A review of affordable best-sellers.
Song : We add up to Infinity (2:57) by Evyn Charles (2014).
LCD: 96 by 31 dot matrix + annunciators
Normal font: 5 by 7 (4 lines of 16)
Menu font: 5 by 6 (eight 5/6 char. labels)
Small font: 4 by 6 (e.g., exponents)
Matrix computations up to 3 by 3
Numerical integration and differentiation
Handles fractions and decimal overbar
No programmation, graphing or CAS
9 variables: A,B,C,D,E,F,X,Y,M
Internal precision: 12 digits (10 shown)
11½ or 12-digit predefined constants
"417 functions" advertised ("Plus" models)
Codes for units & constants shown inside cover
Tabulates 2 formulas at once ("Plus" only)
35ドル.65
(street price 21ドル)
manual
fx-991EX calculator (2015) by Casio
LCD: 192 by 63 dot matrix + annunciators
Normal font: 9 by 14 (4 lines of 17)
Small font: 5 by 9 (6 lines of 32)
Annunciator for solar power (991EX)
Nice
Matrix computations up to 4 by 4
Nice
Numerical integration and differentiation
Handles fractions and decimal overbar
No programmation, graphing or CAS
9 variables: A,B,C,D,E,F,X,Y,M
Internal precision: 13 digits (10 shown)
Nice
11½ or 12-digit predefined constants
Poor
"552 functions" advertised (no GCD
Poor )
Constants & unit conversions via menus
Tabulates 2 formulas at once
Spreadsheet (5 columns & 45 rows)
Nice
Displays QR codes for smartphones
Released in the Spring of 2015, the Casio fx-991EX (above right) improves in speed and screen resolution over its predecessors of the ES series.
A larger memory allows a greater functionality in the EX series (4 by 4 matrices are now supported, instead of 3 by 3 for ES calculators) and a slightly better precision (1 more digit). However, some beloved functions have been dropped: The GCD function is absent from the fx-991EX (552 functions) but survives in the German fx-991 DE X (696 functions).
Besides a more intuitive interface and the support of "engineering symbols" (prefixes) there are two noteworthy innovations: Spreadsheet calculations and the generation of QR graphics readable by smartphones.
Although outdated calculators in the Casio ES series are still available, Casio is mostly marketing one calculator powered in two different ways:
Those calculators have been advertised as featuring "403 functions". Their upgraded "PLUS" versions (2012) offer "417 functions".
Such calculators are best viewed as several specialized calculators combined into one unit. The specialization is determined by the number of the "mode" selected in response to the menu that pops up when you hit the MODE key (located at the top right of the keypad, next to the ON button). The choice for normal computation is 1.
The downgraded calculators of the series, which are advertised to offer "249 functions" (fx-82es, fx-83-es, fx-85-es, fx-300es and fx-350es) offer only 3 modes. So do the slightly better "252 functions" fx-82es Plus, fx-85es Plus and fx-350es Plus:
1:COMP 2:STAT 3:TABLE
The fx-95es Plus has "274 functions" and a menu with 6 modes:
1:COMP 2:STAT 3:EQN 4:TABLE 5:INEQ 6:RATIO
The standard "403 functions" versions the fx-115es, fx-991-es or fx-570es have 8 modes:
1:COMP 2:CMPLX 3:STAT 4:BASE-N 5:EQN 6:MATRIX 7:TABLE 8:VECTOR
The PLUS version ("417 functions") of those same calculators flaunt three additional choices (for a grand total of 11 modes) that appear on an extra menu that comes up from the aforementioned main MODE menu when you press the lower part of the big round navigation button:
1:INEQ 2:VERIF 3:DIST
Tutorial Videos: Thanks for the memories (9 variables: A,B,C,D,E,F,X,Y,M) by T. Jefferson Definite integration by T. Jefferson
What complex numbers really are.
Tutorial Videos: Complex Numbers
Tutorial Videos: Basics of Statistics | Regressions
At a time of my life when I was doing assembly programming daily, I once spent a lot of money on a specialized calculator that was doing only that!
