mnvr / mrmr.io Goto Github PK

In

and the following paragraphs you stated that integers are not real numbers; that they are "comparable" but not "equatable". This is incorrect. Every integer is a rational number, and every rational number is a real number. Mathematically speaking 1.0 and 1 are exactly the same number. You are correct that the integers make up an infinitesimal portion of the real numbers (the random selection example) but this could be said of most subsets of the real numbers. For example, the odds of randomly selecting a number where no digit is 7 (in your choice of base ≥ 8) from the set of real numbers is zero; it will never happen. Yet there are infinitely many numbers meeting that criteria, and your inability to randomly select them from the set of real numbers does not mean they are not real numbers.

Floating point numbers are an extended subset of the rational numbers (they cannot represent non-rational real numbers) with power-of-two denominators within a fixed range, augmented with negative zero, positive and negative infinity, and various forms of NaN. Because they are a subset many operations on rational numbers can only give approximate results, but double-precision floating point math can at least give exact answers for addition, subtraction, and multiplication of all 32-bit integer operands for which the corresponding integer operation would not overflow as the significand is more than 32 bits. This is how Lua and JavaScript can get by with using only floating point numbers. (The same does not hold for 64 bit integers, however.)