I found a handful of pocket calculators in a drawer. I haven’t used them in ages—Excel, MathCad, the software called Derive, and smartphones have taken their place—but I thoroughly enjoyed rediscovering their science! Once upon a time, they were important and expensive tools, and even today, machines with similar capabilities aren’t cheap. Yet they seem to be representatives of a species on the verge of extinction.
A calculator answers many questions, but it doesn’t foster understanding. I vividly remember how resistant our teachers were to pocket calculators. They tolerated the basic four-function ones, but those that could handle trigonometric functions and logarithms were like waving a red flag before a bull. They wanted us to learn to use the Four-figure tables properly. Eventually, they gave up and accepted the inevitable.
Soon, even more powerful little machines appeared, capable of integration, differentiation, working with complex numbers, and solving equations. As university students, we didn’t mind! Today you can get such 200–500-function calculators for a €10–20, and programmable ones for about twenty thousand. The top-tier graphing calculators with function-plotting capabilities cost €100–250, and about half or a third of that used—if anyone even buys them anymore, since tablets and smartphones are now retiring these devices the same way they once this calculators swept slide rules off engineers’ desks.
The Slide Rule
Just as calculators dumbed us down, slide rules dumbed down students in my father’s school days. He used to tell this story: a kid stood at the blackboard, sweating profusely until he came up with the solution: the square root of one. “And how much is that?” asked the teacher. The student proudly pulled out his slide rule, frantically slid it back and forth, then blurted out: “One… and something else!”
Note: Anyone who doesn’t get the joke should stop reading right now! The rest of this article will be painfully exhausting for them.
The principle of the slide rule is simple. Put two rulers side by side with their zeros aligned. Let’s “calculate” 1.3 + 4.7. Slide one ruler so its zero aligns with 1.3 on the other ruler. Then find the 4.7 mark on the second ruler and read the number opposite it on the first ruler: it will be 6, because the two lengths add up. A slide rule works the same way—except not for addition, but for multiplication. The trick is that the logarithm of a product equals the sum of the logarithms:
log(A × B) = log(A) + log(B).
If the rulers’ scales are logarithmic rather than linear, you can multiply with them. The basic scales, labeled C and D, run from 1 to 10. To multiply 1.3 × 4.7, align the movable inner scale’s 1 with 1.3 on the fixed scale, then read the result at 4.7: about 6.11. If it doesn’t fit on the scale, use the other end (10) as the starting point and shift left. Whether it’s 1.30 × 4.70, or 0.13 × 0.047, or 1300 × 470, the process is the same. You must express numbers as A.AA × 10^X and B.BB × 10^Y, and add the exponents in your head, adjusting by 1 whenever the cursor runs off the left end.
Division works similarly, only subtract instead of add (since log(A/B) = log(A) – log(B)). To try this yourself without an actual slide rule, visit the Virtual Calculator Museum (www.arithmomuseum.com) or download a slide rule app for Android!
Slide rules are more accurate the longer they are and the finer their graduations. Most are about 25 cm long and accurate to 2–3 digits—enough for engineering work, since input data weren’t more precise anyway. Around 1900, Germany produced a 2-meter steel slide rule for astronomical calculations that could be read to six decimal places using a microscope. The world’s longest working slide rule, 107 meters, was built in 2001 by Skip Solberg and Jay Francis in Fort Worth, Texas.
From the 1700s, slide rules were made of hardwood, bamboo, or metal, but they only became the emblem of engineering in the 1950s when companies like Faber-Castell and Reis began mass-producing affordable, high-quality plastic versions. In Europe, they were usually 25 cm long with two-decimal accuracy, though 12.5 cm pocket models and 50 cm office models also existed. In Hungary, Gamma Works made copies of Faber-Castell slide rules using political prisoners at Vác Prison. Slow work was punished, so they rushed—resulting in many sloppy, loose, inaccurate rules that often produced results like “one… and something else.”
