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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Casio Calculators
Casio fx-991es  =  fx-570es  =  fx-115es
and their successor, the Casio fx-991ex.

A review of affordable best-sellers.

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Related articles on this site:

Related Links (Outside this Site)

NCEES-Approved Calculators:
Casio FX-115 series, HP-33s and HP-35s, TI-30X and TI36X series.
Casio fx-115 ES PLUS Review  by  Eddie Shore.
Casio fx-991EX Classwiz Review  by  Eddie Shore.
Keisan high-precision online calculator   " ke!san "   courtesy of Casio.

Videos :

Song :   We add up to Infinity  (2:57)   by  Evyn Charles  (2014).

$17.99   (street price $13)   manual
 fx-115ES PLUS calculator (2012) 
 by Casio

Casio fx-115ES Plus
fx-991es Plus  &  fx-570es  Plus

(46 keys + D-pad)  school calculators
with 2-dimensional Input/Output
and/or up to 4 lines of regular text
 Natural Textbook Display

 Natural V.P.A.M.

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)

$35.65   (street price $21)   manual
 fx-991EX calculator (2015) 
 by Casio

Casio fx-991EX
and  fx-570ex

(46 keys + D-pad)  school calculators
with 2-dimensional Input/Output
and/or up to 6 lines of small text
 Natural Textbook Display

 High Resolution

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

92.2 g   (127.8 g)
80 x 162 x 13 mm   (83.5 x 164 x 15.6)
Weight with LR44 battery (and cover)
Dimensions  (with cover)
93.5. g   (130.5 g)
76.8 x 165 x 12 mm   (81.5 x 167 x 17.3)

(2015-12-03)   Casio's EX Series of Scientific Calculators
Hardware improvement over the ES Series, at a higher cost.

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.

(2012-10-08)   Casio's ES Series of Educational Scientific Calculators
Many features, without graphing or programmation, for less than $20.

Although outdated calculators in the Casio ES series are still available, Casio is mostly marketing one calculator powered in two different ways:

  • The  fx 570es  is powered by one  AAA (R03) battery which would last for about two years  (17000 hours)  if the calculator was just left "on", displaying a flashing cursor.
  • The other version comes with "dual power":  a solar cell backed up by one small LR44 (GPA76) button battery, which is good for no more than 3 years.  That model is labeled either  fx 991es  (in Europe)  or  fx 115es  (in North America).

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:


The fx-95es Plus has "274 functions" and a menu with 6 modes:


The standard "403 functions" versions the  fx-115es, fx-991-es  or  fx-570es  have 8 modes:


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:


(2012-11-09)   MODE 1: Ordinary Computations
The Basics.

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

(2012-10-14)   MODE 2: Complex Numbers
Manipulating complex numbers.

What complex numbers really are.

Tutorial Videos:   Complex Numbers

(2012-10-14)   MODE 3: Statistics
Elementary statistics and regressions.

Tutorial Videos:   Basics of Statistics   |   Regressions

(2012-10-08)   MODE 4: Base-N Integer Arithmetic
In radix ten (DEC) sixteen (HEX) two (BIN) or eight (OCT).

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)

(2012-10-09)   MODE 5: Algebraic Equations (degree 3 or less)
Linear, quadratic and cubic equations.

Tutorial Videos:   Solving Quadratics

(2012-10-09)   MODE 6: Matrix Manipulations (dimension 3 or less)
Three special variables have values that are matrices: MatA, MatB, MatC

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

(2012-11-24)   MODE 7: Tabulate a function  (or a pair of functions).

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).

(2012-10-25)   CALC :   Repeated Calculations
Work out a formula for many values of the input variables.

Tutorial Videos:   Using the CALC feature

(2012-10-19)   CONST :   Fundamental Scientific Constants
Numerical values of 40 physical constants, in SI units.

