The Museum of HP Calculators

HP Forum Archive 11

Minimal Scientific Calculator: A Modest Proposal (long!)
Message #1 Posted by Michael F. Coyle on 14 Mar 2003, 7:12 p.m.

## Introduction

A recent thread here asks, "What is the most useless function on a scientific calculator?" Many people have answered and a spirited discussion has ensued. One argument against functions such as x^2 and % is that it's easy to calculate the function without needing a special key.

That got me to wondering: how few functions can we use to build a scientific calculator. Right now I'm looking at my 11C: a beautiful machine, to be sure, but loaded with keys and functions. The engineer in me asks, "How can I reduce its complexity? What can I get rid of?" The 10C was a step in the right direction but in my opinion was a failure because it did not go far enough.

Let's see how much of the extra functionality we can remove. True, we will have to learn some new key sequences but HP users are used to doing things a little diffenently, what with RPN and all.

Let's call this new machine the HP-11C-. (Although some other names, like HP-11D- or HP-11F may come to mind too.) Readers of the other thread will be happy to learn that the 11C- does not include %, x^2, Grads, n!, or hyperbolics; thus, it should make everyone happy.

## Hello, Numbers, Goodbye -, /, Trig and Logs

First, let's mention a few things we'll keep. All of the number entry and stack manipulation keys: [0]-[9], [.], [CHS], [EEX], [ENTER], [<-], [x<>y], [R Down], [R Up], [LST x]. Display Mode keys: [FIX], [SCI], [ENG]. Shift keys: [f], [g]. Not to mention [ON].

Now let's get rid of some stuff. We still need to add, so [+] stays. But with [CHS] to negate numbers we don't [-] any more; use the sequence [CHS] [+] to subtract.

Similarly, we may eliminate [/] if we keep [1/x], which is useful in contexts other than just straight division, in forming negative exponents, for example.

All trig functions will be done in radians so [DEG], [RAD], [GRD] go away. We don't need the trig functions either: angle [ENTER] 1 [->R] gives us the cosine in x and the sine in y. They can be divided to get the tangent. [->P] gives the arctangent; the other inverse trig functions can be calculated from the arctangent with the appropriate formulas. [0] [1] [CHS] [->P] [x<>y] gives us pi, so there goes another key. With pi still available, we don't need [->DEG] or [->RAD].

Let's hold onto [LN] and [e^x]. We can then calculate common logs and antilogs, and arbitrary powers and roots, so we don't need [LOG], [10^x], [y^x], [x^2] or [sqrt x]. Hyperbolic functions and their inverses are defined in terms of exponentials and natural logs, so they go out the window, too.

## Other Functions

Memory ([STO], [RCL]): Keep.

[%], [Delta %]: Not necessary, compute using formula on back panel of calculator.

[ABS]: Trivial to do by hand or in a program.

[CLEAR Sum], [CLEAR PRGM], [CLEAR REG]: Don't need. Get rid of them.

[CLEAR PREFIX]: Useful for correcting incorrect prefix key press. Keep.

[CLx]: Not sure about this one. Let's get rid of it and put it back later if it proves necessary.

[RAN #]: Too hard to duplicate. Keep.

[Py,x], [Cy,x]: Can be computed with factorials. Get rid of them.

[x!]: Get rid of it unless you need the gamma function.

[->H.MS], [->H]: Drop; easy to implement.

[FRAC], [INT]: We don't them both. Keep [INT] and compute fractional part with Frac(x) = x - INT(x).

[USER]: Just a conveinence function. Don't need it.

[MEM]: Why bother? Get rid of it.

Statistics: [Sum+], [Sum-], [x bar], [s], [y hat, r], [L.R.]: Can all be computed by hand. Get rid of them.

## Programming Functions

We will keep all of the programming stuff with the following exceptions:

[BST]: We can just go back to the start of memory and [SST]. Drop.

[DSE], [ISG]: Can rewrite with other sequences of instructions. Get rid of them.

Flags: We can duplicate their effect using regular memory registers. Out they go!

Conditionals: We certainly don't need eight of them! First, get rid of the four "compare x to 0" tests. We can just put the zero in ourselves and use the "compare x to y" tests. Next, [x=y] and [x!=y] are exact opposites; we only need one. Let's keep [x=y]. Finally, [x<=y] and [x>y] are also opposites; let's keep [x>y]. This eliminates six of the eight conditionals and leaves just two nonredundant ones.

