The best scientific, graphing, and CAS calculator apps for Android: Comprehensive comparative review

[Skyica]

Introduction to my app reviews on the XDA forum

This is the first of a series of such comprehensive app review threads that I have planned (the others will probably be shorter however).

While I will publish the reviews later in other places also (like my website and medium.com), I have decided to publish them here first in the hope of getting suggestions for improvement from many interested people, and maybe from the developers of the apps also.

Introduction to this review of calculator apps

In this article, I will attempt to find, evaluate, and compare the best advanced calculator apps for Android, as comprehensively and objectively as possible.

I also include high-quality “basic” scientific calculator apps with features like scientific constants, units, and base-n calculation. Further, I include advanced "learning calculators" like Photomath, but I evaluate only their ability to solve mathematical problems, not their pedagogical value.

I hope the article will be helpful to anyone who wants to do more complicated calculations on Android devices, and also to the developers of the calculator apps, and to programmers considering the development of another calculator app.

I decided to write it because there are now many good and useful advanced calculator apps for Android, but not one thorough review (at least I couldn’t find one). While there are many “reviews” of the best such apps on the web, those lack

• systematic search for available apps,
• thorough tests of the apps,
• comparison of features,
• comparison of data from Google Play,
• examination of other reviews.

Accordingly, they do not provide an objective overview of the best apps. Also, they seem to be written mostly by people with no special love for math and programming.

While I am myself not a mathematician, I have studied mathematical physics and computer science (PhD), and have developed complex Android apps (an advanced 3D modeling app, and App Finder, a sophisticated search engine for apps on Google Play, which I have used for this review).

Please note that this is a VERY long and detailed review. If you just want to know what the best calculator apps are, you may directly read my recommendations here.

If you are mainly interested in this as an example for the usage of App Finder, you may just read the section about the searches.

A shortened version of the review is available at Medium.

If you are new to this forum, please note that you can change the style from dark to light via the "XDA-Dark" button at the very bottom of the page. Also, you can hide the left pane using the three-line button top left.

If you like this article, please read the post at the bottom of this page.

 

[Skyica]

Classification of apps and compilation of features

Kinds of calculators

Since for different people and different purposes, different types of calculator apps will be optimal, let us first make a classification.

Traditional handheld calculators may be classified as follows (excluding completely basic and financial calculators):

• Basic scientific calculators
• Advanced numeric scientific calculators
  (distinguished from the basic ones by e.g. equation solving, calculus, and matrix calculation capabilities)
• Programmable calculators
• Graphing calculators
• CAS (Computer algebra system, i.e. symbolic computation) calculators
  (also including the functionality of advanced numeric, programmable, and graphing calculators)

However, if one needs more than a basic scientific calculator at all, it will probably be useful to have symbolic as well as numeric calculation, and also graphing (and possibly programming).

This also seems to be what most calculator app developers have thought, since nearly all good apps with a functionality exceeding basic scientific calculators include at least basic symbolic computation and graphing features.

Therefore, I will classify the apps here simply as “basic” or “advanced” calculators for feature comparison.

Nevertheless, a more detailed classification is useful and will be used to structure this review:

• Advanced general-purpose calculators
  Apps with a wide range of advanced numeric and symbolic computation and graphing that are convenient for both quick and more complicated calculations.
• Graphing tools
  Apps specializing in graphing, whether or not they include other functionality.
• CAS answer engines
  Apps with powerful symbolic computation, that also accept input in natural language, and can give not only calculation results, but also information on mathematical definitions and concepts (WolframAlpha is currently the only such app).
• Advanced learning calculators
  Apps designed for learning mathematics, often with symbolic computation, showing intermediate calculation steps, and able to recognize written problems from photos.
• Advanced calculators of other types
  E.g., apps with a “notebook interface” (similar to Mathematica), computer algebra systems with a command-line interface, and apps with programming / scripting.
• Intermediate scientific and graphing calculators
  Apps with some advanced features like 2D graphing, basic numeric / symbolic calculus, numeric equation solving. Up to the level of the TI-84.
• Basic scientific calculators
  Apps without advanced mathematics like calculus, that are nevertheless suitable for science and engineering (with features like scientific constants and units, and base-n calculation).

