Nostalgia & Fun With Calculators

By Béranger

Despite having developed a dependency of my ~200 installed Android apps and ~150 installed Windows programs, I still use a hand-held scientific calculator. Most of the time I’m using it for innocuous tasks–making sense of the monthly expenses, which I categorize and track in time, for which the fact that such a calculator has a 2-display line is of a great help.

I’m actually using two calculators during the categorization of the expenses. To subtract items from a receipt while not interrupting what I’m doing with the physical calculator, I use an Android app. But I still need the first one! (I’m a person who likes to touch and to feel, hence I very much appreciate a physical object. I’m a person who wears a wristwatch to know the time.)

Speaking of apps: the stock Android calculator is an unusable piece of ugly crap; after having tried dozens of alternatives, I settled for the most convenient one: CalcTastic’s Scientific Calculator, the free and ad-free edition. The paid edition is worth the money if you need more themes and a programmer’s calculator layout, but I find the programmer’s approach much less practical than KebStudio’s BitCalculator.

The thing is, sometimes I remember that in high school and college I was able to do more, such as solving simple everyday questions by modeling them so that a simple integral calculus would provide the answer. But tell me how on Earth would you do that with a regular scientific calculator, being it physical or an app?

OK, the first one is actually a matter of symbolic calculus. It’s 4x when x=1, so it’s 4. But the second one… how do you so it? How do you enter it on your calculator?

We’re not in the 1980s anymore, and the reference is not CASIO fx-82 anymore. Today we have matrix displays dubbed Natural Display, Natural Textbook Display, Visually Perfect Algebraic Method (V.P.A.M.) or Natural V.P.A.M.–the idea being that one can input formulas exactly as they are normally written on paper or in a textbook. And yet, most such calculators are used in K-12 to operate with fractions, to simplify them and to give results in fractions or generally in symbolic writing instead of numerical values, e.g. 2√2/3 instead of 0.942809041.

Now, simplifying fractions is an extremely dumb thing, but this makes Texas Instruments and CASIO sell quite a lot of devices to kids in the States. What’s worth the money is a device able to solve such things:

Of course, it can also display 0.5.

Notice however that this is a symbolic result that is “nice” enough to have a perfect precision. What happens when a calculator doesn’t solve the integral before computing it, but it actually use an approximation method, such as the Gauss–Kronrod quadrature formula? Let’s compare the fabulous CASIO fx-991EX ClassWiz with the ugly TI-36X PRO and with an Android app, Philip Stephens’ Scientific Calculator:

Wow. If the physical calculators agree on 7 significant digits (of which 4 decimals: 9281), the app is severely wrong: only 4 correct digits!

That’s because it doesn’t solve the integral. Let’s ask Wolfram Alpha:

It solves it well, but it only gives 5 digits. Let’s compare the two ways of getting a numerical result (approximated vs. solved), and also ask CalcTastic about the value of the solved expression:

717.92810945 are indeed correct significant digits. So it’s crucial to be able to solve the integral first–but there aren’t many scientific calculators able to do that. Under Android, LisyThien’s four calc apps and Nhi Thien Studio’s five calc apps (all 9 are identical, except for the theming) are displaying 717.92810945099, which is fabulously accurate:

Let me now introduce you my everyday device, a cheap clone of something (we’ll talk later about clones and not-quite-clones of CASIO devices), the Olympia LCD-8310 that was sold by LIDL under the pseudo-brand UNITED OFFICE; it can’t solve, but it can calculate:

We don’t have the exact digits 717.92810945 on the screen, but 717.9281095 instead–the same value displayed by CASIO fx-991EX ClassWiz and fx-82DE PLUS both when solving the integral or calculating the logarithmic expression.

All the scientific calculators are internally using a better precision, they only display a rounded value. So by subtracting what we see on the screen (ANS- 717.9281095) we can find out what’s left inside: -0.000000049 in the case of my LCD-8310. Not convenient, let’s switch to ENG: -49.0082E^-09. Since it’s negative, we should have actually tried ANS- 717.9281094 and add this result to the original one; by doing so, we get the internal representation of the calculus: 717.9281094509918. The digits in bold are identical to Wolfram Alpha’s result, so we have 14 correct significant digits, although only 10 are displayed (of which the last one is rounded)!

