More = Less (x2^(Time/Halftime)


No, this isn't a sermon on the spiritual hazzards of collecting too many HP calculators! The relation dN/dt=kN occurs all over the place in the physical and biological sciences -- in first order chemical reactions, radioisotope decay, simple biological turnover, microbial population growth, the absorption of electromagnetic radiation by matter, the discharge of a capacitor, etc. etc. I used to teach a course on the applications of radioisotopes to biology (back in the sixties), and my students and I would spend hours with slide rules and semi-logarithmic paper, calculating and plotting the decay or build-up of materials. There are four variables in the integrated solution, N2 = N1*e^kt, so one has four different possible cases to solve for. Since there are so many different applications of this "compound interest" relationship, one can wind up with a library of equations as one roves around the various sciences.

But with the solve routine of the 19B or 27S or 17B etc. calculators, one can cover all the possibilities with the one statement: MORE = LESS x 2^(TIME/HFTM), or if one prefers the natural base, with MORE = LESS x EXP^(TIME/TRTM), where HFTM stands for half-time, and TRTM stands for turnover time (=1.44xhalftime). It doesn't matter which way time is flowing in the actual case, whether towards decrease or increase -- you know which it is, and the calculator doesn't care. With the numerical solve routine, you give the calculator three of the factors and it solves for the fourth, whichever it is. And of course "time" and "half-time" can actually stand for "distance" and "half-distance" if that is the nature of the physical problem.

I find it almost indecent that so much can be done with so little -- but that is also the joy, to one brought up in the age of trig and log tables, of the little pocket calculator.

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