Features and test problems

To be able to objectively compare the apps, I have made lists of important features and some test problems.

The test problems are of course quite arbitrary and cover only a small part of the features, and a still smaller part of the range of important problem types. Nevertheless, I think they give some helpful insight into the power and reliability of the apps.

The features that I consider essential are in blue and will be used as the baseline for the reviews. More advanced features, which fewer people will need, are in parenthesis. Even more advanced features are in double-parenthesis. The test problems are in green.

Advanced calculator features and test problems​

As a basis I have taken the functionality of advanced hardware calculators like the TI-89 or TI-Nspire CAS, while considering the capabilities of modern Android devices.

Numeric and symbolic calculation
(in addition to the basic functionality of all scientific calculators):

• Complex numbers: Rectangular and polar form, common operations and functions
• Symbolic computation essentials:
  • Calculation with exact values (fractions, roots, pi, etc.)
    • e^pi/e^(2pi) should be simplified to 1/e^pi or e^-pi
  • Automatic simplification
    • 2x/(x^2-1)-1/(x-1) should be simplified to 1/(x+1)
  • (Factorization and expansion of polynomials)
  • ((Equation manipulation))
  • (Assumptions, e.g. domain constraints)
• Equation solving:
  • Numeric
    • Ln(x) = -x should be solved as x = 0.567143
      sin(x) = x/2 should be solved as x = {0, ±1.89549}
  • Symbolic
    • sin(1/x)=0 should be solved as x = 1/(n*pi)
      ln(x*y) = y*ln( y) should be solved as x = y^(y-1)
  • Real and complex solutions
    • (X+1)^2 = -1 should be solved as x = -1±i
  • (Equation systems)
    • x²+y² = 2 and x²*y² = 1 should be solved as x = ±1 and y = ±1
  • (Specification of variable ranges)
  • (Inequalities)
• Differentiation and integration:
  • Numeric
  • Symbolic
    • (e^x+1)/(e^x-1) should be integrated to 2*ln|e^x-1|-x
      1/(1+sin(x)) should be integrated to e.g. tan(x/2-pi/4), or -2/(tan(x/2)+1), or (sin(x)-1)/cos(x)
      x/(1+sin(x)) should be integrated to e.g. 2*ln(cos(-x/2+pi/4))-x*tan(-x/2+pi/4)
      x/ln(x) should be integrated to Ei(2*ln(x))
  • Multiple / nested operations
• Other calculus and analysis:
  • Finding minima and maxima, numeric (and symbolic)
    • Minima of sin(1/x) should be given like 2/(4pi*n-pi)
  • Sums and products of sequences, also infinite, numeric and symbolic
  • Limits, also one-sided, numeric and symbolic
  • (Taylor expansion)
  • ((Differential equations solving))
• Matrices and vectors:
  • Standard operations and functions
  • (Vector calculus, esp. gradient, divergence, curl)
• Basic number theory and combinatorics: random, prime factorization, gcd, lcm, factorial, nCr, nPr
• Base-n calculation: conversion and bitwise operations for binary, octal, hexadecimal numbers
• Statistics and probability
  • One and two variable statistics: with frequencies, (histograms, box plots)
  • Regression: Linear, quadradic, logarithmic, exponential, …, with plots
  • ((Confidence intervals calculations))
  • ((Statistical tests, e.g. t, z, F, chi², ANOVA))
  • (Probability distribution calculations)
• Calculation with units and unit conversion
• Scientific constants

2D and 3D graphing

• Display of multiple graphs in one coordinate system, also of different types
• (Multiple workspaces, i.e. coordinate systems)
• Graph types:
  • Cartesian function plots
  • 2D: Polar function plots
  • Parametric plots, in 3D both curves and surfaces
  • (Implicit plots)
    • 4*(x^2+y^2+z^2)+16*x*y*z = 1 (Cayley's cubic surface)
  • ((Differential equations plots))
  • ((Complex plots))
• Display:
  • 2D: (Customizable) grid, adapting to zoom (with major and minor lines)
  • 3D: Labeled coordinate axes, (optional semitransparent x-y-plane, optional x-y-grid)
  • Custom colors for individual graphs
  • (3D: Surface display: solid with variable transparency, and as wireframe)
  • (3D: Both orthographic and perspective projection)
  • ((3D: Quality settings))
  • (Animation of parameter variables)
  • Portrait and landscape orientation
• Navigation:
  • Equal and independent scale of axes
  • 2D: Simultaneous pan and zoom with 2 fingers
  • 3D: Complete and intuitive navigation, e.g., orbit around center with one finger, pan and zoom with 2 fingers
  • (3D: Automatic rotation)
• Analysis:
  • 2D: Tracing
  • 2D: Display of coordinates of intersections, zeros, extrema

