Physical Quantities, Units, and Dimensional Analysis

What is a Physical Quantity

A physical quantity is a characteristic or property of an object that can be measured. It is a fundamental concept in science and engineering that allows us to describe and understand the world around us in a systematic way.

The measurement of a physical quantity is quantified by a magnitude and a unit. For example, the physical quantity of length can be quantified as 10 meters, where 10 is the magnitude and meter is the unit.

Base and Derived quantities

Physical quantities are divided into two categories: Base quantities and Derived quantities.

  • Base quantities are physical quantities that can be measured using different instruments and measuring tools, and they form the foundation or base for other physical quantities. The units used to measure base quantities are called base units. These quantities are Length, Mass, Time, Electric Current, Temperature, Amount of a Substance, and Luminous Intensity.
  • Derived quantities are all other physical quantities that can be calculated or “derived” from base quantities, like speed, area, and volume. The units used to measure derived quantities are called derived units.

Units and Measurements

Units are standardized values used to measure and express physical quantities. They provide a common language for scientists, engineers, and individuals to communicate measurements effectively.

There are two main systems of units in use today: the International System of Units (SI), and the English system, also known as the customary or imperial system. The English system is mainly used in the United States, while the SI system is practically used everywhere else in the world, and it’s also the standard system agreed upon by scientists and mathematicians.

SI Base Units
QuantityUnitSymbol
LengthMeterm
MassKilogramkg
TimeSeconds
Electric currentAmperea
TemperatureKelvink
Amount of substanceMolemol
Luminous intensityCandelacd

SI Unit Prefixes

An SI unit prefix is a prefix that precedes a basic unit of measure to form a multiple or subdivision of the unit. Each prefix has a unique symbol that can be prepended to any unit symbol. For example, the prefix kilo can be added to the unit meter to express a multiple of 1000, thus one kilometer is equal to 1000 meters. Similarly, the prefix milli can be added to the unit second to express a division by one thousand, thus one millisecond is equal to one-thousandth of a second.

SI Unit Prefixes
NameSymbolFactorEnglish Word
quettaQ1030nonillion
ronnaR1027octillion
yottaY1024septillion
zettaZ1021sextillion
exaE1018quintillion
petaP1015quadrillion
teraT1012trillion
gigaG109billion
megaM106million
kilok103thousand
hectoh102hundred
decada101ten
Smaller quantities or sub units
decid10-1tenth
centic10-2hundredth
millim10-3Thousandth
microμ10-6millionth
nanon10-9billionth
picop10-12trillionth
femtof10-15quadrillionth
attoa10-18quintillionth
zeptoz10-21sextillionth
yoctoy10-24septillionth
rontor10-27octillionth
quectoq10-30nonillionth

Scientific Notation

Imagine if you wanted to count the number of cells in a human body, which is a very large number, approximately 35,000,000,000,000 and if you were to measure the mass of one of those cells, you would get a very small number, approximately 0.000000000001 kg. You can see that writing very big or very small numbers is not super convenient, and they are also hard to evaluate and compare because of all the zeros. That is where scientific notation can come in handy, instead of using a very long string of digits to represent numbers, scientific notation provides us with a shorter and easier way to represent the same numbers.

Scientific notation form
Scientific notation form - Note (1 ≤ a < 10)

The way we write numbers in scientific notation is pretty straightforward. First, we rewrite only the digits with the decimal point placed after the first digit, then we multiply it by base 10, raised to a power that indicates the number of places to move the decimal point to give the number in long form.

Scientific Notation Example 1.1

The distance from Earth to the sun is approximately 150,000,000 km. Rewrite that number in scientific notation.

Step 1) We write only the digits with the decimal point placed after the first digit followed by x10 raised to an n power as:

1.5 x10n

Step 2) We count the number of all the digits we had after the decimal point to get the base 10 exponent n:

1.5 x108 km

And we are all done, but let's just verify that 1.5 x108 is indeed equal to 150,000,000:

108 = 100,000,000 1.5 x 100,000,000 = 150,000,000

Scientific Notation Example 1.2

The mass of a dust particle is approximately 0.000000000753 kg. Rewrite that number in scientific notation.

