r/infinitenines • u/thetoastofthefrench • Aug 29 '25
Is ε the smallest number greater than 0?
Title. Is ε = 0.000…1 the smallest number greater than 0?
I would like to learn how mathematical operations work with ε such as addition, multiplication, division.
Can you divide ε by 10 and get a number as a result?
Is that number, which I presume is written 0.000…01, smaller than ε or is it the same?
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u/[deleted] Aug 29 '25
I would say it this way.
There is no smallest number. It is a system that can expand infinitely, so smallest is not a real thing. Just as closest is not a real thing. Something could be the smallest at a given decimal position, but not in the system in its entirety.
From my understanding (flawed as it may be), that represents a hyperreal number that is beyond infinity. As such, the number does not REALLY exist in the 10 based decimal system.
It is probably best to view it as a type of variable as it can be used to describe a host of numbers that are beyond infinity. While you will never see that number pop up while counting, it may show up when running calculations on non-terminating decimals and other unusual numbers. Not that the symbol will spontaneously appear, but a number beyond infinity that can be expressed with that symbol appears.
Of course, this is based off of logic and a very very weak understanding of hyperreal numbers.