Environment and user interface

• Variables and functions:
  • Saving values to variables
  • (Custom-named variables for storing values and for use in symbolic computation)
  • (User-defined functions that can be used in numeric and symbolic calculations)
  • (Piecewise defined functions), ((Programmed functions))
• (Intermediate calculation steps)
• ((Programming environment))
• Input and output:
  • Display and input of expressions in standard mathematical notation, like in a textbook
  • Copy and paste, also parts of expressions
  • Undo & Redo
  • (Automatic calculation without tapping enter key)
• (Multiple workspaces)
• (“Quick Info”: Display of various useful information about the entered expression or result with one tap, e.g.: prime factorization for integers, alternative formats for numbers, alternative forms for expressions with variables, solutions for different variables for equations, graphs for expressions and equations with 1 or 2 variables)
• History of calculations
• Import / export, (sync and share) of variables, functions, (and history)
• Adaption and customization:
  • Convenient interface for small phones, good use of the space available on large phones (and tablets)
  • Adjustable display text size (and keyboard size)
  • (Keyboard layouts for different screen sizes), ((Custom keyboard layouts))
  • (Light and dark mode, different themes, custom colors)

Documentation and source

• Complete user guide (with full text search), (in-app and external)
• (Function / command reference with full text search)
• (Info / description of the algorithms used)
• (Open source)

For Basic scientific calculators, I use a separate list:

Basic scientific calculator features​

• Calculation
  in addition to the basic functionality of all scientific calculators:
  • Basic number theory and combinatorics: random numbers, prime factorization, gcd, lcm, factorial, nCr, nPr
  • (Calculation with fractions, roots, pi)
  • (Base-n calculation: conversion and bitwise operations for binary, octal, hexadecimal numbers)
  • (Complex numbers: Rectangular and polar, common operations and functions)
  • (One-variable statistics)
  • (Calculation with units and unit conversion)
  • (Scientific constants)
• Memories/variables, functions
  • Enough memories/variables
  • Using variables directly in expressions, not just inserting the value
  • (Saving functions)
• Input and output:
  • (Standard mathematical notation, like in a textbook)
  • Otherwise, multi-line
  • Copy and paste, (also parts of expressions)
  • (Undo & Redo)
  • (Automatic calculation without tapping enter key)
• Number formatting:
  • Custom number of decimal places, round/cut modes
  • Scientific and engineering formats
  • Thousands separator
  • (Repeated decimals)
• History of calculations
• (Import / export of variables, functions, and history)
• Adaption and customization:
  • (Different keyboard layouts, custom keyboard layouts)
  • Light and dark mode, (different themes, custom colors)
• User guide

 

[Skyica]

The search

To search Google Play, I have used App Finder, an advanced app search engine that has data for nearly 2.4 million apps.

I have also quickly searched the following alternative app stores directly:

• Samsung Galaxy Store
• Huawei AppGallery
• Amazon App Store
• F-Droid
• Aurora store
• APKMirror
• AppBrain
• SlideMe
• Aptoide

However, I did not find additional apps that can be recommended.


App Finder

For our purpose, App Finder has several advantages over Google Play’s build-in search:

