>>315>>316So to answer the question of whether there's a number system with a number ε such that
¥ 0 < ε < 1¥ 0 < ε < 0.1¥ 0 < ε < 0.01¥ 0 < ε < 0.001,and so on, the answer is yes, there are many. There are the hyperreal numbers which are useful as an alternate way to do calculus. There are the surreal numbers which arose out of game theory. And you can even just postulate such a number ε and add it to the reals sort of like how we add i to make the complex numbers. In the case of the complex numbers, you get the complex numbers from all the linear functions of i. If you add ε, your new set of numbers consists of all the rational functions (polynomial divided by polynomial) of ε.
But we can see that a nonzero infinitesimal number like ε will cause trouble if we want our numbers to be expressed by infinite decimals. The whole idea behind infinite decimals is that we can identify a number by comparing it with finite decimals. If we allow ε into our system, then we can no longer distinguish ε from 2ε, or 0.4 ÷ 0.03 from 0.4 ÷ 0.03 + ε, or √2 from √2 + ε by comparing them with finite decimals. We would need something else. For example, in the surreal numbers, ε is the simplest number between {0} and {1, 1/2, 1/4, 1/8, ...}, and 2ε is the simplest number between {ε} and {1, 1/2, 1/4, 1/8, ...}.
Another definition:
¥ Two numbers are infinitesimally close if the difference between them is infinitesimal.There are many equivalent ways to construct the real numbers, but all of them are designed so that we never construct a real number infinitesimally close to another real number. I don't think anyone knows whether it's possible for two real-life quantities (such as distances, times, weights, and volumes) to be infinitesimally close to each other yet not equal. If it is possible, then when we model these infinitesimally close quantities with real numbers, we are choosing to ignore any infinitesimal difference and assign the same real number to both.
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