Finding it is just the same as "rounding off" that finite number to the "nearest real number," a concept much more intuitive than finding derivatives and integrals as limits using the standard epsilon/delta definition.) Robinson calls that the "standard part" of the finite number. (The constant is just the ratio of the leading coefficients. Ratios with numerators of the same degree as their denominators are finite numbers that differ at most from some unique constant by some infinitesimal. And, every such ratio with a numerator of greater degree than its denominator is infinitely large. In fact, every polynomial ratio with a denominator of greater degree than its numerator is an infinitesimal. This set (call it †R is an ordered field that is a superset of R, the "standard" real numbers and proper subset of Robinson's hyperreals, *R, with the constant functions identified with the standard reals.ĭoes †R contain any infinitesimals? Yes, j( n) = 1/ n is less than every positive constant function for all values of n greater then the reciprocal of the constant, and is still greater than zero. One needs to think a bit outside the standard box to find a rigorous model of nonzero infinitesimals.Ĭonsider the set of rational functions, ratios of one polynomial to another, in a single positive integer variable (an index n) with real coefficients, where f g iff, for some n 0, n> n 0 implies f( n) g( n), respectively. The only such standard number is zero, itself. This is a shame, because everything one needs to establish a rigorous "hyperreal" extension of the real numbers, complete with (nonzero) infinitesimals and their infinite reciprocals, can be found in high school level algebra.Īn infinitesimal is a number with an absolute value (or size) less than every positive standard real number, no matter how small. The usual approach to nonstandard (or infinitesimal) calculus (based on Abraham Robinson's work, Non-standard Analysis, 1962) does indeed depend on ultrafilters and the Axiom of Choice, and is usually reserved for third year university level or even post graduate courses. I would prefer staying with the graphs with holes, and the procedures of filling the holes. Some axiom of choice demanding hyper real numbers. And what is the more intuitive idea supposed to be?. So a more intuitive idea could also be used in teaching. Now some people claim that this conventional idea of defining the derivative with a limit, which is the same thing as filling a hole in the graph, is not intuitive enough, and can confuse students. IMO the idea, that a derivative can be defined by filling a hole in the graph, is pretty much as intuitive as it can get. The first examples of limits are examples where you have some continuous function f:\to\mathbb, and the derivative f'(x) is obtained, when you fill the hole in the graph at the point (0,f'(x)). I cannot understand this talk about infinitesimals being intuitive
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