If e gets smaller than there are more 9's. And then talk about what happens if e gets smaller. I think that your teacher should have said that 10-e = 9.9999.9 (that is the number of 9's end.
then 10 times x would equal 99.999., that is 10x = 99.999.īut if 9x = 90, that is 9 times some unknown number equals 90, then the unknown number, x, must be 10. If we say that the special number x = 9.999. exponents and eccentricities of the infinitesimal periodic orbits around. Just remember that when you multiply a number by 10 you move the decimal to the right 1 place. Here is a standard proof that a 4th grader should be able to understand. The idea is as e gets smaller and smaller and smaller then 9.999. I suspect that the answer to your question has something to do with the equality of the real part of two hyper-real numbers, but that is really not much more than a guess.Ĭlick to expand.1st I need to applaud you for finding this website, asking your question and thinking about this problem while only in the 4th grade. I know very little about non-standard analysis. What then do those mathematicians mean when they say So for those mathematicians, they will say your question is every bit as meaningless as whether unicorns' tails make the best dust mops. And, as I said, once you get over the initial weirdness, it makes certain branches of mathematics sort of accord with common sense.īut other mathematicians say that they do not want to deal with infinitesimals at all. That's weird, but the mathematics that arises from accepting that idea is called non-standard analysis. An infinitesimal, if you accept the idea at all, is a number that is not zero but sometimes acts like zero and sometimes does not act like zero.
Infinitesimals 0.999 pdf#
The idea of an infinitesimal makes some things much easier, but it is a super-slippery concept. This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999 and 1, even after they have seen and understood logical arguments for the 1 Highly Influenced PDF View 4 excerpts, cites background and methods Infinitesimal analysis without the Axiom of Choice K. The history of the idea of infinitesimals is one of tentative acceptance during the 17th and 18th centuries, rejection during the 19th and early 20th centuries, and grudging acceptance since 1962. In nonstandard analysis, there are positive infinitesimals, so consider 1-inf where inf is an infinitesimal. So my original answer shown below is almost certainly incomprehensible on a stand-alone basis. I did not initially notice that you were in 4th grade.
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