How about the integral shown on the first physical calculator of this post, the German no-name but excellent GENIE 92 SC? Wolfram Alpha gives this:

It’s a totally unimportant calculus, but for the sake of estimating the accuracy of a rather no-name device, let’s compare some displayed results for 0.5*(4*pi-1)*sin(2 in degrees), the digits identical to Wolfram’s in bold:

  • GENIE 92 SC: 0.2018302566
  • LCD-8310: 0.201830256 (0.2018302565578661 internally)
  • Android’s stock calculator (8.1): 0.20183025655 (0.20183025655786604703 internally, so the last digit 5 should have been rounded to 6)
  • CalcTastic: 0.201830256558 (0.2018302565578660470356108 internally)
  • The 9 Vietnamese calculators, both as integral and as direct calculus: 0.20183025656 (0.20183025655786598564062 internally)
  • The HiPER Scientific Calculator Pro: 0.20183025655786604703561075211129582461158426 (OMG!)

The last digit in WolframAlpha’s result isn’t necessarily correct, as it could have been rounded up (as it’s the case with any last displayed digit), but in this case it seems correct, as it’s followed by 0.

For once, Philip Stephens’ Scientific Calculator doesn’t display inaccurate results, but somehow it decides to show a different number of digits depending on what it’s told to calculate: the integral, or the direct formula:

Somehow, these errors are reminding me of the first calculator I’ve ever used, the original TI-30 (1976) with LED display (based on how fast it was depleting the 6F22/MN1604 battery, initially I thought it uses Nixie tubes). This bulky chap made history and it was extremely popular even in Europe (it cost about 30 DM at the time):

From memory, while it displayed Pi as 3.1515927, the internal representation was 3.1415926536. Otherwise, it was funnily inaccurate, but of the computational errors I discovered at the time only this one about e is mentioned online:

[1][INV][lnx] = 2.7182818 is displayed. Subtract 2.71 and multiply by 100: .82818301 is displayed; .82818284 are the correct digits. Why compute three extra digits that are not quite accurate?

No, it was to old to have been affected by this huge logarithm bug!

Either way, here’s a fact: the purely software calculators are usually more accurate than the hardware ones. For some reason, the chip designers went cheap on the methods and the tables used for floating point operations (anyone remembers the Pentium 5 FDIV bug?), and of course they couldn’t make use of any arbitrary-precision library.

This is why some people thought of a unified method of testing the accuracy of the handheld calculators. Enter Mike Sebastian’s Calculator Forensics page, now unfortunately outdated, and the most popular way of generating a calculator’s signature: the result of arcsin (arccos (arctan (tan (cos (sin (9) ) ) ) ) ). On modern devices or apps that use Postfix Algebraic notation, the formula is entered as above, on classical Prefix Algebraic ones it’s entered 9 sin cos tan arctan arccos arcsin. Note that 9 is meant to be in degrees, not in radians!

Some signatures for famous models:

  • 8.99999863704 → Casio fx-82MS, fx-82TL, fx-115W, fx-115MS, fx-270W PLUS, fx-300MS, fx-300W, fx-570MS, fx-570W, fx-991MS
  • 9.00001568547 → Casio fx-82 SOLAR, fx-82SX, fx-95, fx-991D, fx-991S, fx-991N, fx-580
  • 9.0000157179 → Casio fx-900, fx-990, fx-991, fx-991M, fx-995, fx-961, fx-950
  • 9.0000765511 → Casio fx-82D, fx-82L, fx-250C, fx-250D, fx-350M
  • 9.177087103 → TI-30 (the original; did I mention how inaccurate it was?)
  • 9.114640577 → TI-30 LCD, TI-30-II, TI-30-III
  • 9.00000229461 → TI-30X, TI-30X Solar, TI-30Xa, TI-30Xa Solar
  • 9.000003512065 → TI-30XIIB, TI-30XIIS
  • 9.000001077372 → TI-30XS MultiView
  • 8.99999863704 → TI-34
  • 9.000003512065 → TI-34II
  • 9.000417403 → HP-10C, HP-11C, HP-15C (maybe the most famous RPN calculators)
  • 9.000000000 → HP-30S (“I believe this device uses true floating point for its transcendental calculations, instead of the traditional BCD (Binary Coded Decimal) of most calculators.”)