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

Values of Some Physical Constants
Description & Symbol Casio fx-115es PlusCODATA 2010Unit
Mass of the Protonmp01 1.672621777 10-27 1.672621777(74) 10-27kg
Mass of the Neutronmn02 1.674927351 10-27 1.674927351(74) 10-27kg
Mass of the Electronme03 9.10938291 10-31 9.10938291(40) 10-31kg
Mass of the Muonmm04 1.883531475 10-28 1.88353475(96) 10-28kg
Bohr radiusa005 5.291772109 10-11 5.2917721092(17) 10-11m
Planck's constanth06 6.62606957 10-34 6.62606957(29) 10-34J/Hz
Nuclear magnetonmN07 5.05078353 10-27 5.05078353(11) 10-27J/T
Bohr magnetonmB08 9.27400968 10-24 9.27400968(20) 10-24J/T
Dirac's constant h-bar 09 1.054571726 10-34 1.054571726(47) 10-34J.s/rad
a10 7.29735257 10-3 7.2973525698(24) 10-3  
1/a 2hce0 / e2 137.035999074(44)
Classical electron radiusre11 2.817940327 10-15 2.8179403267(27) 10-15m
Compton wavelength lc12 2.426310239 10-12 2.4263102389(16) 10-12m
Gyromagnetic ratio
of the proton
gp13 267522200.5 267522200.5(63) rad/s/T
gp / 2p 2 mp / h   42.5774806(10) MHz/T
Compton wl for proton lc,p14 1.321409856 10-15 1.32140985623(94) 10-15m
Compton wl for neutron lc,n15 1.319590907 10-15 1.3195909068(11) 10-15m
Rydberg's constantR¥16 10973731.57 10973731.568539(55)1/m
Hartree energyEh 2hc R¥  =  a2 me c2 4.35974434(19) 10-18J
Atomic mass unitu17 1.660538921 10-27 1.660538921(73) 10-27kg
Proton magnetic momentmp18 1.410606743 10-26 1.410606743(33) 10-26J/T
Electron magn. momentme19 -9.2847643 10-24 -9.28476430(21) 10-24J/T
Neutron magn. momentmn20 -9.6623647 10-27 -9.6623647(23) 10-27J/T
Muon magnetic momentmm21 -4.49044807 10-26 -4.49044807(15) 10-26J/T
Faraday's constantF22 96485.3365 96485.3365(21)C/mol
Charge of a protone23 1.602176565 10-19 1.602176565(35) 10-19C
Avogadro numberNA24 6.02214129 1023 6.02214129(27) 1023 1/mol
Boltzmann's constantk25 1.3806488 10-23 1.3806488(13) 10-23J/K
Molar volume, 0°C, 1 barVm26 0.022710953 0.022710953(21)m3/mol
Molar volume, 0°C, 1 atm 273.15 R / 101325 0.022413968(20)
Ideal gas constantR27 8.3144621 8.3144621(75)J/K/mol
Einstein's constant (c)c028 299792458 299792458m/s
1st radiation constantc129 3.74177153 10-16 3.74177153(17) 10-16W.m2
2nd radiation constantc230 1.438777 10-2 1.4387770(13) 10-2m.K
Stefan's constants31 5.670373 10-8 5.670373(21) 10-8 W/m2/K4
Electric constante032 8.854187817 10-12 8.85418781762039 10-12 F/m
Coulomb's constant1 / 4pe0   8.9875517873681764 109 m/F
Magnetic constantm033 1.256637061 10-6 4p 10-7H/m
Ampere's constantm0 / 4p  10-7 H/m
Quantum of fluxF034 2.067833758 10-15 2.067833758(46) 10-15 Wb
Normal gravityg35 9.80665 9.80665 N/kg
Conductance quantumG036 7.748091735 10-5 7.7480917346(25) 10-5 S
Z of vacuum = m0 cZ037 376.7303135 376.730313461770655468... W
Ice point  =  0°Ct38 273.15273.15K
Newton's constantG39 6.67384 10-11 6.67384(80) 10-11 N.m2/kg2
Normal Pressureatm40 101325 101325Pa
Standard Pressurebar  100000Pa
The  standard convention  used here is that the digits between parentheses at the end of a measured quantity
indicate its experimental uncertainty  (one standard deviation)  expressed in units of the least significant digit.

