Measurement, approximation and interval arithmetic (I) | Real numbers and limits Math Foundations 81

แชร์
ฝัง
  • เผยแพร่เมื่อ 11 ม.ค. 2025

ความคิดเห็น • 2

  • @KipIngram
    @KipIngram 2 ปีที่แล้ว +1

    15:40 - Totally agree with you; for a rigorously logical treatment we should differentiate between the natural numbers and the integers. In fact, though, you're absolutely right - they are "so close to being the same thing" that some significant hardware efficiencies can result from mingling them. In particular, the "twos complement" representation for integers, where the most signficant bit is zero for positive numbers and the magnitude of the number results from a "powers of 2" representation being given to the various bits, combined with the algorithm for negating a number (invert all bits and then add 1, treating the number as a natural number) is an extremely effective way to handle the situation. It makes the same hardware (for adding values) work regardless of sign.
    To make such a process "allowable" in a rigorous analysis would involve actually *proving* that the aforementioned steps actually produce the expected results. That would actually be a nice little exercise, once you'd established all of the necessary concepts.

    • @njwildberger
      @njwildberger  2 ปีที่แล้ว

      @KingIngram Yes I agree that we should incorporate more computational procedures into theoretical pure mathematics, especially at the level of numbers and data structures.