wolframalpha.com |

#1: Wolfram Alpha

Accessible at wolframalpha.com for free, Alpha is one of the best mathematics services I've used, even regardless of price. This tool can conduct nearly any calculation for you, be it in arithmetic, trigonometry, calculus, or linear algebra, and it can even manipulate symbolic expressions. Moreover, it's capable of making 2D and 3D graphs, with almost all usual coordinate systems and graph types supported. It can convert units, factorize integers, solve algebraic equations with both symbolic and numeric results, and still more, with some functions I probably haven't even seen before.

But most of all, Alpha combines all these functionalities with Wolfram Research's work on heuristic language processing, letting it "think" about what you're asking it. For example, I can type in "Differentiate sine of sine of x" verbatim, and Wolfram Alpha can figure out what that means, and calculate and spit back out the correct answer of cos(sin(x))cos(x). The ability to be lax in what you're asking of the program and still get the answer back is incredibly relaxing, and the language processing also helps you access Wolfram's encyclopedia-like database of mathematical, scientific, and even general information.

Some shortcomings of the program are that, while the language processing is incredibly strong, some times it can mess up and interpret only half of the input, just because you missed a parenthesis somewhere in a long mathematical expression. Additionally, it can be frustrating to do a large number of calculations with the service when you're somewhere with shoddy internet, since it's still hosted on the internet. Some features are restricted to Pro members of the service, such as exporting graphs, and the $2.99 price tag on the app version of the service, while still an absolute steal, is incongruous with the free nature of the internet-accessible version, especially since it still has to communicate with Wolfram's servers to work. Finally, since it's a web service, it's not going to be of much help during in-class tests, unless your professors are much more lenient than those I've had. Still, for homework or for just exploration of mathematics, Alpha is an utter godsend.

desmos.com |

#2: Desmos

Like Alpha, this isn't a program or physical calculator, but an online service available at desmos.com, and also like it it's free to use. Though its calculation functions are limited, the graphs it produces are top-quality, better in some cases than those produced by Alpha, much less the black-white pixelated ones on the screen of a graphing calculator. Multiple graphs of varying types can be plotted on the same axes, and non-axis variables can be introduced and modified by sliders, in a very easy-to-use and intuitive way.

Multiple graphs can depend on the same parameters, which lets you visualize how complicated systems of equations in several variables work incredibly easily. For the statistically inclined, the user can also import data and form regression graphs. Some of Desmos' weaknesses are its lack of 3D plotting capability, and of course that it's again a website and isn't going to be very useful on class tests. Nevertheless, Desmos lets you very easily make quite beautiful graphs, all for free.

Casio.com |

#3: Casio fx-115ES Plus

If you're actually in need of a calculator for use in class and on tests, this is one that I've found to be very robust. The 115ES, while being a scientific and not graphing calculator, nevertheless comes with nearly all the functionality of a graphing calculator that students typically use. It's capable of equation solving, application of most elementary functions, numerical differentiation and integration, iterated summation and multiplication, matrix manipulation, one-variable statistics and regressions of many types, application of PDF, CDF, and inverses of normal, binomial, and other distributions, and much more.

The display is pixel-based rather than seven-segment based in both the input and output fields, and almost all the mathematical structures are displayed symbolically: you can type in and read off expressions on the 115ES just as you would on a piece of paper. And even with all these functions, the calculator is solar-powered, so you don't have to worry about the battery dying in the middle of a test! It is necessary to note again that, since it's not a graphing calculator, it doesn't have actual graphing or programming functions. However, since it's infrequent that tests actually require you to make a graph on the calculator, rather than the test sheet, that doesn't come up as much as you might be led to believe. Some of the input methods are fairly finnicky, though that's the case for any calculator. If you're used to the input methods standard on other brands of calculators, such as TI, you may wind up tripped up by muscle memory a few times.

The best part, however? The 115ES Plus has a list price of only $18, already miles below that of most graphing calculators, and you can often find new 115ES Pluses on websites or in supply stores for still less, usually around $15. Especially for statistics classes, where calculations are often more necessary than in more theoretical classes such as calculus, the breadth of tools and affordability of the 115ES make it a screaming deal.

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