Lattiaarvojen

The term lattiaarvojen is the Finnish plural form of lattiaarvo, which refers to the floor function in mathematics. The floor function maps a real number to the greatest integer less than or equal to that number, and is usually denoted by the symbols ⌊x⌋. For a real number x, the floor value ⌊x⌋ is defined by the condition ⌊x⌋ ≤ x < ⌊x⌋ + 1. In other words, the floor function “rounds down” a real number to the nearest integer not exceeding the original value. For example, ⌊3.7⌋ equals 3, ⌊-1.2⌋ equals -2, and ⌊5⌋ equals 5.

Lattiaarvojen has many useful properties that are frequently employed in number theory, analysis, and computer science. It is a stepwise function that is constant on each unit‑length interval of real numbers and discontinuous at integer points. The floor function is related to the ceiling function ceil(x) = –⌊–x⌋, and together they provide a pair of complementary rounding operations. The identity ⌊x + n⌋ = ⌊x⌋ + n holds for any integer n, confirming that the floor function behaves linearly with respect to integer translations. The fractional part of a real number x is given by {x} = x – ⌊x⌋, which lies in the interval [0,1).

In algorithmic applications, lattiaarvojen is used for integer division, determining array indices, and discretizing continuous domains. In number theory, it underlies the definition of the fractional part, Dedekind sums, and the Beatty sequence, and appears in formulas for counting lattice points within geometric shapes. The function is also central to modular arithmetic, because the difference x – ⌊x⌋ is congruent to x modulo 1. Overall, lattiaarvojen is an elementary yet powerful tool with broad applicability across mathematics and related disciplines.