Onion Information

Page Clicks: 0

First Seen: 04/26/2024

Last Indexed: 10/23/2024

Domain Index Total: 84

[ G.Polya ] Can you derive the result differently? When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution. Can you derive the result differently? Can you see it at a glance? Yet even if we have succeeded in finding a satisfactory solution we may still be interested in finding another solution. We desire to convince ourselves of the validity of a theoretical result by two different derivations as we desire to perceive a material object through two different senses. Having found a proof, we wish to find another proof as we wish to touch an object after having seen it. Two proofs are better than one. "it is safe riding at two anchors." 1. Example Find the area S of the lateral surface of the frustum of a right circular cone, being given the radius of the lower base R, the radius of the upper base r, and the altitude h. This problem can be solved by various procedures. For instance we may know the formula for the lateral surface of a full cone. As the fur frustum is generated by cutting off form of cone a smaller cone, so its lateral surface is the difference of two conical surfaces; it remains to express these terms R,r,h. Carrying through this idea, we obtain finally the formula s=pi(R+r)sqrt(R-r)^2+h^2) Having found this result in some way or other, after longer calculation, we may desire a clearer and less roundabout argument. Can you derive the result differently? Can you see it at a glance ? Desiring to see intuitively the whole result, we may begin with trying to see the geometric meaning of its parts. Thus, we may observe that sqrt((R-r)^2+h^2) is the length of the slant height (the slant height is one of the non parallel sides of the isosceles trapezoid that revolving about the line joining the midpoints of its parallel sides, generates the frustum; see fig 12.) Again, we may discover that pi(R+r) = (2piR+2pir) / 2 is the arithmetic mean of the perimeters of the two bases of the frustum. Looking at the same part of the formula we may be moved to write it also in the form pi(R+r) = 2pi(R+r)/2 that is the perimeter of the midsection of the frustum. (we call here mid-section the intersection of the frustum with a plane which is parallel both to the lower base and to the upper base of the frustum and bisects the altitude.) r . ~*****====_. .*=._._.C___-~*\ /. | h | \ / . ____L____ \ /. -~.*** | * R ~--.\ ............L~~~~~~~~~~~~~ ****~~~~~~~~~~**** Having found new interpretations of various parts, we may see now the whole formula in a different light. We may read it thus; Area = Perimeter of midsection * slant height. We may recall here the rule for the trapezoid; Area = Middle line * altitude. (the middle line is parallel to the two parallel sides of the trapezoid and bisects the altitude.) Seeing intuitively the analogy of both statements, that about the frustum and that about the trapezoid=, we may see the whole result about the frustum "almost at a glance." That is, we feel that we are very near now to a short and direct proof of the result found by a long calculation. 2. The foregoing example,l.e is typical. Not entirely satisfied- satisfied with our derivation of the result, we wish to improve it, to change it. Therefor, we study the result, trying to understand it better, to see some new aspect of it. We may succeed first in observing a new interpretation of a certain small part of the result. Then, we may be lucky enough to discover some new mode of conceiving some other part. Examining the various parts, one after the other and trying various ways of considering them, we may be led finally to see the whole result in a different light, and our new conception of the result may suggest a new proof. It may be confessed that all this is more likely to hap- pen to an experienced mathematician dealing with some advanced problem than to a beginner struggling with some elementary problem. The mathematician who has a great deal of knowledge is more exposed than the beginner to the danger of mobilising too much knowledge and framing and unnecessarily involved argument. But, as a compensation, the experienced mathematician is in a better position than the beginner to appreciate the reinterpretation of a small part of the result and to proceed, accumulating such small advantage, to recasting ultimately the whole result. Nevertheless, it can happen even in very elementary classes that the students present an unnecessarily complicated solution. Then, the teacher should show them at least once or twice, not only how to solve the problem more shortly but also how to find, in the result itself, indications of a shorter solution. See also REDUCTIO AD ABSURDUM AND INDIRECT PROOF