## Examples

Now let's put it all together. First some simple examples (all to 4 decimal places):

Square root of 625: 625 [LN] .5 [X] [e^x] (Answer: 25.0000)

Common log of 2: 2 [LN] 10 [LN] [1/x] [X] (Answer: 0.3010)

Common antilog of 2.775: 2.775 [ENTER] 10 [LN] [X] [e^x] (Answer: 595.6621)

2^5: 2 [LN] 5 [X] [e^x] (Answer: 32.0000)

cos 0.75: .75 [ENTER] 1 [->R] (Answer: 0.7317. [x<>y] gives sin 0.75 (0.6816))

And finally, a larger example:

Arc sin 0.6, in degrees: [LN] [LST x] [x<>y] 2 [X] [e^x] [CHS] 1 [+] [LN] .5 [X] [e^x] [->P] [X] 180 [X] 0 [ENTER] 1 [CHS] [->P] [X] [1/x] [X] (Answer: 36.8699)

Isn't this more satisfying than hitting just two keys, like on other calculators? Here you have a real sense of ownership of your answer, having worked hard for it. The HP-11C- will give you many opportunities for this kind of satisfaction.

## Conclusion

The appeal of the proposed HP-11C- is clear. Fewer built-in functions mean less ROM and fewer keys. This translates to lower manufacturing costs and higher reliability. Fewer keys also means that the calculator can be made smaller without sacrificing key size or spacing, preserving the excellent Voyager series ergonomics. And as can be seen from the description above, the calculator provides a built-in never-ending refresher course in math, as opposed to other calculators which only act as a mental crutch the more you use them. This aspect alone could make the 11C- a hit in the educational market.

Comments and suggestions are always welcome.

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #3 Posted by Gordon Dyer on 15 Mar 2003, 3:30 a.m.,in response to message #2 by Patrick The humour seems to have got lost in the ROM with no key to access it !

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #4 Posted by Christof on 15 Mar 2003, 4:24 a.m.,in response to message #2 by Patrick Acknowledging the possibility of lcd keys, I'd like to suggest something else- which may seem odd, but has potential. more single line LCD displays above the next two rows of keys. this gives you a 3 row soft menu effect very easily, without having to worry about the keys themselves going bad. but looking at all these answers- I find that most solutions ahve been used in various HP calculators- just not all at the same time. overlays- good ones like the 41 had, don't exist for the pioneers. Soft menus like you have for the 42, don't exist for the 41. (overlays aren't the ideal answer, since they are seperate and therefore losable pieces, but they are better than nothing, and I don't trust the quality of the lcd keys in a field calc.) a display that displays 4 stack levels isn't on any machine previous to the RPL era. the way the % key is implemented that works so well for VAT would be even better done as a program if you had a (minimum)two line display and a user configurable keyboard. This makes the need for the dedicated % key less. it also applies to many other keys. x^2 would not be such a concern on stakc conservation if the stack was configurable. natively supporting both keystroke and RPL programming models on the same machine would solve many problems.(such as dynamically adding to the functions usable by the keystroke programming 'language') - personally, I like the 32Sii. I *like* the 2 shift keys. give me that keyboard layout, the functionality size of my 42S, the display of a 28, and 64K or better RAM and I'd probably keep quiet. (well, IR I/O and the ability to flash the OS would be nice)

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #5 Posted by bill platt on 15 Mar 2003, 10:46 a.m.,in response to message #4 by Christof Yes, right on my thoughts with: stack configurable, and RPL/keystroke support, and 32sii is mighty nice generally (E-bay seems to be proving that!) I think IR will be uneccessary in the future--look at what is on the market right now for memory transfer at very low cost.... The business (further up the thread) about saving lines by using x^2 etc only matters when the memory is rather "hard" rather than configurable. With memory more "soft" like the 32sii (the old 10-c series was on the right track already with the automatic line allocation ) --the newer idea of having a pot of memory that can be used for anything, that makes the number of lines per se unimportant.

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #7 Posted by Patrick on 16 Mar 2003, 1:54 a.m.,in response to message #6 by Michael F. Coyle Hey, sorry if I missed the humour... I think I gave up half way thru... mea culpa. When you look at reliability, I bet a lot of failures are not reported (uh... it kind of ... well ... slipped out of my ... uh ... shirt pocket as I was ... uh ... flushing the ... ) I guess my point is that human error is likely a huge problem with reliability. I realize mechanical parts are the major culprits otherwise. Bet a lot of keys fail, though, because they're trying to work through that layer of Coca-Cola that got laid down betwen them and the key sensors.

