The meaning of 0.999… is a tricky concept, and depends on what we allow a number to be. Math can be about questioning assumptions, pushing boundaries, and wondering “What if?”. If we allow the idea of “infinitely close numbers”, then yes, 0.999… can be less than 1.With these rules, 0.999… = 1 since we don’t have a way to represent the difference. A common assumption is that numbers cannot be “infinitely close” together - they’re either the same, or they’re not.The meaning of 0.999… depends on our assumptions about how numbers behave.It’s a rare thing when people wonder about math: let’s use the opportunity! Instead of bluntly offering technical definitions to satisfy some need for rigor, let’s allow ourselves to explore the question. No, there’s a broader, more interesting subtext: What happens when one number gets infinitely close to another? #Positive and negative infinitesimals seriesWhen people ask about 0.999… they aren’t saying “Hey, could you find the limit of a convergent series under the axioms of the real number system?” (Really? Yes, Really!) “You’re in a balloon!” he hears as he drifts off into the distance. “Where am I?” he shouts desperately through the wind. He sees a lighthouse approaching in the fog. A famous joke illustrates my point:Ī man is lost at sea in a hot air balloon. 999… = 1? The question invites the curiosity of students and the ire of pedants.
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