I suppose those were displayed values, not internal values. Since the results stopped being updated in 2002, there’s a lack of reports for the calculators that can be bought today. Let’s add some extra results then, starting with apps (I’ll list even devices I wasn’t able to test yet):

  • Windows Calc, Android 8.1 Calculator, CalcTastic, HiPER Scientific Calculator, the 9 Vietnamese apps: 9
  • Philip Stephens’ Scientific Calculator: 8.99999999983257
  • Olympia LCD-8310, 229 functions, 2 lines: 8.999999998 (8.999999998078897 internally)
  • STYLEX 38181, 240 functions, 2 lines, 10 €: 8.999999998 (8.999999998078897 internally)
  • GENIE 56 SC, 228 functions, 2 lines, 9 €: 8.99999986 (same value internally)
  • GENIE 82 SC, 240 functions, 2 lines, 13 €: 9.000000002 (9.000000002123857 internally)
  • GENIE 92 SC, 580 functions, Natural Display, 18 €: n/a
  • Casio fx-82MS 2nd ed., 240 functions, 12 lines, 12 €: n/a
  • Casio fx-85MS 2nd ed., 240 functions, 12 lines, 24 €: n/a
  • Casio fx-82DE Plus (fx-82ES Plus for Germany), 252 functions, 96×31 pixels Natural-V.P.A.M., 20…23 €: 9.000000007 (9.00000000733338 internally)
  • Casio fx-85DE Plus (fx-85GT Plus for Germany), 252 functions, 96×31 pixels Natural-V.P.A.M., 26 €: n/a
  • Casio fx-991ES, 403 functions, Natural Display: n/a
  • Casio fx-991ES Plus, 417 functions, Natural Display: n/a
  • Casio fx-87DE Plus, 510 functions, Natural-V.P.A.M., 28 €: 9.000000007 (9.00000000733338 internally)
  • Casio fx-87DE X (dumber edition of fx-991DE X, no matrix, vector and complex numbers) ClassWiz, 192×63 pixels Natural-V.P.A.M., 593 functions, 30 €: 9.000000007 (9.00000000733338 internally)
  • Casio fx-991DE X (fx-991EX for Germany) ClassWiz, 192×63 pixels Natural-V.P.A.M., 696 functions, 32…40 €: 9.000000007 (9.00000000733338 internally)
  • HP-10s+ (NW276AA), 240 functions, 2 lines, 10 € MSRP: 9.000000002 (9.000000002123857 internally)
  • HP 300s+ (NW277AA), 315 functions, 96×31 pixels Natural Display, 16 € MSRP: 9.000000001 (9.000000000881497 internally)
  • Casio fx-82SOLAR II, 144 functions, 1 line, 17 €: 9.000007164 (9.0000071639 internally)
  • Sharp EL-501X, 130 functions, 1 line: 8.999998637
  • Sharp EL-506X, 469 functions, 2 lines: n/a
  • Sharp EL-W506X WriteView, 556 functions, 96×32 pixels Natural Display: n/a
  • Sharp EL-W550XG WriteView, 335 functions, 96×32 pixels Natural Display: 9.000000099 (9.000000098906 internally)
  • TI-30 ECO RS, 1 line: n/a
  • TI-30X IIS, 2 lines: n/a
  • TI-30X Plus MathPrint, 4 lines, 29 €: 8.99999986 (8.999999860032 internally)
  • TI-30X Pro MathPrint, 4 lines: n/a
  • TI-30XS MultiView, 4 lines: n/a
  • TI-36X Pro, 4 lines: n/a

Note that from the above physical calculators, only Casio fx-991EX (and DE X) ClassWiz and TI-36X Pro are able to perform both Differentiation and Integration and Matrix Calculus; that Differentiation and Integration can also be solved by GENIE 92 SC, Sharp EL-W506X WriteView, Casio fx-991ES (and Plus); that not a single of them is programmable or graphing; but that almost all 2-line models are capable of statistics (Standard Deviation, Quadratic Regression).

Despite the fact that a result starting with 9.000… is aesthetically more pleasant than one starting with 8.999…, displayed values of 8.999999998 and 9.000000002 both mean the error is 2e-9. Sorting the devices by the approximate errors (absolute values), we notice that some of the devices that start with 9.0 are actually less precise than the others:

  • 0.7e-9: Casio fx-82DE Plus, fx-87DE Plus, fx-87DE X, fx-991DE X (likely true for non-DE editions too)
  • 1e-9: HP 300s+
  • 2e-9: Olympia/United Office LCD-8310, Stylex 38181, GENIE 82 SC, HP-10s+
  • 99e-9: Sharp EL-W550XG
  • 140e-9: GENIE 56 SC, TI-30X Plus MathPrint
  • 1363e-9: Sharp EL-501X
  • 7164e-9: Casio fx-82SOLAR II

So far, the worse of all is Casio fx-82SOLAR II, but once I try the TI-30 ECO RS, things might change. Either way, Texas Instruments remains one of the worst choices ever.

