1.02 Dimensionless numbers

Most of the numbers you will use in scientific calculations are dimensions. e.g. you may enter values for the distance travelled by an object and the time taken to travel this distance into an equation in order to determine the average speed. However equations can also contain what are known as dimensionless numbers.

There are two types of dimensionless numbers
Numerical constants: these are values without any corresponding units of measurement.
Dimensionless units: these are numbers which represent a measureable value but the units of measurement do not have any equivalent base dimensions

Numerical constants

Example
The equation for energy of a moving object is E = ½ mv2

E represents the objects Energy (kinetic energy) and is measured in units called Joules
m represents the objects mass and is measured in Kilograms
v² represents the objects velocity squared and is measured in (metres per sec)²
but ½ is just a numerical constant, i.e. it does not represent any dimension and has no units associated with it.

Other examples include
Π in the equation for the area of a circle A = Π r²

Area (A) has units of metre squared
radius (r) has units of metres (so r² has units of metres squared)
Π has no units of measurement it is just a numerical constant.

Note: The dimensions on both sides of the equation must be balanced. In the equation above A and r²both have units of m² so it is clear that PI must have no units if the dimensions are to balance.

Dimensionless units

The numbers used to quantify the size of angles are also dimensionless numbers. However angular measurements do have units of measurement (i.e. radians). The radian however is what is known as a dimensionless unit. We can illustrate this by showing how the radian is derived from the base dimension of length.

The equation for calculating an angle in radians is:
Angle in radians = arclength / radius
As previously stated the dimensions on either side of an equation must be the equal!

Therefore
Dimensions of (Angle in radians) = Dimensions of (arclength / radius)
But the dimensions of the arclength and radius are both length! (L)

Therefore
Dimensions of Angle in radians = L/L
So the dimensions on the right hand side are (L/L) which cancels out leaving the right hand side dimensionless.
Therefore radians must also be dimensionless units!