A Peculiar Pepper Grinder
Typical slide rules also had extra scales for squaring, cubing, roots, and trig functions. Few know you can even add and subtract with the formulas:
x + y = (x/y + 1)·y
x – y = (x/y – 1)·y
Accuracy was still only about three digits—fine for engineers, not for bookkeepers, who didn’t care about tangents but needed lots of additions and subtractions to 8–12 digits. The first truly useful adding machine was built by Blaise Pascal in 1642. Six- and eight-digit versions were made, and about fifty survive. Building on this, French inventor Charles Xavier Thomas de Colmar created the Arithmomètre in 1820, capable of all four operations. About 5,500 units were made between 1850 and 1915, priced at 300 francs (which today might correspond to €3,500–€7,000). WWI ended its production, and postwar labor shortages prevented restarting. Copies were made into the 1940s in the U.S., among others. You can admire the original and try a virtual model at Arithmometre.org. Mechanical calculators thrived from the 1890s until WWII, though they were made into the 1970s. Some Hungarian offices still had working units after the regime change; clever people could grab beautiful specimens.
In the 1930s, young Curt Herzstark began designing a pocket-sized calculator. This was the Curta. Completed in 1948, it looked like a tiny pepper grinder. You entered numbers via sliders on its side; after turning a crank, the result appeared on top. Besides the four basic operations, it could square, extract roots, and compute negative powers. It was made until 1972, with 150,000 units sold for $125–175 each. Electronics killed it, and the company went bankrupt. Today the Curta enjoys a renaissance; see www.curta.de for original manuals and a working simulator. Rumors of $30 Chinese replicas are false—you can’t make such a fine mechanism of 600+ tiny parts cheaply. Real units sell for over €1200–1800 on eBay.
The Legendary HP
In 1974, Hewlett-Packard launched the HP-65 at $795 (about $3,770 today), likely the most expensive pocket calculator ever. It gained fame for being used on the 1975 Apollo–Soyuz mission. Its red, 12-digit LED display showed only stick-like digits, no letters. It could be programmed with up to 100 steps, and programs could be stored on magnetic cards. Not the first electronic pocket calculator, but perhaps the most iconic.
Development of electronic calculators began at Texas Instruments and Cal-Tech in 1965. The first integrated-circuit prototype appeared two years later, and by 1970 the first market-ready model was out. By the early ’70s, U.S. and Japanese electronics firms churned out calculator chips, enabling feature-rich devices with very few components. MOS technology allowed thousands of transistors per chip and low power consumption, so calculators could truly be battery-powered handheld devices. Visit the HP Virtual Museum (www.hpmuseum.org) to trace their evolution—from the HP-35 to the final model, the HP-50g with a 203 MHz ARM processor.
The cutting-edge chips were on the COCOM list, meaning they were forbidden for export to COMECON countries and China. According to an urban legend, the piano-shaped reservoir of the Gellért Hill Water Storage was designed using an HP-65 at the Fluid Mechanics Department of the Technical University. In reality, we didn’t have such devices back then; for the reservoir, they used a programmable desktop calculator (possibly an HP 9100) that the department had received as a gift.
Despite the embargo, by the late 1970s, domestic counterparts of scientific pocket calculators appeared in Hungary. Older engineers might vividly remember the TK-1023 and the PTK-1023 (programmable up to 102 steps), produced by the Híradástechnika Cooperative. These were sold in neat boxes with leather cases—the cooperative didn’t actually manufacture the calculators themselves, only the case, the manual, and the “Made in Hungary” label. The machines were, in fact, rebadged National Semiconductor models: the Novus 4520 and Novus 4525. How they made their way from the USA to Hungary remains a mystery.
When I was a student, the PTK-1050 was the big hit—actually a rebadged Texas Instruments TI-57. It could only handle 50 program steps, but it supported parentheses, which was a big deal at the time! Earlier machines used Reverse Polish Notation (RPN), where the operator follows the operands, eliminating the need for parentheses. In RPN, 3 + 4 = looks like this:
[3][ENT][4][+].
For 3 × (4 + 5), you’d type:
[3][ENT][4][ENT][5][+][×]
(pressing Enter between numbers). How about this one: ((3 + 4) × 5 + 6) / (7 + 8 + 9)? Here’s the sequence:
[3][ENT][4][+][5][×][6][+][7][ENT][8][+][9][+][÷].
Not a bad brain teaser, right? No wonder the parentheses on the PTK-1050 felt like a lifesaver! If you’d like to try your hand at an RPN calculator, I recommend the HP-15C simulator (http://hp15c.com). Besides downloadable versions for Windows and Mac OS X, there’s an online one as well, which even shows the internal registers if you enlarge the pop-up window.
