Considering two observer, one inside a spaceship traveling from the earth to a near by start; and the other standing over the surface of the earth. In the drawing above, it is shown that the space ship will use a single clock to measure the events described below. The drawing also shows a fictitious "rule" use by the observer on the Earth to measured the distance from the Earth to the start. However, this observer will need a sequence of synchronized clocks to determine the time for the different events on the journey of the space ship. Accordingly with the number of clocks need to do the physical measurements, the time measured by the spaceship is a proper time while the time measured by an observer on the Earth is an improper time.
For distance starts, the Earth/Sun system is at rest, in other words, the distance Earth-Start is fixed. Therefore, this distance can be measured with a "rule" like the one shown on the diagram. Distances and lengths that can be measured with "rules" are called Proper Lengths.
In the case presented in the above drawing, the observer on the Earth will measure a time for the trip given by the regular formula relating distances, velocities and time. For the Earth observer, the spaceship is moving at a constant velocity v. Thus, the time taken by the spaceship is . Where LE is the length measured by the Earth with the "rule". Notice that based on the previous definitions of proper time and improper as well as proper length, the observer on the Earth measures an Improper Time and a Proper Length.
How can the observer on the spaceship measure the distance from the Earth to the start? Knowing that for this observer the Earth is moving with a velocity v but to the left, see drawing above, the observer on the spaceship can measure the distance from the Earth to the start by just measuring the time it will take since it saws the Earth under the spaceship and the time when the start is under the spaceship. This method of measuring lengths or distances is called Improper Lengths.
Again, the time measured by the spaceship requires only one clock. Therefore, this time is a proper time. Thus, the distance between the Earth and the start, as measured by the spaceship, is .
The relation between the times measured by the Earth observer and the spaceship observer is the relation between proper times and improper times. In this case, the time measured by the spaceship is the proper time and the time measured by the Earth is the improper time, , with and . Replacing those relations, a formula relating the length measured by the Earth and the length measured by the spaceship can be obtained,
Since the distance measured by the spaceship is an Improper Length and the length measured by the Earth is a Proper Length, the general relation between proper and improper lengths is
It is important to remark that there is not such a thing as a proper frame of reference or an improper frame of reference. Both frame of reference may measure a proper and an improper quantity, it just depends of what they are measuring. In the present situation the following table shows the quantities that each observer is measuring,