Best of breed, 2008.

I wanted to have a good non-graphing calculator, either non-programmable or minimally so, to supplement my TI-89 Titanium, in case a teacher had a problem with such things. I wasn't sure which non-graphing, scientific calculator was the best, so I bought several TI's, Sharps and Casios to try them all out.

The clear winner, based on several different criteria, is the Casio fx-115ES, which also goes by the name fx-991ES for models sold in Asia.

The power supply is a solar panel supplemented by one LR44 button battery.

Features.

All the usual scientific calculator functions are present, plus a few that you don't often see, such as log to any base, absolute magnitude, mixed and improper fractions with conversion to decimal, Reimann sums, numerical integration and differentiation.

The integral function doesn't use Simpson's rule, but rather Gauss-Kronrod quadrature.

The base of a logarithm can itself be a function, yep.

The Casio fx-115ES will hold equations (for example, Y=X-AsinX for multiple calculations of Y for different values of X, so that points (X,Y) can be generated quickly by repeatedly pressing the CALC button and entering X when prompted. The equals sign (=) used with functions is in the upper left corner of the calculator panel as ALPHA CALC; it is not the equals sign at the lower right corner.

It will also SOLVE functions in reverse. For example, given this same function with the calculator in radian angle mode, pressing 2nd SOLVE will prompt you to input Y and A, after which the calculator would solve for X. If Y=4 and A=0.8, then X=3.626850697. Note that solving a transcendental function for the value of the transcendental variable is not a problem for the Casio fx-115ES.

The CALC and SOLVE features give calculators having them a tremendous advantage in ease of use over calculators that don't have them. Entered functions, integrals, and derivatives can be edited by using the directional button (the big circular one) and the DEL key.

The Casio fx-115ES provides 40 physical constants (though I question the order in which they are indexed) as well as 40 units conversions. It will also convert from decimal degrees (or decimal hours) to DD.MMSS format and the reverse. And you can put a displayed number into engineering notation, if you want to.

The statistics mode offers seven two-variable regression models (linear, quadratic, logarithmic, e exponential, ab exponential, power, and inverse), plus the usual single variable statistics function is also available.

The vector & complex calculations can be swapped between rectangular and polar coordinates.

The available alternate bases, besides decimal, are hexidecimal, binary, and octal. The math capabilities of the fx-115ES are truncated in any of the bases other than decimal.

Calculation accuracy.

The accuracy of calculation is excellent: significantly better than most other calculators that might be considered as being in the same class.

One standard test is to compute, in radian angle mode, the tangent of 355/226. The nearly exact answer is -7497258.18532558711290507183. Relative errors on this calculation for various calculators are most often in the 1E-3 to 1E-7 range. The fx-115ES gives -7497258.44001983, which has a relative error of -3.4E-8.

(The most accurate calculator that I know on this calculation is the BASE 8 DG1000, a very "el cheapo" calculator from Dollar General. It has lousy ergonomics, but costs only $5. It gave -7497258.1853255631, which has a relative error of only -4.3E-11. This is, however, very unusual. The only other calcuator to return a more accurate answer than the Casio fx-115ES is the Palm Pilot Rom Calculator IEEE-754, which have a relative error of 8.4E-10.)

Another standard test is evaluating sin(1E8). The nearly exact answer is 0.9316390271097260. The Casio fx-115ES returns 0.9316391026, which has a relative error of 8.1E-8. Other calculators in the same class commonly have relative errors ranging from 1E-2 to 1E-7. (However, the BASE 8 DG1000 returns 0.9316390271097260, which has no apparent error at all.)

Room for improvement.

First. There is no good reason why a calculator can't use any positive base (other than 1) directly with any of its functions. Nor is there any reason why alternate base results can't be displayed in fractional or post-dot digital (shouldn't say "decimal" if the base isn't 10) form. I've seen e-calculators for computer desktops use the full-blown any-base-with-fractions calculation mode, and if it can be done for the PC, then it can also be done for calculators. A scientific calculator ought to be able to display fractional complex numbers in base pi.

Second. The matrix operations are restricted to a maximum size of 3x3 matrices. Students who apply the theory of relativity need at least 4x4 matrix operations. And bigger, of course, would be better.

Third. The factorial key should be replaced, or, better, supplemented by a gamma function key.

Fourth. The 40 physical constants aren't ordered such that the most often used constants occur first. A sensible ordering of the constants would be

1. speed of light
2. gravitational constant
3. Planck constant
4. Planck constant divided by 2 pi
5. Boltzmann constant
6. Stefan-Boltzmann constant
7. atomic mass unit
8. electric charge unit
9. Avogadro's number
10. The permittivity of free space
11. The permeability of free space

And then the rest. Instead, the fx-115ES orders the above constants as numbers 28, 39, 6, 9, 25, 31, 17, 23, 24, 32, 33, respectively, which isn't easy to remember.

Also, the gravitational constant should be updated at least to the CODATA recommended value for 2006. The built-in value has a relative error of about one tenth of one percent.

Fifth. Among features not included that would be nice to have are:

1. Lagrange Interpolating Polynomials