No longer useful

We don’t need to look up things in tables of numbers nowadays.

Our tutor had a stray thought this past week, a memory of his own High School experience.  In those days if one needed to do a highly-accurate calculation, beyond the three figures or so of a slide rule, one pulled out a table of numbers: logarithms or trig functions or whatever.  A four-place set of tables was a small pamphlet; five places meant a book.  Six-place tables were almost unknown, since few tasks required that level of accuracy (final design of optical systems was one of those).  In any case, one could squeeze out one more decimal place by interpolation, setting up a proportion between the listed numbers and the one you wanted.  Our tutor hadn’t actually done an interpolation in many years, so he was gratified to find that he could still do one properly.

But this is a useless skill, not even allowing a John Henry moment.  A painfully worked-out five figure number doesn’t compare with the ten figures thrown up effortlessly by any calculator.  No math teacher these days would consider requiring students to learn the arcane art (and we suspect few are aware of its existence).  So our tutor went on to other things.

One of these was helping a student use a graphing calculator to solve a problem with derivatives.  She had trouble because the machine had constructed the graph by plotting a large, but finite, number of points; that is, it simulated a continuous graph by making a discrete one.  The point she wanted, at x = 2, wasn’t there.  As it turned out, the points on either side were close enough for her purposes.  There was no need to fiddle and re-plot, and certainly no need for a manual interpolation.  But the lesson remains: the calculator has limits, just like any log table.

We live in a digital world, masquerading as a continuous one.  There are enough plotted points that we don’t notice it often, but interpolation is happening all the time.  It is unconscious or behind the scenes, but it’s there.  Generally it makes no difference.  The fact that you’re actually going a little faster than the 35 miles per hour on your speedometer has no real effect on your arrival time.  But an important part of setting up any computer calculation is working out what digitization does to its accuracy; there is a population of scientist and engineers devoted to the task.  And as calculations become more complex and ramified, their importance increases.

Our tutor is not about to teach interpolation to any of his students.  He is, however, pondering how he can get across the idea of the limited accuracy of the digital world.  Especially to those who will spend their lives immersed in it.