Fracp1

Fracp1 is a parametric generalization of the fractional-part operator used in mathematics to extract the remainder of a real number when divided by a positive parameter p1. It yields a value in the interval [0, p1), and for p1 = 1 it reduces to the ordinary fractional part.

Definition: for a real input x and a positive parameter p1, fracp1(x) is defined as fracp1(x) = x − p1 · floor(x / p1).

Properties: fracp1 is defined on all real x; its range is [0, p1); it is periodic with period p1; on each interval [kp1, (k+1)p1) it is linear with slope 1; it is non-decreasing over the real line.

Examples: with x = 3.7 and p1 = 1, fracp1(3.7) = 0.7. With x = 3.7 and p1 = 0.5, fracp1(3.7) = 0.2.

Applications: fracp1 can be used for phase wrapping in digital signal processing, tiling and texture generation in computer graphics, and in classroom demonstrations of modulo arithmetic and torus-valued dynamics.

History and notation: fracp1 is a generalized construct found in some instructional materials and software libraries as a variable-scale modulo operator. It is not a widely standardized term in formal literature.

See also:

- fractional part

- modulo operation

- phase wrapping

- torus

- periodic functions