• Search algorithm: Google Play uses a complex undocumented search algorithm that takes many factors into account, and prefers the apps it thinks “you’ll have a good experience with” (see here). While this is often very good for quickly finding some of the best apps, it is not suitable for a comprehensive search.
  In contrast, App Finder matches your search query against the descriptions of the apps in a clear and easily comprehensible way (see below).
• Search operators: In addition to the usual search operators, App Finder has some special operators. For example, to require specific words to occur in title or summary, which can be very helpful to exclude irrelevant apps.
• Filters: While Google Play’s search has only filters for a user rating above 4 or 4.5 stars and for paid apps, App Finder has a filter for user rating in 0.1-star steps, and many other filters.
• Sort options: Like Google Play, App Finder sorts the results by “relevance” by default. That is, by a score calculated from the match with the query, the average user rating, and the popularity. However, App Finder also has the option to sort by a single characteristic like the average user rating or the number of ratings, which gives us better orientation for comparing the results.
• Result count: App Finder shows the exact result count, so we can add keywords or filters if there are more results than we want to inspect.
• Result list: Unlike Google Play’s search, App Finder shows all essential data, summary, and screenshots directly in the result list.
• Additional data: App Finder shows
  • the average user ratings with two decimals, which is relevant for high ratings (consider two apps rated 4.85 and 4.94 stars respectively, which are both rounded to 4.9 stars by Google Play),
  • the number of downloads with two significant digits, with is quite an improvement over Google Play which rounds them down to powers of ten times 1 or 5,
  • World-average user ratings in addition to country-specific ratings.

Here’s a how App Finder matches search queries against the app descriptions:

• In the simplest case, if you just enter keywords separated by spaces, it returns exactly those apps where all the words (or forms of them, like the plural) appear in the description (or title or developer name).
  .
• Words inside quotes are required to appear in sequence (different forms of the words are still allowed).
  .
• + in front of a word (or quoted words) requires it to occur in title or summary, not just anywhere in the description.
  (The summary is a short description limited to 80 characters that the developer must provide).
  .
• - can be used as usual to exclude words. It can be combined with + to exclude them from title and summary only.
  .
• The usual OR operator is supported. Additionally, / (a slash) can be used instead of OR. Contrary to OR, / is evaluated first, which makes it often possible to avoid parenthesis and repetitions. Also, / can be used inside quotes. No spaces are required around /.

Simple Search

Since scientific calculator and graphing calculator are universally used terms, we would expect each app intended as such a calculator to have one of these terms in title or summary.

With App Finder, we can search for this using the query +”scientific calculator” OR +“graphing calculator” (see above). Using the / operator, it can be simplified to

Query 1: +“scientific/graphing calculator”

Without filters, this query gives 432 results.

Note that all searches in this article were done in Mai 2023 for the United States.

To not miss anything, I have looked through all results without filters.

Already from the screenshots, it is clear for most of the 432 apps that they are very low quality, or that they are not what we’re looking for.

Investigating the promising apps further, and quickly testing the still promising ones, I found that all the best apps probably have at least 4.5 stars average rating in the US for the full-featured version, at least 100 ratings, and were updated within the last year.

(Which is of course roughly what one would expect anyway.)

With these filters, we get 33 results. The following screenshots show the result list, ordered by number of ratings, which gives us a good orientation.

I have cut away some irrelevant apps, and placed the ad-supported and paid versions of the same apps together (future versions of App Finder will place apps by the same developer together automatically ).

Reading App Finder’s result list

The numbers left of the colorful stars are the average user ratings calculated by Google Play, the top one for the United States, the bottom one the whole world. (App Finder can also show the ratings for any other country supported by Google Play.)

The stars represent the rating distribution: The size (that is, area) of the stars is proportional to the number of 5-, 4-, 3-, 2-, and 1-star ratings (from green to red). Between different apps, the star-size is correlated with the total number of ratings at a logarithmic scale.

Paid apps without an indication do not contain ads. For apps with in-app purchases, the price range is shown in parentheses.

Spoiler: Result list

Search by functionality

Since there may be advanced calculator apps that are not called scientific or graphing calculator by the developers, I have done a search by functionality mentioned in the description also.

It is of course very difficult to predict which features a developer would choose to include in the description and what words would be used.

But if we only refer to functionality that is very important and popular, and consider alternative descriptions, we can expect to miss only few good apps, if any.

Choosing differentiation, equation solving, and graphing as such functionality, we arrive at a query like

Query 2: differentiate/derivative equation/algebra graph/plot

(Remember that App Finder equates different forms of a word. E.g., calculate also searches for calculator, differentiate also for differentiation, and integrate also for integration and integral.)

(With the simpler query derivative equation graph we miss two relevant apps, Wolfram Alpha and Algeo.)

Without filters, this query gives 264 results.

