I wrote about:
comparing 7^sqrt(8) vs. 8^sqrt(7) instead of 1/13 vs. 0.13
In case it was lost in the details it's worth pointing out that both comparisons are easily obtained by looking for a "simple" interpolant c between the two numbers a,b. Here "simple" means it is simple to show say a<c, c<b so a<b by transitivity. For 1/13 < 0.13 a simple interpolant is c = 1/10 = 0.10 For 8^sqrt(7) < 7^sqrt(8), I chose c = 7^(31/29 sqrt(7)), and then I further employed the following simple interpolants: (7^5/2^14)^6 > (16800/16400)^6 = (42/41)^6 > 1 + 6/41 > 1 + 1/7 Of course this is a very trivial example of breaking up a complex problem into "simpler" steps - a process that becomes almost subconscious to well-trained mathematician. That student math teachers lack such basic skills seems to indicate that there is a major gap in the teaching of mathematical problem solving. No doubt some of the blame lies on the widespread availability of computers and calculators. As Gauss once said "the purpose of calculation is insight". In the pre-calculator days one developed good insight and problem-solving skills because one was forced to in order to make hand/mental calculations tractable. But nowadays most students never develop such insight because they've been limping so long on computer crutches. They will be at a loss as soon as they are presented a problem that doesn't fit mindlessly into the prepackaged set of problems soluble by brute-force by their symbiotic partner. It should come as no surprise then that US math students keep performing poorly vs. other nations. They are the victims of the ubiquity of the technology and it's rampant abuse in the US educational system. Until this problem is rectified, calculator brain drain will continue to sap their minds. Btw, here's a precise link to the 1/13 vs 0.13 article http://learningcurves.blogspot.com/2004/12/from-final.html --Bill Dubuque