But infinitesimals are, by definition, uncountably infinite. Moreover, if , then , , etc… can be “much smaller than” itself (by many “orders of magnitude”). 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 (To be continued.) There are also infinities of the nature presented previously as unlimited numbers. Using Infinitesimals. Any number which is not Infinite is called a Finite number– including the Infinitesimals. Infinity and infinitesimals are both undefined quantities. Sometimes we think of this result as saying the real numbers are the points on a line with no gaps. 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. Infinitesimal, in mathematics, a quantity less than any finite quantity yet not zero. 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): . 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. Infinitesimals are not real numbers, and therefore don't live on the real number line in the first place. For instance, consider we want to integrate the expression $3x(x^4+1)^3$. When we take a line with no gaps and add lots of infinitesimals around each point, we create gaps! 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 … One exception is a recent reconstruction of infinitesimals — positive “numbers” smaller than every real number — devised by the logician Abraham Robinson and … The Standard Part Theorem says all the limited hyperreals are clustered around real numbers. The real numbers are Dedekind complete. The multiplicative inverse of an infinity is an infinitesimal and vice versa. 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 … Question from a couple of my students that I couldn't answer and can't find using google. There is also no smallest positive real number! Given any real number , the number , but . Not sure if this is the right place. 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