Again looking through the list and testing promising apps I found that all the best apps here probably have at least 4.2 stars average rating in the US, at least 1 million downloads for the free version, and were updated within the last two years.

With these filters, the query gives 14 results. Below you see the complete list ordered by “relevance”.

While I had to try some different keyword combinations and filters to arrive at this, the result is quite impressive:

The list probably includes all the best advanced calculator apps, no irrelevant apps, only few apps which I would not recommend, and the app that I would rate as the best overall is first in the list.

Spoiler: Result list

Additional searches

I have also repeated the query with -graph/plot instead of graph/plot to check if there are good advanced calculator apps without graphing, but this seems not to be the case.

I have searched for graphing apps specifically with graph/plot function trace and graph/plot function 3D. The latter query yielded 3 additional interesting 3D graphing apps (Grapher– Equation Plotter, Visual Math 4D, and Graphing Calculator 3D).

I have searched for CAS apps specifically with "computer algebra system", which yielded Maxima for Android and Xcas Pad.

 

[Skyica]

Summary of the review and recommendations

Since the introduction of Android in 2008, independent programmers from around the world, as well as small and large companies, have developed hundreds of calculator apps (as we have seen, there are currently at least 432 apps with the phrase scientific calculator or graphing calculator in title or summary available on Google Play).

There is currently no single app with as comprehensive, reliable, and convenient calculation functionality as the best hardware calculators (like the TI-89, TI-Nspire, or Casio fx-CG500 / ClassPad).

However, for many purposes there are perfectly suitable apps, which are sometimes even preferable if one has a hardware calculator at hand (e.g., for graphing).

Note that there are also emulators available for Casio’s and HP’s flagship CAS calculators (but not TI’s), which however greatly suffer from emulation related problems.

For almost all purposes, a combination of the best native Android apps (e.g., Hiper Calc, GeoGebra, and WolframAlpha, see below), is probably the best choice.

Advanced general-purpose calculators

There seem to be essentially 3 advanced general-purpose calculator apps currently (see my definition above).

Of these, I would rate Hiper Calc as the best overall. Its numeric and symbolic calculation functionality is rather comprehensive and seems to be quite reliable. It is very easy to use and often very efficient. Important disadvantages are weaknesses with symbolic computation and small bugs, and very limited graphing.

Scientific calculator plus 991 has a more extensive calculation functionality than Hiper Calc (e.g. for statistics), and its symbolic calculation is more powerful in some areas (simplification and integration). However, it has some bugs and inconveniences, not all functionality is well organized and documented, and 3D graphing is missing.

(There are some very similar apps that may possibly be used instead of the Plus 991.)

Scientific Calculator by Philip Stephens may be chosen for 3D graphing (see below), some user interface advantages, and maybe for statistics. Its numeric calculation functionality seems to be good, but symbolic calculation is very limited. While the user interface has some great features, it also has significant issues, and documentation is missing.

Apps for difficult symbolic and numeric calculations

In general, WolframAlpha (based on the leading mathematics software Mathematica) can be recommended as the most powerful tool for difficult calculations on Android. It can also give further information on definitions and concepts, and often show intermediate calculation steps. However, it does not work offline, and expressions cannot be entered in textbook notation.

Maxima for Android is the long-established open-source computer algebra system Maxima with a command-line interface. Xcas Pad is another open-source computer algebra system with a command-line interface .

The best alternatives seem to be GeoGebra, Symbolab, and maybe Maple Calculator, which all support entry in textbook notation. GeoGebra works offline and is based on the open source CAS Xcas/Giac, but does not give intermediate calculation steps or further information. Symbolab and Maple Calculator often give intermediate calculation steps, but no further information.

For equation solving, Hiper Calc seems to work well also, but it has problems with simplification and symbolic integration. For symbolic integration, Plus 991 seems to work well also, but it has some bugs and problems with equation solving.

WolframAlpha sometimes shows the results in a more complicated form than other apps, and in one of my tests it did not return all minima of a simple function, while Hiper Calc did this. So, it may sometimes be a good idea to check with different apps.

Apps for graphing

For 2D graphing, Plus 991, Graphing Calculator + Math, Desmos, and GeoGebra are probably the best, with different advantages and shortcomings.