 Re: Why stop there?Message #8 Posted by Mike Rivera on 14 Mar 2003, 10:57 p.m.,in response to message #1 by Michael F. Coyle Great ideas! While you're at it, let's get rid of the "8" key. I hardly ever use it and if I need to I can always use: 7 1 + Happy Friday! - Mike

 Re: Why stop there?Message #9 Posted by Michael F. Coyle on 15 Mar 2003, 7:24 p.m.,in response to message #8 by Mike Rivera Hi Mike! Thanks for your suggestion. I will give all the consideration it deserves. :) - Michael

 Re: Why stop there?Message #10 Posted by mapet on 17 Mar 2003, 3:50 a.m.,in response to message #8 by Mike Rivera In fact the calc can run only in binary mode, so all number keys we need is "0" and "1" :-))))))). Sqrt and ln are not needed as well as we can program the calc with Taylor series based algorithms...

 ...and here's MY brainstorm on that... ; )Message #11 Posted by Jeremy on 14 Mar 2003, 11:23 p.m.,in response to message #1 by Michael F. Coyle By the time you put all that thought into doing what we now consider basic calculation, why not completely eliminate the calculator? If we instead used slide rules, we would never worry about batteries and corrosion. There would be no buttons to wear out. With today's technology, an accurate and rugged slide rule could be produced for mere pennies. I would think that the type of person who would eliminate the minus sign and common log would be all over slide rules. If you stop to think about it, slide rules are beautiful in their simplicity and reliability. With pen and paper, some nice laminated log tables, and a good slide rule or two and we'd be all set. People would be more well educated in basic math, but by the time they got anywhere, they will have had their fill. I'm thinking of it, and I'm trying to imagine jumping through all those hoops just to find a simple sine in degrees. (gasp) Talk about increasing your chance for erros, esp. math novices... Anyway, I'm sure you've thought of all this and maybe the purpose of the post was just to illustrate that which features are really necessary is a matter of personal taste and knowledge base... Good thoughts! -Jeremy

 Re: ...and here's MY brainstorm on that... ; )Message #12 Posted by Christof on 15 Mar 2003, 4:27 a.m.,in response to message #11 by Jeremy Binary! all math on ten fingers.

 Re: ...and here's MY brainstorm on that... ; )Message #13 Posted by db(martinez,california) on 15 Mar 2003, 5:46 p.m.,in response to message #12 by Christof cristof - thats not binary: fingers are "digital" :-) - d

 Re: ...and here's MY brainstorm on that... ; )Message #14 Posted by Michael F. Coyle on 15 Mar 2003, 8:41 p.m.,in response to message #11 by Jeremy Jeremy wrote: I'm thinking of it, and I'm trying to imagine jumping through all those hoops just to find a simple sine in degrees. (gasp) Talk about increasing your chance for erros, esp. math novices... You find 28 keystrokes for an arcsine excessive??? Gee, way back when, men were made of sterner stuff. :) Anyway, I'm sure you've thought of all this and maybe the purpose of the post was just to illustrate that which features are really necessary is a matter of personal taste and knowledge base... Good thoughts! I hadn't thought of it that way but that sounds about right. I have used slide rules and while they're fun to play with, I sure wouldn't want to give up my calculators for them. Thanks for your reply. - Michael

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #15 Posted by Karl Schneider on 15 Mar 2003, 12:44 a.m.,in response to message #1 by Michael F. Coyle Mr. Coyle put together a nice tongue-in-cheek discussion (particularly the conclusion) that made for an interesting quick read. Patrick might have missed the point, I think... I believe that the 10C (mentioned in Coyle's post) failed commercially (1982-84 lifespan) because its functionality was stripped down too much in the interest of creating an "entry-level" model. 12C-like programming capability and lots of stuff taken out for an \$80 price tag in 1982? More like a marketing gimmick -- wise salesmen would have directed customers to the 11C and 15C. ("Yeah, it's cheaper, but you don't really want it -- here's why...") In those days, I'll bet those units not bought through mail order were sold by knowledgeable clerks at electronics counters.

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #16 Posted by Michael F. Coyle on 15 Mar 2003, 7:37 p.m.,in response to message #15 by Karl Schneider You're right about the 10C, of course. If someone's in the market for a Cadillac, they're just not going to be interested in a stripped-down Cadillac. Likewise with HP calcs. I like your "sell-up" theory. I think you've something there. - Michael

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #17 Posted by glynn on 15 Mar 2003, 1:25 a.m.,in response to message #1 by Michael F. Coyle Thank you Michael... LOL, at the wit of it all. (Um, his title should have been a clue-- Jonathan Swift's famous essay "a Modest Proposal", about breeding babies as a potential food crop, was actually castigated and Swift reviled for his cruelty and insensitivity when it was first published. Neither Swift, nor Coyle, are suggesting what they SEEM). The purpose of ANY tool is to make efforts EASIER. Of course you can make a tool much more complex than it needs to be. You can also turn out beautiful work with stone axes and bone awls. You nearly ALWAYS have to know what the heck you are doing, with whatever tool you choose. The question of when the learning curve of the tool exceeds the user's willingness to learn it, and the equally relevant question of when does automation interfere with understanding of the process--- hey, SOME of this was decided long, long ago when mathematicians invented various symbolic representations of concepts which we are now expected to know and understand enough about to IMPLEMENT when necessary. We ALL understand the concepts behind and the rules regarding addition (+) and subtraction (-) of numbers; but it has become second-nature to us primarily because we were TAUGHT this symbol-manipulation from an early age. If you have had experience with little kids, you realize that even these concepts require a bit of training for a mind to grasp. Anyone remember having to memorize their multiplication tables? I remember at the age I was introduced to division that I mused it wasn't as "clean" as multiplication (*I don't remember how exactly I expressed that*) because if division had Remainders, so should multiplication... And we go through schools and more and more ARBITRARY ABSTRACT SYMBOLIC EXPRESSION is thrust upon us; I'm not condemning such, but trying to make the point that concepts such as negative numbers, SINs, degrees, logs, exponents, "imaginary" numbers-- all are attempts by the mathematical community to take real-world observations or logical extensions of them, and make a "language" of symbols and syntax whereby you can easily manipulate them IF you understand what they do, or at least what you can do WITH them. On my 15c, I note that it does imaginary numbers. I frankly have never used them, and don't know HOW to. In my rather mundane life, I have never been asked to solve a problem that required them (or I am ignorant of the need I have, and have thus missed out on a potential use of my tool). This is the truth of ANY function on a calculator: if you don't know WHERE to use it, you won't. Percents and such "derived" operations are no more, no less a "convenience" than automated conversions or, as Coyle has so skillfully pointed out, many other mathematical operations. The whole purpose of them is to let the user avoid the derivation process by automating it to one little keystroke. The PERFECT calculator, in fact, might just have one key: "Answer". ;-)

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #18 Posted by Gordon Dyer on 15 Mar 2003, 3:37 a.m.,in response to message #1 by Michael F. Coyle Of course there would have to be 10 volumes of spiral bound 1" thick manuals which would add \$200 to the price! The only point I disagree on is reliability, you say it will be increased with fewer keys, but lots of keys used infrequently will wear out much more slowly...

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #19 Posted by Michael F. Coyle on 15 Mar 2003, 7:50 p.m.,in response to message #18 by Gordon Dyer LOL. I forgot about the manuals! You do have a point about reliability. A calculator with a very small number of keys will get "used up" quickly. But a very large number of keys with little-used functions will not slow down the wearout of the commonly-used keys like [ON], [+] and [ENTER], but do add more points of potential failure. Clearly there is a minimum to this curve where there are enough keys to spread out the wear but no more. I suspect (but with no actual evidence) that current calculators are fairly close to this point already. Maybe the way to improve reliability is to add redundant keys for the ones most prone to early wearout. How about 3 [ON], 2 [ENTER], 2 [+], etc.? Not much point though since most calculators now are throwaways. - Michael

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #20 Posted by hugh on 15 Mar 2003, 1:24 p.m.,in response to message #1 by Michael F. Coyle this idea has merit. recently i asked myself a similar question. given a fixed number of keys (eg 36) some of which are allowed to be shift keys. what is the best function layout that leads to the minimal number of keypresses on average for scientific functions. thus a function hidden away in a submenu or on multiple shifts cannot have an alternative shorter computation. so the idea of using chs + to perform subtract could only pay out if `wasting' a key on minus meant making other popular operations longer. clearly not in this case. in a gray area are operations like x^2. x^2 is actually quite common, but its only 2 keys if not present, one if present. does it justify that key? also, some functions can share the same key. like pi and eex. early commodore used to have a 1 key memory. "M" is sto after = and rcl otherwise. a bit quirky yes, or should sto be f rcl or should rcl be f sto or should they have their own keys? so this is a different minimization goal.

 Re: Minimal Scientific Calculator: A Modest Proposal (long!)Message #21 Posted by Michael F. Coyle on 15 Mar 2003, 8:28 p.m.,in response to message #20 by hugh Hugh wrote: recently i asked myself a similar question. given a fixed number of keys (eg 36) some of which are allowed to be shift keys. what is the best function layout that leads to the minimal number of keypresses on average for scientific functions. Thanks for your reply. Clearly some functions are more important than others. No one would even dream of making [+] a shifted function, yet it seems perfectly reasonable for, say, hyperbolics. So what you do is assign a weight to each function based on how ofter it's used. For instance, give each of [0]-[9], [.], [+], [-], [X], [/], and [ENTER] a weight of 0.03 (3%). Then we're saying that taken all together these keys account for half the keystrokes. [STO] and [RCL] are also popular, but not quite as much, say 0.02 each. And so on, down to, say, inverse hyperbolics, at perhaps 0.005 each. (When we're done hopefully the weights add up to 1.0.) Then design a hypothetical keyboard arrangement and score it by counting the keystrokes required for each operation and multiplying it by that operation's weight. Now your job is to arrange the keyboard so as to minimize the score (which is the weighted average of the number of keystrokes to perform all operations). My apologies if the above is already obvious to you. I would be very very surprised if the calculator manufacturers were not doing this already. The layout of any given calculator reflects the engineers' attempt to minimize the keystrokes for the machine's expected use. Note that on some HP machines, square root needs a shift key to operate (some other operation was deemed more important), while on others, square root is unshifted (i.e. square root is very important). This reasoning also applies to the question of what functions to include in the first place. Is 10^x so important that we must include it, even though y^x can be used instead? You get the picture.

 Minimal Scientific CalculatorMessage #22 Posted by Nenad Vulic (Croatia) on 16 Mar 2003, 5:57 a.m.,in response to message #1 by Michael F. Coyle If I might add something to the discussion, IMHO, what I would need in my everyday life (an would for sure carry it always with me) is an RPN scientific calculator (0-9, +, -, *, /, ENTER, RollDown, x>HMS, >H, STO, RCL, SIGMA+ and nothing else) built into my mobile phone, which should be designed and manufactured in an oldHP fashioned way... Or perhaps a mobile phone built into a HP11C :)

 Re: Minimal Scientific CalculatorMessage #23 Posted by Michael F. Coyle on 18 Mar 2003, 5:33 p.m.,in response to message #22 by Nenad Vulic (Croatia) Hello Nenad, That's an interesting suggestion -- RPN calc + cell phone. The good news is that from a technological standpoint, it's easy. The bad news is that you would have to convince a cell phone manufacturer that there's a market for it. :( Thanks for writing. - Michael

 Re: Minimal Scientific CalculatorMessage #24 Posted by James M. Prange on 19 Mar 2003, 9:12 p.m.,in response to message #23 by Michael F. Coyle RPN calc + cell phone? Sure there'd be a market for it. I'd bet students would love it, at least until the instructors figured out what was going on and banned them from the classroom. Regards,James

 Re: Minimal Scientific CalculatorMessage #25 Posted by Paul Brogger on 20 Mar 2003, 1:38 p.m.,in response to message #22 by Nenad Vulic (Croatia) . . . and add a programmable, universal remote control capability. If I'm going to carry a phone around, it'd be nice if it could substitute for the always-misplaced VCR or TV control! Universal remotes are cheap, small, and not too hard on batteries. One could easily be built into a Personal Multi-Function Device.

 Have you heard about the Turing machine?Message #26 Posted by Andrés C. Rodríguez (Argentina) on 16 Mar 2003, 1:21 p.m.,in response to message #1 by Michael F. Coyle Just a comment by a not-so-convinced minimalist :-)

 Re: Have you heard about the Turing machine?Message #27 Posted by Michael F. Coyle on 18 Mar 2003, 5:09 p.m.,in response to message #26 by Andrés C. Rodríguez (Argentina) Hi Andrés, Yes I do know about Turing Machines. Of course a calculator is already a T.M. and can compute anything any other T.M. can, within the limits of its memory (and the patience of its owner). I was thinking of building a T.M. but I couldn't find out where to buy the infinite tape. :) Thanks for writing. - Michael

 Re: Have you heard about the Turing machine?Message #28 Posted by David Smith on 18 Mar 2003, 5:59 p.m.,in response to message #27 by Michael F. Coyle Have you tried www.infinitetapes.com? There seems to be a specialized seller out there for just about anything else you could imagine.

 Re: Have you heard about the Turing machine?Message #29 Posted by Andrés C. Rodríguez (Argentina) on 19 Mar 2003, 6:51 p.m.,in response to message #27 by Michael F. Coyle Michael: I enjoyed the thread, and some of your points are valid indeed, but... don't get too minimalist! Best wishes!

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