You are watching: What is one divided by infinity
I know that $0.999 cdots$ (repeating) = $1$. For this reason wouldn"t $1 - frac1infty = 1$ as well? due to the fact that $frac1infty $ would certainly be infinitely close come $0$, maybe as $1^-infty$?
So $1 - 1^-infty$, or $frac1infty$ would certainly be equivalent to $0.999 cdots$? Or am I lacking something? Is infinity something that can even be supplied in this type of chathamtownfc.netematics?
you can say that $frac1infty = 0$, therefore $1-frac1infty = 1$. But then, you"re extending the definition of division past breaking suggest - division as you recognize it isn"t identified for infinity, for this reason the prize is undefined. Otherwise you can easily get yourself into a pickle and end up saying 1=2.
Arithmetic operator - add, subtract, divide, multipy, raise to the power of - are characterized on a particular collection of numbers: such as genuine numbers, or complicated numbers.
The set you use for definition, will determine what girlfriend can and also can"t speak meaningfully. Commonly (but not always), infinity is excluded from that set.
If we take the collection of real numbers, and look in ~ "raise come the strength of", then $1^x$ is equal to 1 for any x, as x -> infinity. Therefore in the case, you might have a convention of saying the $1^infty = 1$ . Yet $frac11 = 1$, for this reason $1^-infty$ would likewise equal 1. However, once you go around defining these brand-new conventions, you have to be extremely careful - sometimes, a convention will certainly seem obvious, yet if girlfriend run v it, you end up seeming to prove 1=2, which means that your convention wasn"t that helpful.
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Let"s to compare with increasing to the strength 0.5, i.e. Acquisition the square root. $-1^0.5$ is undefined when we room working top top the reals - so, simply as separating by infinity, girlfriend can"t incorporate it in her arithmetic. Only as soon as you increase to the complicated numbers, and extend your definition of the arithmetic operator to cope, can you to speak something meaningful around $-1^0.5$
Similarly, the reals and the facility numbers every exclude infinity, so arithmetic isn"t defined for it.
You can extend those to adjust to include infinity - yet then you have actually to extend the definition of the arithmetic operators, come cope through that expanded set. And then, you need to start thinking about arithmetic differently. If you want to learn much more about that, then there are several friendly places on the internet to gain into the occupational of Cantor on the different types of infinity. (of i beg your pardon there are an infinite number of different infinities)