I know the Arithmetic proof and everything but how to explain this practically to a kid who just started understanding the numbers?
In: 3202
The meaning of 0.999… depends on our assumptions about how numbers behave. A common assumption is that numbers cannot be “infinitely close” together. With these rules, 0.999… = 1 since we don’t have a way to represent the difference. If we allow the idea of “infinitely close numbers,” then yes, 0.999… can be less than 1. Those numbers would be infinitesimals.
Infinitesimals are quantities that are closer to zero than any standard real number but are not zero. They do not exist in the standard real number system but can exist in other number systems such as the surreal number system and the hyperreal number system. Infinitesimals were introduced in the development of calculus, where the derivative was first conceived as a ratio of two infinitesimal quantities. However, as calculus developed further, infinitesimals were replaced by limits, which can be calculated using standard real numbers.
tldr: 0.999… both does and does not equal 1 depending on how you evaluate the expression. It’s a neat thought experiment but in most any real world application you would place reasonable limits to avoid the complexities of infinity.
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