Re: [Phys-L] sig figs versus rational numbers

[John Denker <jsd@av8n.com>]

On 09/15/2017 11:33 AM, Karim Diff wrote:

> When students ask me about that at the beginning of first semester physics
> (the ones who've had physics in high school are particularly worried that I
> don't bring it up) I usually tell them that I'll let my Chemistry
> colleagues beat them on the head with that (and they do)😝

That's fine as far as it goes.  Continuing down that road,
it may help to go into more detail about what they /should/
do (not just what they shouldn't do).

 *) Separate the idea of number from the idea of uncertainty.
  For example, 2.54 is just a number.  In fact it is a
  rational number, 254/100, exactly, by definition.  This
  number may describe some physics approximately and some
  other physics exactly (e.g. 2.54 cm per inch) ... but 2.54
  by itself is just a number, with no "implied" uncertainty
  or any other kind of uncertainty.

  Any finite or repeating decimal represents a rational
  number, exactly.  Conversely, any rational number can
  be represented by a finite or repeating decimal, exactly.

  You learned in third grade that 2.54 equals 254/100, and
  the rules of arithmetic have not changed.  Ask your math
  teachers, if you dare, whether 2.54 equals 254/100 exactly.
  They will wonder why you had to ask.

 *) Very often, in class and in the real world, it is not
  necessary to quantify the uncertainty, so long as it is
  known to be /small enough/ i.e. within tolerances.  Unless
  and until you hear otherwise, you (the student) should
  report temperatures within ±2 °C, mass, length, and time
  within ±2%, et cetera.  Use as many or as few digits as
  you please, so long as the bottom-line answer is within
  those tolerances.

  The tolerance-based approach allows everybody to focus on
  the physics, rather than on useless side-issues.

 *) On the special occasions when we care about uncertainty, we
  will use at least two numbers to describe the distribution,
  perhaps in the form A±B, where A is the nominal value and
  B is the uncertainty.

 *) In all cases:
    -- Separate the idea of roundoff error from overall uncertainty.
    -- Roundoff error is an error.
    -- Digits that serve primarily to reduce roundoff error
     are called /guard digits/.  Always use plenty of guard
     digits, to ensure that roundoff error is small compared
     to other contributions to the overall uncertainty.  This
     usually easy.  "Extra" guard digits are harmless.
    -- Using too few digits is often catastrophic.

 *) Not all correct answers will be digit-for-digit identical.
  This is normal.  There is no problem, so long as they are
  all within tolerances.  Scatter in the data is normal and
  unavoidable.  Rounding off to the point where everybody
  got the same answer would replace a small amount of scatter
  with a much larger amount of roundoff error, which would
  be very very bad practice.

      Note to teachers: To facilitate grading, the answer key
      should specify a /tolerance band/, i.e. the /range/
      between the smallest and largest acceptable answer
      ... rather than specifying a single numerical answer.

 *) When using a calculator, leave intermediate results in the
  machine.  Use the memory features if necessary.  If you wish
  to write down intermediate results, that's fine, but leave
  them in the machine also.  This provides roughly 15 digits
  or more, which is usually plenty.

 *) The usefulness of sig figs is exceedingly limited:
  -- In cases where we don't need to quantify the uncertainty,
   sig figs is not worth the trouble.
  -- In more demanding situations, the sig figs representation
   is too coarse.  This degrades the uncertainty.
  -- Sig figs requires you to round off too aggressively,
   introducing intolerable roundoff error.  This degrades
   the nominal value.
  -- It is madness to assume that roundoff error is the
   dominant contribution to the overall uncertainty.
  -- It is madness to try to represent two numbers (A±B)
   using one numeral.

 *) In real-world research labs, engineering labs, kitchens,
  et cetera, people do not use sig figs.  Don't worry about
  sig figs.  We have more interesting stuff to attend to.
  Important stuff.  Useful stuff.  Physics!