To switch between bases, use the four keys at the left of the third row, which are clearly labeled in blue for DECimal, HEXadecimal, BINary and OCTal.
For the upper hexadecimal digits A,B,C,D,E,F use the keys of the fourth rowm that are so labeled (in red).
The display shows the current computation on the first line, the radix (Dec, Hex, Bin, Oct) on the second and the (last) result on the third line.
The basic functionality of the calculator is available for integer arithmetic this way (including STO, RCL, M+ and M-). Division is performed by discarding the remainder if there's any. Negative results are understood to be in two's complement with a 32-bit word-size. They are shown with a negative sign in decimal or binary but are displayed "raw" in hexadecimal or octal (where the negative of 1 is, respectively FFFFFFFF and 37777777777). In other word, arithmetic is performed modulo the thirty-second power of two.
Two capitalized bitwise unary operators (Not, Neg) and four lowercase bitwise binary operators (and, or, xor, xnor) are provided in the specific menu that pops up when taping Shift-3 in this mode.
Tutorial Video: Base-n (Spanish)
Tutorial Videos: Solving Quadratics
This is clearly for educational purposes only. The matrices are cleared when you switch modes and you cannot do anything with the results. Matrix elements can only be real numbers (complex values are not allowed).
Tutorial Videos: Matrix Multiplication | Inverse of a Matrix | Matrix Calculation
You are prompted for an expression for f (x) and g (x). (Press "=" when prompted for g if you only want to tabulate one function.) Then you're asked for a starting value of x, an ending value and an incremental step.
The classic version of this calculator only allowed one function to be tabulated at a time. If that's all you want from your "plus" calculator, you can avoid any prompting for "g" via the option labeled "5: TABLE" in the second screenful of the SETUP menu (to access that, press Shift-SETUP, next to the ON button, followed by a down arrow from the big round navigation button).
What will be displayed as a result is a table with 3 or 4 columns, showing a line number, a value of x and the corresponding values of f (x) and (possibly) g (x). Use the navigation button, down or up, to view many lines.
Tutorial Videos: Graphing Functions with the Casio fx-115 ES (not "plus" in this demo).
Tutorial Videos: Using the CALC feature
Calculators in the Casio ES series provide the values (in SI units) of 40 physical constants. Those are obtained via the CONST key (Shift-7) followed, as prompted, by a two-digit identifier from 01 to 40 (according to the menu printed inside the calculator's protective cover). This combination of 3½ keystrokes results in a symbol that can be evaluated by itself or as part of any algebraic expression.
As of this writing, the "fx-115es Plus" uses the latest (2010) self-consistent values of physical constants recommended by CODATA. The links to the NIST database provided below may show slightly different values after 2015 or so (the CODATA recommendations are regularly revised with a periodicity of about 4 years and the new values become widely available in the second part of the year following the nominal date).
The values of a few physical quantities are khown exactly de jure because of the way modern units are defined (for example, the speed of light is exactly 299792458 m/s by virtue of the SI definition of the meter). Otherwise, the most precisely known physical quantity is the following dimensionless ratio, expressing the value of the magnetic moment of the electron using the Bohr magneton as a unit.
me / mB = -1.00115965218076(27)
Neither me nor mB is known that precisely (see below) but their ratio is.
The runner-up is the frequency of the hyperfine transition of protium, which stood for nearly 40 years as the most precise measurement ever performed:
1420.4057517667(9) MHz
The yellow highlighting for the electric constant indicates that the built-in fx 115es value should have been rounded up to 8.854187818 pF/m. We have similarly highlighted Casio's dubious choice of the symbol c0 instead of c for the speed of light in a vacuum (Einstein's constant) which they obscurely grouped with the historical radiation constants.
The Casio engineers may have been afraid of a possible confusion with the variable "C", although they do display the latter in UPPER case. Faced with the same dilemma, competing manufacturers simply made sure to use a smaller font for Einstein's constant than for the names of variables (which those other folks display in lower case, by the way). That's a much better solution than Casio's artificial subscript which nobody else ever uses (especially in calculators intended to contribute to the educational experiences of their users).
