This file contains a series of articles by Wlodek
Mier-Jedrzejowicz. They were published in DATAFILE,
the jounal of the HPCC (please join!).

2.02 ln e^{x}

gives the result 2. instead of 2.02 but this bug was soon removed. Another bug
is that the sine of some small angles comes out completely wrong -- this was
dealt with later than the other bug, but newer models have neither bug.
* At first the HP-35 was made only in the USA (at the Advanced Products
Department in Cupertino -- the calculator division moved subsequently to
Corvallis). Later on, the HP-35 was made in Singapore too, which is what HP do
with all the handheld calculators they make.
* The electronics inside the HP-35 was updated as well during the lifetime of
the product. I did not take any of these HP-35s apart to show this though!
All this should be very clear then -- a really early HP-35 have the red dot,
will have the 2.02 bug, have key function captions above the keys, have a label
which says "HEWLETT.PACKARD", and will have been made in the USA. This was not
the case with the HP-35s on show, though! The one with the red dot did not have
the bugs, and had a label saying "HEWLETT.PACKARD 35". The only one with the
2.02 bug was the latest model, with the new-style keyboard!
There are two explanations for this confusing state of affairs. First of all,
some people sent their HP-35s to HP to be fixed. The ROM chips in these will
have been updated, so the bugs are no longer there. At the same time, if the
label was coming loose, it will have been replaced with a new one. (HP did the
same to me once -- I sent in an HP-25 to be fixed -- it was returned with a
note saying that they no longer had the required spare parts -- but the label
had been overlaid with one which says HP-25C.) Secondly, the calculators were
assembled from parts held in large bins -- it is quite possible that some older
parts were used in the assembly of newer calculators.
Anyway, the four HP-35s exhibited had all these features between them -- a red
dot and none, labels above keys and on them, bugs and no bugs, made in the USA
and in Singapore. What's more, they all work -- 20 years later! That's HP
quality for you!
I included some other items. Unlike modern HP calculators, the old models came
in a plastic case, containing the calculator, a charger cable, the manual with
a list of corrections, and even some "PROPERTY OF" labels so you could identify
your calculator. You could buy spare battery packs and chargers. If you
really wanted to secure your HP-35, you could buy a security cradle, which
could be screwed down to a surface, or held in place with a security cable.
Well, that's all! I hope people had fun looking at them -- fortunately I got
all four back at the end of the day -- not all of them were mine! (Many thanks
to Dr Bob Speer of the Spectroscopy group in the Physics Department at Imperial
College, where our club meets, who introduced me to the HP-35 more than 20
years ago, and who loaned me some of his 35 collection to show.)
x^{2} - 2 = 0

SQRT(2) is an irrational number, because it cannot be expressed as the ratio of
any two integers. The discovery that this is so caused ancient Greek
mathematicians great grief, but that is another story.
On the other hand, pi is a transcendental number, because no expression can be
written of the type:
a_{n} x^{n} + a_{n-1} x^{n-1} +
... + a_{1} x + a_{0} = 0

to which x=pi is a solution, so long as a finite number n of terms is used. Pi
can only be expressed exactly if n is allowed to be infinite.
Pi is the solution to the equation:
C = 2*pi*r

where C is the circumference of a circle and r is the radius, but getting pi
exactly from this equation again requires an infinite number of steps. In this
case, this is because measuring C exactly with a straight ruler requires that
an infinite number of infinitesimally small pieces of the circumference be
measured.
Why a straight ruler? A practical reason is that straight edges are
comparatively easy to make -- and a straight edge can be converted into a ruler
just by laying it next to another straight edge which has already been marked
out, and copying the marks. The insistence on a straight ruler comes from the
ancient Greek foundations of geometry, based on their philosophical notions,
but in this case, the philosophy was based very sensibly on the practical
limitations of their technology. Indeed the whole business of treating
transcendental numbers as special in some way comes largely from philosophical
notions.
In practice, calculating any irrational number, whether transcendental or not,
is done by means of a series of repeated approximations. If you have only a
simple four-function calculator then you can calculate SQRT(2) or pi by making
a set of approximations. The earliest handheld electronic calculators could be
used to calculate square roots and so on only this way. In fact, it was
simpler to carry round a set of printed mathematical tables with your
calculator, look up a square root, log, sine, or whatever, and type it into the
calculator to use it. In effect the calculator had to be used with a book of
tables. (These books were commonly called "log tables", though they usually
contained tables of trigonometric functions and square roots as well.)
Then a few calculators were designed which could calculate percentages as well
as carrying out the four basic operations + - * / . The step after came with
calculators which automatically calculated square roots. This could be done
because a short program was built into the calculator to make the required set
of approximations. A program to do this can be very short, involving
repeatedly halving the difference between the square of the current
approximation and the number whose square root is to be found.
The HP-35 took the next step, providing keys to calculate not only square
roots, but also transcendental functions -- trigonometric functions, and
inverse trigonometric functions, exponentials and logarithms. Packing all this
into a handheld calculator required far more programming! HP managed to fit
their programs in a small, low-power, handheld unit by using an exceptionally
clever and fast method of calculating these functions -- the Cordic technique.
All HP calculators since the HP-35 have used this same technique to provide
compact and fast (and therefore low-power) calculation. If you want to know
more about the method, see the article from the HP journal which introduced the
HP-35. [ Note from Craig: see the bibliography for another reference to an
article on Cordic techniques. ]
It was this ability to calculate transcendental functions automatically and
rapidly which made it a huge success, and led to HP's further advances in
calculators. One thing to note is that the HP-35 was designed to work for the
users who previously carried a table of functions with their four-function
calculators. Instead of looking up a function and typing it on the calculator,
the user could now type in a number and press a button to get the function
value. The HP-35 was really a combined calculator and set of log tables -- it
was only later models which began to go beyond this to a fuller exploitation of
the things this made possible. But it was the HP-35 which began it all.
Meditate transcendentally on that if you will!
Now for an article which I wrote tonight -- it's one I have been planning for a long time, and your expression of regret at not getting the ones I had not yet writen encouraged me to write it down!

The Calculator Reference by Rick Furr (rfurr@vcalc.net)

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