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:

  • 2p re   =   a lc   =   a2 2p a0   =   a3/ 2R¥
  • me lc   =   mp lc,p   =   mn lc,n   =   h/c  =  2.210218901(98) 10-42 kg.m
  • me mB   =   mp mN   =   e h / 4p   =   8.44805 10-54 J.kg/T
  • u NA   =   0.001 kg/mol
  • e NA   =   F  Klaus von Klitzing 
 (b. 1943)  Niels Bohr 
 (1885-1962)  Max Planck 
 (1858-1947)  Amedeo Avogadro 
  • k NA   =   R
  • c1   =   2p h c2
  • c2   =   h c / k  
  • s     =   ( 2 p 5 k 4 /  ( 15 h 3 c 2 )   =   5.6704 10-8 W/m2/K4
  • e0 m0 c2   =   1   (electromagnetic wave propagation).
  • F0   =   h / 2e   (magnetic flux quantum).
  • G0   =   2 e2/ h   =   4a / Z0   (conducance quantum, Landauer 1957).
  • Z0   =   (m/ e)½  =   mc   (characteristic impedance of the vacuum).

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

(2012-10-19)   CONV :   Physical Units and Conversion Factors
40  predefined unit conversions  (a few inaccuracies and one mistake).

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:

  • Odd-numbered conversions translate non-metric units into metric ones.
  • Even-numbered conversions translate metric units into something else.

Conversion Factors
Standard quantity Casio fx-115es PlusExact ValueUnit
Int'l  inch (1959)1 in01 2.542.54cm
Int'l  foot (12" = 12 in)1 ft03 0.30480.3048m
Int'l  yard (3' = 3 ft)1 yd05 0.91440.9144m
Gunter link (0.22 yd)7.92 in  20.1168cm
Gunter chain (22 yd)66 ft  20.1168m
furlong (220 yd)1 fur  201.168m
statute mile (8 fur)1 mi07 1.6093441.609344km
nautical mile (nmi)1 NM09 18521852m
acre1 acre11 4046.8564046.8564224m2
Winchester gallon (US)1 gal13 3.7854123.785411784L
fluid ounce (gal / 128)1 fl oz  29.5735295625mL
Imperial gallon1 gal (UK)15 4.546094.54609L
Imp'l fluid ounce = 1 gal(UK) / 160  28.4130625mL
parsec1 pc17 3.085678 1013 3.08567758149... 1013km
astronomical unit1 au (p / 648000)  pc 149597870700m
light-year   9460730472580800m
kilometer per hour1 km/h19 5/185/18 = 0.277777...m/s
knot  (NM/h)1 kt  463/900 = 0.514444...m/s
mile per hour1 mph  0.44704m/s
meter per second1 m/s20 18/518/5  =  3.6km/h
grain  =  1 lb /70001 gn  64.79891mg
carat  =  1 g / 51 ct  200mg
troy ounce (480 gn)1 ozt  31.1034768g
ounce (avoirdupois)1 oz21 28.3495228.349523125g
pound  =  16 oz1 lb23 0.45359240.45359237kg
normal pressure1 atm25 101325101325Pa
standard pressure1 bar  100000Pa
millimeter of mercury1 mmHg27 133.3224133.322387415   Pa
Torr  (101325 Pa / 760)1 Torr  133.322368421...Pa
horsepower1 hp29 0.74570.74569987158227022kW
kgf / cm2 31 98066.598066.5Pa
kilogrammeter  (kgf.m)1 kgm33 9.806659.80665J
lbf / in21 psi35 6.8947576.89475729316836...kPa
Fahrenheit (°F)
vs. Celsius (°C)
°F37 (F+40)  =  1.8 (C+40)°C
calorie 1 cal404.1858 4.184J
15° calorie4.1855 (3)
American calorie4.1858
IST calorie4.1868

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.

(2015-11-19 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!

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