Graphing Calculators
From the late 1970s onward, countless models of pocket calculators flooded the market. Ambitious companies such as Casio, Motorola, Sharp, and later Commodore—better known for its computers—joined the industry. The largest online calculator database, MyCalcDB, lists 668 brands and over 7,000 models.
The first calculator capable of plotting functions, a true graphical model, was Japanese: the Casio FX-7000G, introduced in 1985. The Americans weren’t about to be left behind—Hewlett-Packard soon came out with its own graphing model, the HP-28C, followed two years later by Texas Instruments’ TI-81.
Texas Instruments quickly became iconic in the graphing segment, especially with the TI-83 (1986), built around the Zilog Z80 processor, and the TI-89 (1989), which ran on the Motorola 68000 CPU. These machines could handle symbolic math, simplify systems of equations, find roots, integrate, differentiate, solve differential equations, perform calculations in arbitrary number systems, and work with complex numbers and matrices. Of course, they were programmable, and later models featured USB ports and SD card slots.
Back in the day, I used a Sharp EL-9300. I still have it, and it still works—the CR-2032 lithium battery has preserved the memory for two decades! At the time, it felt revolutionary to see integrals displayed on screen exactly as they appeared in textbooks. I loved that machine and wouldn’t sell it for anything. Today, well-preserved units fetch about €15–30 on eBay—roughly the same as their original price, except that back then it equaled a month’s net salary here.
Nowadays, I run a TI-89 emulator on my phone, purely out of nostalgia. If you’d like to try one of these calculators, this emulator is excellent and realistically simulates many models from the TI-73 to the TI-84 Plus. It requires the ROM file from the actual calculator, which you have to supply yourself. The same goes for phone emulators—the ROM isn’t included because the calculator’s internal program is protected by copyright and remains the intellectual property of Texas Instruments. If you own a real device, you can extract the ROM using tools from ticalc.org. If not, well… there’s always Google.
The ticalc.org site is a treasure trove of technical, scientific, financial, and even gaming programs for these machines, plus manuals, documentation, and forums—you could browse for days. Texas Instruments even produced data loggers and various sensors for school experiments, along with a teacher’s version of the calculator designed for overhead projectors.
From 2007 onward, Texas Instruments introduced the TI-Nspire series with color displays, but these models never really gained widespread popularity. The company started adding features reminiscent of tablets, but this was more of a desperate attempt—much like when Faber-Castell tried to save its slide rule business in the early 1970s by embedding tiny digital calculators into its rulers.
The era of programmable pocket calculators is fading; today’s motto is simple: “If you have the internet, you have everything.” There’s no need to cram computing power into a tiny handheld device—any smartphone with a browser can do the job. In 2009, Illinois-based Wolfram Research launched Wolfram Alpha, a computational knowledge engine that acts as a scientific search tool with advanced mathematical capabilities. It’s close to being able to solve high school math word problems autonomously, but using it requires a different kind of skill than consulting the traditional Four-figure tables. The key here is knowing how to ask the right question.
Of course, after a natural disaster or a societal collapse, when there’s no electricity and no internet, these tools will be useless. That’s when the old folks will reach for their slide rules again!
A Thought on Accuracy
Most people take for granted that whatever result the calculator displays is absolutely correct—but that’s not always true. Part of the error comes from the fact that calculators typically work with only 10–15 significant digits, so the last digit can always be off by ±1. In multi-step calculations, these tiny errors accumulate, making the final result less accurate. Manufacturers mitigate this by performing internal calculations with higher precision than what’s displayed. For example, Casio displays results with 10 digits (9 after the decimal point) but computes internally with 15-digit precision. Usually, this means only the last digit differs due to rounding, but certain operations can produce more noticeable discrepancies.
Take the sine function as an example. By definition, the sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse, so sin(30°) = 1/2. Its inverse, arcsine, returns the angle given the ratio: arcsin(1/2) = 30°. Obviously, arcsin(sin(30°)) = 30°, because we’re applying the function and its inverse. The same logic applies to cosine, tangent, and their inverses. So, we might expect that: arcsin(arccos(arctan(tan(cos(sin 9°))))) = 9°. Try it and see! Surprisingly, on a Casio, the result is 9.000000007.