For 3D graphing, GeoGebra and Scientific Calculator by Philip Stephens seem to be clearly the best, also with different advantages and shortcomings.

GeoGebra and Graphing Calculator + Math may also be used as a “general-purpose calculator”, however the keys are quite small on phones for quick calculations.

Apps for longer complicated calculations and programming

MATLAB Mobile allows to create and execute MATLAB scripts in the cloud. MATLAB is a full-featured programming language, extensive numeric and symbolic calculation functionality is available, and 2D and 3D graphs can be created (but not navigated).

For tablets, a “notebook interface” is also available, where the code is placed on an interactive document together with the calculation results, graphics, and formatted text. While this concept is optimal for longer complicated calculations, there are serious problems with the interface of the app currently.

micro Mathematics is another calculator app with a notebook interface (for phones also). It seems to work reliably, however the input mechanism is quite inefficient, and symbolic computation and programming are missing.

GeoGebra and Graphing Calculator + Math also have kinds of (more limited) notebook interfaces.

Plus 991 seems to have a full-featured programming language, however there are significant issues currently.

Graphing Calculator (X84) includes TI-BASIC programming.

Scientific Calculator Scalar has limited scripting functionality.

The Casio fx-CG500 / ClassPad emulator might also be chosen for a “notebook interface” (eActivity) and for programming, however there are emulation related inconveniences.

The web-interface of SageMath might also be used, however it is quite inconvenient on phones.

An alternative would be to use a general programming environment with math libraries like SymPy, but this is beyond the scope of this review.

Apps for simple calculations

ChampCalc Scientific Calculator is a very sophisticated and customizable "basic" scientific calculator.

TechCalc Scientific Calculator includes formulas for many important mathematical, scientific, engineering, and financial applications.

Scientific Calculator Scalar allows to define custom functions easily, and has some scripting functionality.

Panecal Scientific Calculator may be very efficient for longer simple calculations.

CalcTastic has comprehensive base-n calculation functionality.

Graphing Calculator – Algeo and Jade Scientific Calculator have a simple and convenient user interface and some advanced features.

If more advanced features are needed, Hiper Calc can be recommended as the easiest to use advanced calculator.

Apps that imitate hardware calculators

Scientific calculator plus 991 has essentially the keyboard layout of the Casio fx-991 calculators, while being much more powerful.

Graphing calculator plus 84 83 has essentially the keyboard layout of the TI-84 calculators, while being much more powerful.

Graphing Calculator (X84) imitates the TI-84 quite closely, still making use of the touch-interface of Android devices.


If you find this review helpful, please read the note at the end of this page.


Introduction to the reviews of individual apps

I list all apps that looked promising to me from the store descriptions (as described in the section about the searches above, I have looked through several hundred apps).

For the apps that I would not recommend after testing, I only give the Play Store data and quickly explain the problems.

For the apps that I would recommend, I give screenshots and more information.

For apps that I would recommend for regular use, I give complete “feature lists”. For the others, I only note distinguishing features.

Reading the feature lists

In the feature lists, I take the “essential” (blue) features from the above lists as the baseline.

As advantages, I note especially good implementations and additional features. As disadvantages, I note bugs, bad implementations, and missing “essential” features. This may be a bit confusing, but I see no better way.

So, to make sense of the feature lists in the reviews, you must read the blue in the lists above.

If you are looking for apps with a specific feature, just use the search of you browser, all reviews are on this page.

I always review the full-featured and ad-free versions. For the good apps these are all available at a very fair (or more than fair) price.

Reading the Play Store data from App Finder

The numbers left of the colorful stars are the average user ratings calculated by Google Play, the top one for the United States, the bottom one the whole world.

The stars represent the rating distribution: The size (that is, area) of the stars is proportional to the number of 5-, 4-, 3-, 2-, and 1-star ratings (from green to red). Between different apps, the star-size is correlated with the total number of ratings at a logarithmic scale.

Paid apps without an indication do not contain ads. For apps with in-app purchases, the price range is shown in parentheses.

 

[Skyica]

Advanced General-Purpose Calculators

Hiper Calc​

This app stands out with quite comprehensive, reliable, and powerful functionality, and an exceptional user interface.

It seems to be very good at equation solving and minima / maxima finding (however there also is a bug).