Step 1) Following the same process, we first write down only the digits with the decimal point placed after the first digit times base 10, raised to an n power:

7.53 x10n

Step 2) Since the number is less than 1, this time we count the number of all the zeros preceding our first digit, to get the base 10 exponent n:

7.53 x10-10 kg

Now, let's verify that 7.53 x10-10 is equal to 0.000000000753:

Scientific notation example verifying that 10 to the power of -10 is equal to 1 divided by 10,000,000,000

Scientific notation example verifying that 7.53 times 1 divided by 10,000,000,000 is equal to 0.000000000753

A positive exponent means the decimal point will be moved to the right, and a negative exponent means the decimal point will be moved to the left.

Dimensional Analysis

Dimensional analysis refers to the process of converting one unit of measure to another unit of measure, using what we call a conversion factor or conversion ratio. The conversion factor describes the relation between two equivalent values. For example, there are 60 minutes in one hour, thus the conversion factor between hours and minutes is 1 hour : 60 minutes.

Diagram showing the algebraic concept of dimensional analysis
Dimensional analysis concept

The idea behind dimensional analysis is to create a conversion factor or a fraction relating the units we want to convert from to the units we want to convert to, in such a way that when we multiply the units we have or want to convert from by that conversion factor, the units cancel out and we are only left with the units we want.

Dimensional Analysis Example 2.1

Convert 5 inches to centimeters, knowing that the conversion factor from inches to centimeters is 1 in : 2.54 cm.

First, we need to identify and write the value and unit we've or want to convert from, which in this case is 5 in, and then multiply it by the conversion factor, where the unit we want to convert to is in the numerator and the unit we want to convert from is in the denominator:

Dimensional analysis example showing how the units cancel out in 5 inches times 2.54 centimeters divided by 1 inch

Notice that the inches cancel out, and we are left with only the centimeters.

5 x 2.54 cm = 12.7 cm

Dimensional Analysis Example 2.2

If a car is traveling at a speed of 50 km per hour, What is the speed of the car in meters per second? Given the following conversion factors:

1 km : 1000 m 1 hr : 60 min 1 min : 60 sec

First, we will start by writing down the value and units we want to convert from, which is 50 km/hr, and then we multiply it by as many conversion factors as needed to get to the unit we want to convert it to:

Dimensional analysis example showing how the units cancel out in 50 kilometers per hour times 1000 meters divided by 1 kilometer times 1 hour divided by 60 minutes times 1 minute divided by 60 seconds

Now, by canceling out the units in the numerator and denominator, we get the speed of the car in meters per second: 13.89 m/s

What are physical quantities?
Physical quantities are the characteristics or properties of an object that can be measured or calculated from other measured quantities.
What are units?
Units are standardized values used to express and compare the measurements of physical quantities.
What is units conversion?
Units conversion is the process of converting between different units of measurement for the same quantity. For example, time can be expressed in terms of seconds or minutes.
What are the fundamental physical quantities?
The fundamental or base physical quantities are Length, Mass, Time, Electric Current, Temperature, Amount of a Substance, and Luminous Intensity.
What is a base unit?
It's a unit defined by measurement and not in terms of other units. The base units in the SI system are meter, kilogram, second, ampere, kelvin, mole, and candela.
What is a derived unit?
It's a unit of measurement derived from the base units. For example, the SI unit of speed is defined as meters per second.
What is a conversion factor?
A conversion factor describes the relation between two equivalent values, and it's used to convert between them. For example, there are 60 minutes in one hour, thus the conversion factor between hours and minutes is 1 hr : 60 min.
Can scientific notation be negative?
Yes, scientific notation is a cleaner way to write down very large or small numbers, and it does not matter if the number is positive or negative. For example 23,000,000 can be rewritten in scientific notation as 2.3 x107, and -34,000,000,000 as -3.4 x1010
Related Pages
References: 1) National Institute of Standards and Technology (NIST). “SI Units”. https://www.nist.gov/pml/owm/metric-si/si-units. 2) Paul Peter Urone. “College Physics. Physical Quantities and Units” OpenStax, 2012. June 21. https://openstax.org/books/college-physics/pages/1-2-physical-quantities-and-units.