HP Calculator History - Trascendental Functions

by Wlodek Mier-Jedrzejowicz Ph.D.

The December 1992 meeting was our club's last chance to celebrate the 20th anniversary of the HP-35. I was sure HPCC members would like to have a close look at some HP-35s, and to push some keys on them, so I brought four to exhibit.

The HP-35 was the first handheld electronic calculator which could calculate transcendental functions automatically. Transcendental functions are those whose values cannot in general be expressed as solutions to an algebraic equation. (Isn't that why we use RPN instead? Sorry, silly joke.) If you want to know more about transcendental functions, look at the appendix to this article.

Several changes were made in the HP-35 design during the time it was made, before it was replaced by the HP-45. The HP-35s I brought showed these changes.

* The earliest 35s had a small hole to the right of the on/off switch. When the HP-35 was turned on, a red spot showed through this hole. This was soon removed - you could see the HP-35 was on anyway, because the display lit up! Even if the batteries were low, you could see the switch was at the side marked ON.

* So long as the HP-35 was the only handheld calculator made by HP, there was no need to put the model number on it; the label below the keyboard just said "HEWLETT.PACKARD". The success of the HP-35 led HP to introduce the HP-80, the first business model, and a model number on the label became necessary. By the way, the HP-80 was a combined calculator and set of financial tables, in the same way as the HP-35 was really a combined calculator and set of log tables - see the appendix for more about this. That is not at all surprising - the first mechanical calculators, such as Babbage's machines, were designed specifically to print actuarial tables. Mind you, the HP-80 already had more features to exploit the availability of built-in functions, and even had sigma+ and sigma- keys.

* All HP-35s have the numeric digits, pi, and the operations + - * / and ENTER printed on the keys, but at first the other operations were printed above their keys, on the keyboard. Later models have the other functions on the keys as well. These later models also have the words OFF and ON moulded into the keyboard, not printed on it next to the on/off key. True HP-35 zealots will point out that these later models also had one silver-painted line below the display, not two.

* Early HP-35s had some bugs! (So, what's new? Or old?) I included an HP leaflet describing these bugs with my exhibits. One bug meant that typing:

                  2.02 ln e

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.)


Transcendental numbers are numbers which are the solutions of transcendental equations - equations which transcend algebraic methods - they cannot be solved purely by algebraic manipulation.

For example, SQRT(2) is not a transcendental number, because it is a solution to the algebraic expression:

x  -  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:

          n     n-1
       a x + a x    + ... + a x + a = 0
        n    n-1             1    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!


This article is part of the WMJARTS file. This file contains a series of articles written by Wlodek Mier-Jedrzejowicz and published in DATAFILE, the journal of the HPCC. The article was reproduced with permission of the author.

Copyright Wlodek Mier-Jedrzejowicz Ph.D.