In the old days, trigonometric functions and logarithms were calculated using printed tables or slide rules. Those tables—and the scales on slide rules—were based on values computed manually from the functions’ Taylor series expansions. But a Taylor series is an approximation, so its accuracy depends on how many terms you calculate. Fortunately, we can estimate the error size, which tells us how many terms are needed for a given precision. Producing a four-digit table was much easier than a seven-digit one, not to mention that manual calculations and typesetting introduced plenty of errors. In fact, faulty tables sometimes caused real disasters—bridges collapsed because engineers relied on incorrect data.
Engineers and scientists typically start from measurements with 0.1–1% accuracy, so calculating beyond four significant digits is often pointless. The push to mechanize scientific calculations wasn’t about getting more digits; it was about ensuring those four or five digits were correct. British mathematician Charles Babbage began working in 1835 on the idea of a punched-card-controlled calculating machine to automate the production of tables. He attempted to build several designs, but due to financial and personal setbacks, never completed them. Their remains are on display at the Science Museum in London. In 1991, his Difference Engine No. 2 was finally constructed using 19th-century materials and techniques—and it worked flawlessly.
Believe it or not, much of the math for the Manhattan Project was still done by hand, and the same was true during the Apollo program. Before John Glenn orbited Earth and Neil Armstrong walked on the Moon, human “computers” crunched the numbers. Margot Lee Shetterly’s book Hidden Figures (later a movie) tells the story of African-American women at NASA whose calculations were vital to America’s space successes. Highly recommended reading!
Modern calculators generally don’t use tables or Taylor series (though some do in special cases). Instead, they employ a clever iterative algorithm called CORDIC (COordinate Rotation DIgital Computer). This method repeatedly rotates a vector in the plane in small steps until the desired value is reached. Each rotation involves multiplying by a 2×2 matrix, which in binary can be implemented with simple shifts and additions—no real multiplication needed, and no hardware multiplier in the chip.
You might guess that rotating a vector can compute trig functions via coordinate geometry, but CORDIC can do much more: logarithms, square roots, and even multiplication and division. A calculator equipped with CORDIC hardware can compute almost anything. The method isn’t the fastest, but it’s extremely hardware-efficient, requiring fewer transistors and less power—perfect for pocket calculators. Even Intel’s math coprocessors for PCs (8087 through 80487) used it. The concept was first developed by Jack E. Volder between 1956 and 1959 for aircraft navigation systems, but similar ideas go back to Henry Briggs in the early 1600s.
In the early 2000s, calculator collector Mike Sebastian came up with a simple method for testing calculators: the already mentioned formula
arcsin(arccos(arctan(tan(cos(sin 9)))))))) = 9?
Although it contains only trigonometric functions, it is generally suitable for assessing the quality of a calculator. Part of the error comes from the formula itself and is independent of the calculator being tested. Mike notes that about five significant digits are lost due to the formula, meaning that for a Casio performing calculations to 15 digits, 10 significant digits remain. The result depends heavily on how many digits the calculator uses in its internal calculations, but even among calculators with the same precision, there can be notable differences because much depends on how the CORDIC algorithm is implemented in the device.
I gathered all the calculators I could find—including my father’s old TK-1023—and ran Mike’s formula on them. I also included mobile and PC applications in the test. Wolfram Alpha is a web-based tool that combines a search engine, mathematical software, knowledge base, and AI. Although available as a mobile app, all computations are performed on the provider’s servers, giving it such an advantage that it’s hardly fair to compare it with ordinary pocket calculators. HiEdu is a mobile app and a good example of how easily a phone can handle a powerful calculator. These apps don’t use the CORDIC algorithm but more accurate methods, which is evident in their results.
The results are shown in the table below. Surprisingly, the “low budget” calculators performed quite well. Mike also observed that more and more cheap Chinese models return a result of 9—or very close to it. It seems the Chinese are using more refined algorithms. True, these calculators generally have the same feature set as the simplest models from big-name brands but cost half as much. The issues likely lie in the mechanics: the quality of the keyboard and casing, and how durable they are. And, of course, speed—which I didn’t measure, but, for example, the Casio FX-991 CE X feels noticeably faster than the Olympia LCD8110.
