Units and Dimensions

Every measurement we take of any value whether it is distance, weight, length, speed, etc. The measurement contains two parts, one is the value part and another is the unit that describes the value. Such as if we say 20 cm. What it tells us, is that we know from our knowledge that a cm is a unit of length so, by saying 20 cm it tells us that 20 is the value of the length and the length is in cm which is 20 cm of length. The number is also called the magnitude of the physical quantity and speaking about dimensions, it has two meanings. Dealing with units and dimensions it is simply the smallest unit of any larger unit. We’ll see both units and dimensions in brief.

What are Units and Dimensions?

Units

Unit’s definition says it is a particular quantity to measure or tell about a magnitude defined by law or convention and which is used as a standard of measurement. It also says that a unit can be converted to any other quantity by the multiple of some values defined by the International System of Units. 

A unit can also be referred to as quantities.

Dimensions

A dimension can be defined as the power to which any physical quantity is being raised to obtain one single unit of that quantity. 

Fundamental/Base and Derived Quantities

Units that are not dependent on any other quantities are called fundamental quantities. Such as meter, kilogram, kelvin, second, etc. 

And units that are dependent on other quantities such as meter per second (m/s), kilogram per cubic meter, etc are called derived quantities.

Fundamental Units/quantities can be further divided into their system of units. Namely, CGS, MKS and FPS. In the CGS system, the length is measured in cms, mass in grams and time in seconds and in the MKS system, length is measured in meters, mass in kilograms and time in seconds and in the last one, length is measured in the foot, mass in the pound and time in seconds. The names of these systems are the short forms of their unit of measurements, like, CGS – Centimeter-gram-second, MKS – Meter-Kilogram-Second and FPS -Foot-Pound-Second.

Some important physical quantities (MKS system) and their units

Speaking about dimensions, it can be better understood by,

Dimensional Formula

It is an expression that represents a relationship between the derived quantities and their fundamental quantities with its dimensional values, such as Density (D) = ML3T0. Every dimensional formula is written in a specific syntax, that is,

Q = MxLyTz

Where,

Q = Derived Quantity,

M = Mass

L = Length

T = Time

x/y/z = Dimensional Values

Why only MLT? Because every derived formula has been derived from either mass or/and length or/and time. 

Let us understand this formula with some examples,

Energy = ML2T-2 which implies that energy is measured in Joules and taking it to its fundamental forms. 1 Joule is equal to 1Kg square meter per square second (kg m2 s-2). 

Similarly, Pressure (P) = ML-1T-2, which is represented in its base form as kg m-1 s-2. 

Other than that, there are some dimensionless quantities. Dimensionless quantities are those quantities that do not have any dimension but contain some particular value of their own. Those quantities may contain units and may not. 

Some examples are

Dimension less with units: Atomic Mass (ATM), Angular Displacement (radian), Abbe’s Number, Archimedes Number, Darcy number, etc. 

Dimensionless quantities without units: Trigonometric values (cos , sin , tan , etc), pie, pure numbers, etc. 

Conclusion

Units and Dimensions are very important parts of any quantity. It gives any quantity or number its value or its worthiness. Imagine if we just say length as 10 or 7. It would not give any meaning to that value, but if those are said as 10 m or 7cm. It would give some meaning to the value of those quantities. 

Similarly, dimensions of units are also important as they determine the power of fundamental/base quantities involved in that particular derived quantity. A derived quantity is represented as the power of base quantities.