Knowing all this, why would anyone use a physical calculator instead of an app or, when sufficient, of the Windows Calculator, and then of something like Excel for what it’s worth? Of course, I don’t know how to work with integrals, differentials, matrices, system of equations and other advanced tasks in Excel or any Windows program, but this is another issue; for now I’m not very familiar with the apps that can do that in Android either–specifically these four calc apps and these other five calc apps. Besides, the most potent smartphone calculator apps are not made by the likes of Casio, but by Mister Nobody of Somewhere, so how can anyone tell anything about the potentially quirky bugs?

To me, the inaccuracies in handheld calculators remind me of the mechanical watches: definitely less accurate than the quartz-powered ones (even the thermally uncompensated quartz is nowadays very satisfactory), and yet some people are huge fans of them. Moreover, there are people who “collect” several “dress watches” (even quartz ones), “weekenders” or sport watches, just so they can wear one or another depending on their mood and apparel. Is there anyone doing the same with pocket or handheld calculators?

What actually makes such devices sell can be resumed in two words: kids and schools. In all of the developed countries there are calculators that have been “certified” and authorized to be used in class and in exams, but what’s worse is that some such devices are strongly recommended or even required, so what can parents do other than purchasing them? It’s less relevant that in my time kids didn’t need any calculator to learn how to simplify fractions or how to perform some other elementary tasks for which nothing is needed but an IQ over 80. Casio, Texas Instruments, Sharp and others must sell their devices, so they lobbied the powers that be. For the sake of the public education, right?

Aaaaand… there are huge differences between countries. You’ve already noticed that Casio has special editions for Germany, with the firmware in German and with a comma instead of the decimal point. Other countries are having their idiosyncrasies too, and let’s take France for one.

Here’s the fx-92+ Spéciale Collège, a device found nowhere on Planet Earth but in France. Being “adapted” to collège (the first 5 years of the secondary education), not lycée (the last 3 years of the secondary school), it doesn’t feature any differential and integral calculus capabilities, but it has rudimentary programming features (plutôt bidon si vous voulez mon avis). But I like the color of the case 🙂

The other major player, TI, aims higher: TI-83 Premium CE is soon to benefit of a Python adapter! But even without Python, Texas Instruments offers online support for various educational activities involving his advanced calculators. Incidentally, this made me learn of the Paradox of the Grand Duke of Tuscany–something I couldn’t find in Wikipedia!

From their materials I’ve learned that Galileo Galilei, while in Florence at the Court of the Duke of Tuscany, wrote up a paper in 1620 to explain a situation that the Duke himself noticed during the parlor games popular at the Court. The Duke had noticed that rolling the dice three times and summing the values led to the sum 10 slightly more often than the sum 9, despite the fact that both sums can be obtained in 6 ways:

10 = 6+3+1 = 6+2+2 = 5+4+1 = 5+3+2 = 4+4+2 = 4+3+3 9 = 6+2+1 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 3+3+3

The students are supposed to use a TI calculator in order to reproduce the situation and explain it. The explanation? Of the 6x6x6 = 216 dice values possible in 3 runs, only 25 lead to the sum 10, while 27 lead to the sum 9:

Nice, but no need for an expensive calculator to solve this problem!

Something about the two HP calculators mentioned above–the HP 10s+ and HP 300+, the only two models of relevance that HP sells nowadays. Their main quality: they’re elegant.

The thing is, these are marketing images, not real photos. You’ll find anywhere (at least on and people complaining that the screen has an abysmal contrast, that everything is reflective and it’s hard to see everything regardless of the level of the ambient light and–most important maybe–that the buttons are simply crappy:

Reports of missed key presses with this calculator are NOT exaggerated. This one is an exam grade killer! … The features are great for the price but it must be watched very closely or your results will be incorrect.
I can’t believe HP actually put their name on this device. I have HP Voyager calculators purchased in the early 1980s which still work flawlessly as they did when new. No missed key presses with them. Evidently HP no longer builds any level of quality in their non-graphing calculators. If Amazon allowed less than one star, this item deserves it. Returning this thing today.

It is very important for your calculator to accept the inputs from buttons 100% of the time, expecially under a time crunch or during an exam. This fails to do just that, where random times, may 75% of button presses actually register, I do believe it has something to do with the way the buttons are made, where the edges of the buttons don’t cause the input detection, but a small dot in the center of the buttons do, so if you’re always very careful, sure maybe it isn’t so bad, but I expect better from HP.