The “Maximum Error” shown next to each test result indicates the difference from 9 including ±1 digit of display rounding, to account for output precision. This matters because 9.00 and 9.0000 both differ from 9 by zero, but the latter is much better: the hidden error could be ±0.01 in the first case and ±0.0001 in the second. As expected, the iconic early-1990s Texas Instruments TI-83 Plus performed well, the Sharp EL-9300 a little worse, and the two modern Casio models landed in between. Older, smaller calculators showed significant errors. The LED-display Híradástechnika TK-1023 was downright pathetic—but then, it’s nearly half a century old. According to Mike, calculator accuracy improved noticeably in the late ’80s, and the large errors are typical of earlier models.
If you’re interested in these old but often still functional devices, check out the Virtual Calculator Museum (www.arithmomuseum.com), where you can browse freely and even learn about vintage Hungarian calculators. And if you’re curious, try testing any calculator at hand with Mike’s formula—it’s even a good idea before buying one!
| Calculator / Software | Display Digits | Internal Precision (digits) | Built-in Functions | Trig Test Result | Max Error |
|---|---|---|---|---|---|
| Wolfram Alpha | 15 (max. 2000) | 20 (max. 200) | n/a | 9.00000000000000 | ≤ 1.00E-14 |
| Microsoft Excel | 15 (max. 30) | 15 | n/a | 9.00000000000001 | ≤ 2.07E-14 |
| Calc Pro HD (Windows App) | 14 | n/a | n/a | 8.9999999997535 | ≤ 2.47E-10 |
| HiEdu (Android App) | 10 (max. 30) | 10 (max. 22) | n/a | 9.000000000 | ≤ 1.00E-09 |
| Auchan 92ES Plus | 10 | n/a | 250 | 9.000000001 | ≤ 2.00E-09 |
| Deli D82ES Plus | 10 | n/a | 252 | 9.000000001 | ≤ 2.00E-09 |
| TESCO no name “12-digit” | 12 | n/a | 56 | 8.999999998 | ≤ 3.00E-09 |
| Olympia LCD8110 | 10 | 12 | 229 | 8.999999998 | ≤ 3.00E-09 |
| Texas Instruments TI–83 Plus | 10 | 14 | n/a | 8.999999997 | ≤ 4.00E-09 |
| Casio FX–220 Plus | 10 | 15 | 181 | 9.000000007 | ≤ 8.00E-09 |
| Casio FX–82MS | 10 | 15 | 240 | 9.000000007 | ≤ 8.00E-09 |
| Casio EF–991ES | 10 | 15 | 417 | 9.000000007 | ≤ 8.00E-09 |
| Casio FX–570ES Plus | 10 | 15 | 417 | 9.000000007 | ≤ 8.00E-09 |
| Casio FX–991 CE X | 10 | 15 | 668 | 9.000000007 | ≤ 8.00E-09 |
| Sharp EL–531THB | 10 | 14 | 272 | 9.000000099 | ≤ 1.00E-07 |
| Sharp EL–9300 | 10 | 14 | 437 | 9.000000099 | ≤ 1.00E-07 |
| Sharp EL–W506TB | 10 | 14 | 640 | 9.000000099 | ≤ 1.00E-07 |
| Sharp EL–501X-VL | 10 | 11 | 131 | 8.999981534 | ≤ 1.85E-05 |
| Sharp EL–514 | 10 | 11 | 38 | 8.999695445 | ≤ 3.05E-04 |
| Sharp EL–531H | 8 | 11 | 32 | 8.9899184 | ≤ 1.01E-02 |
| Texas Instruments TI–30 | 10 | 10 | 15 | 9.114640577 | ≤ 1.15E-01 |
| Híradástechnika TK–1023 | 8 | 8 | n/a | 7.1252182 | ≤ 1.87E+00 |
Updated on August 2, 2025:
Accuracy in the Early Days
In 2021, at the Researchers’ Night (Kutatók Éjszakája) event, there was a presentation that also touched on the accuracy of calculators.
In the early days of computing, pocket calculators were alarmingly inaccurate. Here are a few examples of how they calculated ∛8–2. Everyone knows by heart that ∛8 = 2, since 2³ = 2·2·2 = 8, so ∛8–2 = 0, but these calculators showed something entirely different:
