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

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

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

Note: The dimensions on both sides of the equation must be balanced. In the equation above A and r2 both 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!

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

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!