Number theory
While the word "algorithm" goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in "Elements", together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation formula_78,
or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called
"kuṭṭaka" ("pulveriser"), without a proof of correctness.
Julie Beth Lovins
A stemming algorithm is the process of taking a word with suffixes and reducing it to its root, or base word. Stemming algorithms are used to improve the accuracy in information retrieval and in domain analysis. These algorithms help find variants of the terms being looked up. Stemming algorithms are helpful because it is likely that users who look up a particular word might be interested in related words. Stemming algorithms are prevalent in search engines, such as Google's, which did not implement word stemming until 2003. This means that up until 2003, a Google search for the word "warm" would not have returned related words like "warmth" or "warming".