trichotomy law

The trichotomy law, in mathematics, states that for any two comparable quantities, exactly one of three possibilities is true: one is greater than the other, one is less than the other, or they are equal. This principle is fundamental in arithmetic and forms the basis for ordering numbers. For instance, when comparing two numbers, a and b, either a > b, a < b, or a = b.1

This concept extends beyond simple arithmetic. In Euclidean geometry, the triangle inequality is a related principle, stating that the sum of any two sides of a triangle must be greater than or equal to the third side (a + b \u2265 c). This illustrates that the shortest distance between two points is a straight line. The trichotomy law also appears in more abstract mathematical spaces, such as with absolute values of real numbers or norms of vectors.2