The distance between two real numbers \(a\) and \(b\) is defined as \(|a-b|\) in the real number system \(\mathbb{R}\). This definition reflects the intrinsic property of distance, which remains unchanged regardless of how the interval is partitioned. The discussion highlights that while there are infinitely many numbers between any two distinct real numbers, counting these does not provide a valid measure of distance. The example of measuring distances between integers and real numbers illustrates the difference between counting elements and measuring intervals.
PREREQUISITESMathematicians, educators, students studying real analysis, and anyone interested in the foundational concepts of distance and measurement in mathematics.
Completely agree.PeterDonis said:All you're doing is continuing to say the same thing over and over again in different words. You aren't adding any new understanding.