Arnold Sommerfeld's fine-structure constant (a) is the only listed constant to be dimensionless. Its numerical value would be the same in any coherent system of physical units and it remains a mystery:
a = m0 c e2 / 2h = e2 / 2hce0 = 1 / 137.035999...
The following lengths form a geometric progression of common ratio a :
1 / 2R¥ 2p a0 lc 2p re
That's the first of many noteworthy relations between the above constants:
Electron Magnetic Moment (in Bohr Magnetons) and Fine Structure Constant
by David Hanneke, S. Fogwell, Gerald Gabrielse
g/2 = 1.00115965218073(28)
and a = 1 / 137.035999084(51)
(2008)
Precision tests of QED
Calculators in the ES series don't allow physical dimensions to be attached to numbers (it's a delicate issue even for top-notch calculators by HP or TI).
What they do is provide the ability to perform 40 predefined unit conversions. Those are obtained via the CONV key (Shift-8) followed, as prompted, by a two-digit identifier from 01 to 40 (according to the menu printed inside the calculator's protective cover).
Regrettably, some of Casio's conversion factors differ from the precise legally enacted values. In the table below, we have highlighted in yellow slightly inaccurate conversion factors which should have been stored with full precision (or, at least, better accuracy) given the capabilities of the calculator. Highlighted in red is a plain mistake (Casio uses a deprecated value for the "15° calorie" instead of the proper thermochemical conversion factor for published data involving calories). Only the conversion factors that can be specified exactly with finitely many decimals are tabulated below (for the unlisted reciprocal conversions, usually corresponding to even indices, exact conversion factors would entail repeating decimals or fractions).
The most obvious nontrivial ommission is the pound-force (lbf) whose lack of conversion by NASA once caused a costly probe to crash on Mars:
1 lbf = (0.45359237 kg) (9.80665 N/kg) = 4.4482216152605 N
Also missing is the British thermal unit (Btu) whose official IST value is:
1 Btu = (0.45359237 kg) (2326 J/kg) = 1055.05585262 J
With the sole exception of the last pair of conversions (between calories and joules, with an erroneous conversion factor) the rule is:
According to Casio's documentation [2015 user's guide for the fx-991EX] their questionable conversion factor for the calorie is intended for "values at a temperature of 15°C". Actually they chose an outdated value which is rarely retained, although it appears on page 551 of The Encyclopedia of Physics by Robert M. Besancon (1966, 1974, 1985, 1990). It's also what gives the dubiously rounded value of 105480400 J (100000 59°F-Btu) still quoted commercially. Worthless in a scientific context.
This dubious choice may have been influenced by competing definitions of the calorie. It's just one digit off from the IST "definition" which led to another mistake (perpetuated by other calculator manufacturers for thirty years). The bogus IST conversion factor of 4.1868 J/cal, which did appear in a few misguided publications, is ultimately just an offshoot of the unrelated definition of the Btu (unused in modern science and unknown in most parts of the World). Since 1935, the scientific community has used almost exclusively the thermochemical conventional definition of exactly 4.184 J to the calorie, for all results published in calories.
The numbers highlighted in yellow are needlessly inaccurate (but not nearly as bad as what we just discussed). Let's give just one example:
1000 mmHg is the pressure felt at unit depth (1 m) in a liquid having the conventional density of mercury (13595.1 kg/m3, irrespective of temperature) under a standard gravitational field (9.80665 N/kg or m/s2).
1 mmHg = (0.001 m) (13595.1 kg/m3 ) (9.80665 N/kg) = 133.322387415 Pa
On the other hand, 760 Torr is equal to one atmosphere of 101325 Pa.
1 Torr = 101325 Pa / 760 = 133.322368421052631578947 Pa
At Casio's modest level of precision, 1 Torr = 1 mmHg = 133.3224 Pa. However, that approximation isn't the correct definition of the mmHg.
(2015年11月19日 update) The fx-991EX model introduced in 2015 is ten times more accurate than the fx-991ES class of calculators discussed above. However, the above rounded conversion factors were not revised at all, which makes them totally unacceptable at the new precision!