Equality, Equivalence, and Order

The word 'equality' has been and still is a highly charged word in the political arenas of both the past and the present. It has conjured up a lot of different meanings to a lot of different people. In the late eighteenth century, for example, the French fought a very bloody revolution over passionately held beliefs about its true meaning in the context of that time. Even in the pure and cool abstraction of the mathematical world, the word 'equality' is so keenly important that it is precisely defined in every context where it may occur. Mathematicians demand exactness when it comes to true equality whereas the looser terms of equivalence or approximation are only tolerated in cases where sameness of visual identity or exactness of value is either evidently not true or simply too pedantic.

In as much as 'equality' demands mathematical exactitude, it should not be surprising that the same is true for the word 'inequality'. The latter has given rise to whole systems of order from the linear - the less than and greater than - ordering of the real numbers to the sequential ordering of polynomials by degree, of derivatives by repetitive differentiation processes, and of differential equations inherited from their derivatives.

These tutorials will introduce the concepts of mathematical equality, equivalence and order in the various number systems and will highlight how, even in pure mathematics, they may differ slightly from one to the other.

Last Updated by SAS October 07, 2026, 16:00:45