Simple Examples
The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive regardless of how we count the objects in the set. The quantity – a cardinal number – is associated with the set and is invariant under the process of counting.
An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.
Another simple example of invariance is that the distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand multiplication does not have this property, so distance is not invariant under multiplication.
Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. All circles are similar. Therefore they can be transformed into each other and the ratio of the circumference to the diameter is invariant and equal to pi.
Read more about this topic: Invariant (mathematics)
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