Ellipsis - in Mathematical Notation

In Mathematical Notation

An ellipsis is also often used in mathematics to mean "and so forth". In a list, between commas, or following a comma, a normal ellipsis is used, as in:

To indicate the omission of values in a repeated operation, an ellipsis raised to the center of the line is used between two operation symbols or following the last operation symbol, as in:

(though sometimes, for example, in Russian mathematical texts, normal, non-raised, ellipses are used even in repeated summations).

The latter formula means the sum of all natural numbers from 1 to 100. However, it is not a formally defined mathematical symbol. Repeated summations or products may similarly be denoted using capital sigma and capital pi notation, respectively:

(see factorial)

Normally dots should be used only where the pattern to be followed is clear, the exception being to show the indefinite continuation of an irrational number such as:

Sometimes, it is useful to display a formula compactly, for example:

Another example is the set of zeros of the cosine function.

There are many related uses of the ellipsis in set notation.

The diagonal and vertical forms of the ellipsis are particularly useful for showing missing terms in matrices, such as the size-n identity matrix

The use of ellipses in mathematical proofs is often discouraged because of the potential for ambiguity. For this reason, and because the ellipsis supports no systematic rules for symbolic calculation, in recent years some authors have recommended avoiding its use in mathematics altogether.