numeriline

Numeriline is a conceptual construct used in mathematics and data visualization to represent sequences of numbers along a continuous line. In its simplest form, a numeriline refers to the polyline formed by plotting the pairs (n, a_n) for a sequence (a_n) of real values and joining consecutive points with straight segments.

A related notion is the continuous numeriline, obtained by interpolating the discrete sequence to a continuous function f: R -> R that agrees with a_n at integer arguments. Methods include linear interpolation and spline interpolation, yielding curves that approximate the sequence while enabling calculus-based analysis.

Properties depend on the interpolation: the discrete polyline is piecewise linear with corners at the integer indices, while the interpolated version can be smooth, differentiable, or even analytic depending on the method. The choice affects visual interpretation and analytic use, such as detecting trends or oscillations.

Applications include visualization of time series, exploration of convergence and monotonicity, and serving as a bridge between discrete data and continuous analysis. Numerilines can also be used pedagogically to illustrate how sequences behave and to compare different sequences on a common index axis.

See also: number line, polyline, graph of a sequence, interpolation, time series visualization.