.1 mm.
Convert 0.1 mm to cm, as shown below:
This means that when a value (in cm), is written to the nearest tenth of a millimetre, it should have two decimal places. There is only one choice that shows a number with 2 decimal places.
Using the conversion between cm and mm, you should also note that one-tenth of a millimetre basically means one-hundredth of a cm. That is because any value in cm is always 10 times smaller than the same value in mm (1 mm = 1/10 cm).
The correct choice is B.
Understand why the other options are wrong!
A: the reading is given to one-tenth of a centimetre, not one-tenth of a millimetre.
C: the reading is even to one-thousandth (1000th) of a cm, and that is one-hundredth of a mm, not one-tenth.
D: same as the reading in option C — given to one-hundredth (100th) of a mm.

## Summary

The magnitude of a quantity indicates its size or extent. It has no unit and is only a number (always positive).
The absolute value of a number tells you how far that number is from the zero mark on the number line. In simple words, it just ignores the negative sign.
The absolute value is always positive. The absolute value is denoted by the | | symbol, where the number is written between the two vertical bars.
The absolute value of a quantity represents different meanings in different contexts. For example, it could represent speed in one case and the electric field strength in another. It is important to understand the context to see what the absolute value would represent in a given case.
The standard notation is a concise way of representing numbers that are generally very large or very small. It can also be used even if the number is not either too small or too large, but that is uncommon. The standard notation has two parts: one number and a power of 10. The number must be at least 1 and less than 10; and the power of 10 is a whole number.
Instead of powers of 10 (the standard notation), you can also use prefixes. They perform the same function. For example, 103 can be represented by kilo (k) whereas 10-9 can be represented by nano (n). It does not matter which one you use — prefix or power of 10. It is very important to differentiate between lowercase and uppercase letters: m and M do not represent the same value. For instance, the symbol ‘m’ stands for milli (10-3 whereas the symbol ‘M’ stands for 106.
Keep track of all units while reading and solving questions. You should underline or highlight the quantities so that you can easily extract the magnitude and unit of each quantity while attempting problems.
You should also note the unit in which the answer is required. It is very common to get to the right answer but still lose one or more marks because of incorrect units. You have to realize that if the expected answer is 5 m, and you write 5 cm, your answer would be 100 times smaller than the correct answer! Therefore, it should be obvious as to how important that piece of information is.
Convert all units to the required form before plugging values into an equation. Similarly, rearrange the equation before plugging in values — you should rearrange the equation so that the unknown quantity becomes the subject. You can then easily plug in all known values (with correct units) and calculate the answer.
Make sure you remember all important unit conversions!