On the other hand, there are significant problems with simplification, more complicated integrals cannot be solved symbolically, and statistics is very limited.

Graphing is very limited but still useful.

The functionality is very well organized and efficient and easy to use, but it still has some inconveniences, see the feature list.

Numeric and symbolic calculation​

• Symbolic computation essentials:
  • Bad Already some easy simplifications are not done, e.g., ln(2)+ln(3) is not simplified to ln(6), and 2x/(x^2-1)-1/(x-1) is not simplified to 1/(x+1)
  • Bad In results, polynomials are always given in expanded form, which not always is the simplest one
  • Feature Factorization, but works only with one variable.
• Equation solving:
  • BUG In some cases it incorrectly returns “No solution exists”, e.g., for sin(1/x) = 0, and for ln(x*y) = ln( y)
  • Feature Equation systems, entered one equation per line
  • Feature Inequalities
  • Good Apart from some special cases (see above), equation solving seems to work very well, and quite complicated equations and equation systems can be solved symbolically, and even more complicated ones numerically. Periodic solutions are handled well.
• Differentiation and integration:
  • Bad Some more complicated integrals cannot be solved symbolically even if they are elementary, e.g. (e^x-1)/(e^x+1). Nonelementary integrals cannot be solved symbolically.
• Other calculus and analysis:
  • Feature Symbolic minima and maxima, also for periodic functions.
  • Good It correctly found all minima of sin(1/x), what WolframAlpha did not
• Statistics and probability:
  • Missing Only one-variable statistics
  • Bad Input of data is inconvenient, especially with frequencies.
  • Missing No regression
• Feature High precision calculation with up to 100 digits significand and 9 digits exponent

2D and 3D graphing​

• Missing Graphs are only shown in the “calculation details”, and only one graph can be drawn at a time (or two for equations)
• Graph types:
  • Missing No parametric plots
  • Feature Implicit plots in 2D, but for equations involving periodic functions only one “branch” is shown, e.g. for sin(x+y) = 0
• Display:
  • Good Both 2D and 3D graphs are high quality
• Navigation:
  • Missing 3D: No complete navigation: Rotation only around z axis, inclination cannot be changed
  • Bad 3D: One must change between rotate and translate/zoom “modes” (one finger for rotate and 2 for translate/zoom would be more convenient)
  • Bad Equal / independent scale of axes in 3D does not to work well
• Analysis:
  • Missing Tracing

Environment and user interface​

• Variables and functions:
  • Feature Custom-named variables
  • Feature User-defined functions that can be used in numeric and symbolic calculations
    However, note that variable names other than x and y and function names cannot be typed but must be inserted from a menu. Also, variables other than x and y must be defined in this menu before they can be used in symbolic computation. While this is an inconvenience, one may get used to it.
• Input and output:
  • Missing Parts of expressions cannot be selected and copied, only the whole input or result can be copied.
  • Missing No undo & Redo
  • Good Keyboard is well designed.
  • Bad Since recently, the equal sign must be selected from a menu.
  • Feature Automatic calculation without tapping enter key
• Feature Good “Quick Info”
• Adaption and customization:
  • Feature Tablet layout that makes good use of the space available (but only in landscape layout)
  • Feature Different predefined keyboard layouts and custom keyboard layouts
  • Feature Many high-quality themes
  • Bad For larger calculations (e.g. equation systems) the small display area is inconvenient. When the display text size is reduced, the display becomes also smaller (and the keys larger) so that no vertical space is won.
    (When multi-line display is enabled the keys are much too small on phones. A solution is to define a custom keyboard with more columns which makes the keys smaller and thus the keyboard lower.)

Documentation and source​

• Good There is a well-organized complete in-app help.

 

[Skyica]

Scientific calculator plus 991​

This app stands out with very extensive and quite powerful calculation functionality.

However, it has some worrying bugs (see below).

Simplification and symbolic integration seem to be very good, but there are problems with equation solving and minima / maxima finding. Its statistics features are quite comprehensive.

2D graphing is good but has inconveniences. 3D graphing is missing.

The user interface is sophisticated, but some functionality is not well organized and documented, and there are many inconveniences.

