Trivial (mathematics)

In mathematics, a mathematical object, solution, or proof is called trivial if it is considered to be obvious or extremely simple, usually not needing a detailed proof or explanation in context. Trivial objects often involve the numbers 0 and 1.

Whether or not something is trivial depends on the context. $x=-2$ is not a trivial solution to the trigonometric equation

$\sin {\frac {\pi x}{2}}=0$,

but solving this equation is significantly easier than finding other zeroes of the complex-valued Riemann zeta function, so its solutions (including -2) are considered trivial zeroes and ignored in the context of the Riemann hypothesis.

Examples

In arithmetic and number theory, the trivial factors of an integer are itself and 1.

Many types of problem have trivial solutions. In differential equations, the most common trivial solution is the zero function; for example, the differential equation

${\frac {dy}{dx}}=y$

has the trivial solution $y=0$, contrasting the general family of solutions $y=Ce^{x}$.

In group theory, the group containing only the identity element 0 is called the trivial group. In ring theory, the trivial ring is likewise a ring containing only 0, which is both the additive and multiplicative identity (as both $0+0=0$ and $0\times 0=0$). If the trivial ring is considered a field, it is called the trivial field, but most definitions of field exclude it by requiring that $0\neq 1$.

See also

• Tautology (logic)
• Degenerate (mathematics)