Exploring Interval Union Arithmetic with an Innovative Calculator

Welcome to a unique calculator that operates over unions of intervals rather than just real numbers, implementing the advanced concept of Interval Union Arithmetic. In this system, an interval [a, b] includes all numbers between a and b, and a union like [a, b] U [c, d] represents a disjoint set of intervals. This arithmetic is superior to regular interval arithmetic because it remains closed even when dividing by intervals containing zero, such as 2 / [-2, 1], which results in [-∞, -1] U [2, +∞].

The inclusion property is a fascinating aspect of interval union arithmetic. It ensures that if you select any real number from each input union and compute the expression over the reals, the result will always be within the output union. This feature is particularly useful for representing uncertainty, for example, 50 * (10 + [-1, 1]) results in [450, 550].

You can perform complex calculations using the interval union operator U. For instance, ([5, 10] U [15, 16]) / [10, 100] yields [0.05, 1.6]. Operations can also lead to disjoint unions of intervals, such as 1 / [-2, 1] resulting in [-∞, -0.5] U [1, +∞].

In full precision mode, the calculator functions like a regular calculator but provides interval results that account for floating point precision issues. For example, 0.1 + 0.2 results in [0.29999999999999993, 0.3000000000000001], ensuring the true value is contained within the interval.

The calculator supports various functions and constants, allowing for a wide range of mathematical operations. You can input intervals using bracket syntax or bare numbers, which are interpreted as narrow intervals. This flexibility allows for natural mixing of numbers and intervals, such as 1.55 + [-0.002, 0.002] resulting in [1.548, 1.552].

Future enhancements include splitting full precision mode into two controls, adding an 'ans' variable for previous results, and supporting the input of empty unions. The calculator and its engine, not-so-float, are open-source, inviting contributions and sponsorships on GitHub.