Truth be told, nobody makes hardware the way they used to in the 70s and the 80s. CASIO has the best quality of all, and Sharp is not that bad either (I can’t tell of Canon), but HP is horrendous. Even as they re-released a special edition of the legendary HP 12C Financial Programmable Calculator back in 2012, people started complaining about the total garbage that HP calls buttons. Absolutely anything to do with the old HP dependability and quality. Cheap crap that even the worst fake Chinese clones (think CTTTZEN calculators) don’t have.

This being said, in a world where 90% of the scientific calculators seem to be a clone of some variant of Casio fx-82MS, are HP’s 10s+ and 300s+ really using some chip designed by or for HP, or are they yet another set of clones? The first answer you’ll find: «They are so much of a clone that the menus themselves read CASIO» (with instructions on how to get CASIO on the screen).

Well, not so fast. Remember the “forensic test” that should get back 9 after some trigonometrical functions? In a thread on, someone got for HP 10s+ the result 9.000000002123857 (that’s the internal representation), and «None of the CASIO models listed have this HP signature result.» The same for HP 300s+ (9.000000001 on screeen, 9.000000000881497 internally)–«Again, no CASIO models listed have this signature.»

The explanation: «both HP and CASIO shared a common software base developed by Kinpo TW in China and compiled to run on distinct hardware platforms. This would explains why both have the same menu structures and many common functions.»

Of course, this doesn’t fix the poor screen and crappy HP buttons. Simply put, CASIO is the king, and fx-991EX, whatever you call it, is the mother of all calculators, both in build quality and features:

And it also can do that (the left picture):

Not everything CASIO is that shiny though. fx-82SOLAR II is nicely packaged garbage, and I wasn’t able to test the “2nd editions” in modern but bland casings:

Speaking of editions, TI had the bad idea not only of calling too many models “TI-30 something” but also to use the VW approach: let’s just call them Golf, Polo and Passat, no matter they change in time! Here’s 3 editions of the same TI-30X IIS–which one would you receive when ordering online?

The basic calculator has one more quirk: the solar-only TI-30 ECO RS, supposedly just an “eco” edition of TI-30Xa (the contemporary entry-level TI-30), has a specific design in Germany–guess which one?

The fucking picture in the middle is the German edition! Potthässlich. And it still has a decimal dot, not a comma, as it’s not a Casio!

This company has some of the worst designers–people who have never read The Design of Everyday Things. Sure thing, not all their products are as ugly as TI-36X Pro, but their attempt of a HP-like elegance somehow failed in my opinion:

The only good thing is that the keys are toggling between values: instead of using SHIFT and ALPHA, by repeatedly pressing SIN one would get sin, arcsin, sinh. And yet, these bulky objects are uninspiring… and not that rich in features (which is consistent with the TI-30 tradition, huh).

I’m going to end with a video. Remember the «integrate ln (x^2) from -5 to 99» test in which an Android app failed lamentably? Someone compared the speed of three devices performing nothing else but this calculus! Casio fx-991EX Classwiz vs Canon F-789SGA vs Casio fx-991ES:

  • fx-991EX Classwiz: about 45 seconds (35 seconds, claim 2 comments)
  • fx-991ES: some 3 minutes 12 seconds
  • Canon F-789SGA: about 4 minutes 40 seconds

From the comments:

  • fx-991ES Plus: 2 minutes 54 seconds
  • HP 35s: 52 seconds
  • TI-83 Plus: 26 seconds
  • Sharp EL-W506X: 17 seconds (finally, a fast chip!)
  • Sharp EL-W516X: 15 seconds (unfortunately, discontinued model)

Well, I’m impressed by Sharp’s speed (is the lower precision the price to pay for an Autobahn-like speed?). EL-W506X’s speed and keyboard precision is praised by users here and here. Quite a surprise for an under-25 € device. The HP and TI devices are in a higher price class, and HP 35s has such an impressive list of bugs that makes TI look good. As for all these devices that seem to be running at 4.77 MHz… they might be running at an even lower speed: «The 35s is using 1980’s technology–its 8502 processor core primarily.» And, to no one’s surprise, the keys are not always registering and they’re failing after 2, 10, 12 months…

For the time being, I’m going to play with some Android apps though.érangerNostalgiaTechnicalDespite having developed a dependency of my ~200 installed Android apps and ~150 installed Windows programs, I still use a hand-held scientific calculator. Most of the time I'm using it for innocuous tasks--making sense of the monthly expenses, which I categorize and track in time, for which the fact...BérangerBéranger beranger5ca@yahoo.caAdministratorHomo LudditusRebellious enough to be able to survive without a 3D printer