Not sure if this is the right place. The multiplicative inverse of an infinity is an infinitesimal and vice versa. Since positive infinitesimals are considered to be nonzero entities less than any positive real number the appropriate quantity for an multiplicative inverse would be an entity which is greater than any real number but not equal to infinity, a sort of infinitude. But infinitesimals are, by definition, uncountably infinite. The real numbers are Dedekind complete. They are part of an extension of the real numbers, just as the real numbers are an extension of the rational numbers, and the rational numbers are an extension of the … Given any real number , the number , but . During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, treat them as real numbers. This last statement is easy to prove. There is also no smallest positive real number! The Standard Part Theorem says all the limited hyperreals are clustered around real numbers. Infinitesimals: “Do the math” in a different dimension, and bring it back to the “standard” one (just like taking the real part of a complex number; you take the “standard” part of a hyperreal number … Even though no such quantity can exist in the real number system, many early attempts to justify calculus were based on sometimes dubious reasoning about infinitesimals: derivatives were defined as ultimate ratios One exception is a recent reconstruction of infinitesimals — positive “numbers” smaller than every real number — devised by the logician Abraham Robinson and … So if there are no such real numbers… Infinitesimals are the reciprocal of Infinite numbers, and as such, have an absolute value which is smaller than that of any Real number (except 0, which is considered to be Infinitesimal): . Any number which is not Infinite is called a Finite number– including the Infinitesimals. They are not countable. A hyperreal number consists of a real number and a halo of infinitesimals. For instance, consider we want to integrate the expression $3x(x^4+1)^3$. That's the whole point of the real numbers forming a continuum, it has a cardinality larger than the set of any countable number or sequence of numbers. Using Infinitesimals. Sometimes we think of this result as saying the real numbers are the points on a line with no gaps. Question from a couple of my students that I couldn't answer and can't find using google. Infinity and infinitesimals are both undefined quantities. There are also infinities of the nature presented previously as unlimited numbers. Infinitesimal, in mathematics, a quantity less than any finite quantity yet not zero. Infinitesimals are not real numbers, and therefore don't live on the real number line in the first place. (To be continued.) 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