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# Trivial

The word trivial has several common uses:

1. Something that anyone can understand and explain to others — as opposed to something sublime, transcendental, etc., or simply difficult.
2. Of little significance or value.
3. Ordinary; commonplace; trifling; vulgar.
4. Concerned with or involving trivia.

## Mathematics

In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. For non-mathematicians, they are sometimes more difficult to visualize or understand than other, more complicated objects.

Examples include:

Also, trivial refers to solutions (to an equation) that have a very simple structure, but for the sake of completeness cannot be ignored. These solutions are called the trivial solution. For example, consider the differential equation

y' = y

where y = f(x) is a function whose derivative is y′. Then we have the trivial solution

y = 0, the zero function

and the nontrivial solution

y = ex, the exponential function.

In addition, mathematicians use trivial to refer to any easy case of a proof, which for the sake of completeness cannot be ignored.

## Etymology

The word trivial has a long history dating back to early Latin; its meaning has changed considerable over the millennia. It comes from the Latin words for a place where three roads meet (from the prefix tri-, "three," and via, "road.") Trivium thus meant "the meeting place of three roads, especially as a place of public resort." In the Roman empire, a trivium would often have a tavern (Latin: taverna).

In Roman times, such a place was viewed as common and vulgar, in the sense that we express in the phrase the gutter, as in "His manners were formed in the gutter." The Latin adjective triviālis, derived from trivium, thus meant "appropriate to the street corner, commonplace, vulgar."

Trivia appears to have initially entered Middle English from Anglo-Norman or Latin during the Norman occupation of England sometime after the 11th century AD. Its meaning at the time appears to have been very different from its most common modern one. The earliest known use of the word in English is in a work from 1432-1450 that mentions the "arte trivialle," a reference to the three liberal arts that made up the first three subjects taught in medieval universities, namely grammar, rhetoric, and logic; the remaining four liberal arts of the quadrivium, namely arithmetic, geometry, music, and astronomy, were more challenging. Hence, trivial in this sense would have been "of interest only to an undergraduate".

The first known usage of the word "trivial" in Modern English is from 1589; it was used with a sense identical to that of triviālis. Shortly after that trivial is recorded in the sense most familiar to us: "of little importance or significance."