Numeric and symbolic calculation​

• Symbolic computation essentials:
  • Feature Domain constraints, but they are not exploited sometimes, e.g., Ln(x*y)-Ln(x) with x>0 and y>0 is not simplified to Ln( y). (Input: Simplify(Ln(x*y)-Ln(x), x>0 and y>0))
• Equation solving:
  • BUG When trying to solve an equation by just entering it or using SHIFT-SOLVE, inexact results are often returned, e.g., for sin(x) = x, something like [x=-0.0000083, x=-0.0000017] is returned. Using the Solve function, [x=0] is correctly returned.
  • BUG Even with Solve, equations involving periodic functions are often not correctly solved, e.g.:
    For sin(x) = 0 only 0 is returned as solution; for sin(x) = cos(x) two complicated (although correct) numerical expressions are returned instead of the simple exact solution.
    For sin(x) = x/2, sometimes [x=-1.89…, x=1.89…], and sometimes [x=0] is returned.
  • Feature Equation systems
• Differentiation and integration:
  • Good Powerful symbolic integration, even some nonelementary integrals can be solved with special functions, e.g., logarithmic and exponential integrals and Fresnel integrals (however these functions cannot be plotted).
    Also, for complicated integrals like x/(1+sin(x)), for the same input, sometimes operation is not supported is returned, and sometimes the correct solution.
• Other calculus and analysis:
  • BUG Minima and maxima are often not correctly calculated, e.g., for x^4-x^2, only x=-1/4 is returned as minimum, for periodic functions also only one minimum and maximum is returned.
  • Bad No one-sided limits
• Statistics and probability:
  • It may take some time to learn to use the statistics features, but they seem to work quite well.
  • Bad Data editor shows no row numbers.
  • Feature Extensive distribution calculation functionality with many distributions (I did not test this much).
  • Feature Statistical tests (at least in the Plus 84 83, see below)
  • Feature Confidence intervals calculations (at least in the Plus 84 83)
• Feature It has hundreds of advanced functions that are named as in WolframAlpha and Mathematica, e.g., Series, Fourier, PowerExpand, ApplySides(~600 documented, ~300 I could find in the menus). However,
  • I did not test them and don’t know if they work reliably.
  • The inconvenience of the interface (e.g., that parts of expressions cannot be copied, and that equation systems must be entered in one line) makes more complicated calculations tedious, so that I question the usefulness of many of them.
  • The functions cannot be typed, and there is no alphabetic list from which they can be inserted.
• Feature Programming:
  It seems to have a full featured programming language that can interface the calculation functionality (Symja). However, the print function and graphics output seem not to work (only the result for the last line is printed automatically). Also, files cannot be saved.

2D and 3D graphing​

No 3D graphing.

• Bad The functions to draw must be entered in linear form (i.e. not textbook form), or all in one line separated by :
• Feature 3 workspaces
• Graph types:
  • Feature Implicit plots
• Display:
  • Bad Grid is quite coarse (and cannot be adjusted)
• Navigation:
  • Feature Independent scale of axes
  • Bad No simultaneous pan and zoom with 2 fingers
• Analysis:
  • Good Tracing is well implemented, tangent can also be shown
  • Good Derivative can also be shown, inflection points can also be shown

Environment and user interface​

• Variables and functions:
  • Feature User-defined functions that can be used in numeric and symbolic calculations
• Input and output:
  • Missing Parts of expressions cannot be selected and copied, only the whole input or result can be copied
  • Missing No undo & Redo
  • Feature Automatic calculation without tapping enter key
• Feature Good “Quick Info”
• BUG When copying expressions from the history, it often returns incorrect results, even if it retuned the correct result the first time the expression was entered.
• Missing No import / export
• Adaption and customization:
  • UI is good for both small and large phones, but not tablets
  • Feature Many high-quality themes
• Bad Not convenient for symbolic computation, e.g.,
  • Variable x and equal sign are not accessible with one tap
  • For most symbolic computation one must switch between the main and a 2nd keyboard, whose keys are not well organized
  • Equation systems must be entered in one line

Documentation and source​

• Bad Help is not well organized
• Feature Help can be searched (but not full-text)
• Feature Function / command reference with search (but not full-text) and examples

Source: Although the developers do not state this, the app is based on the open source CAS Symja, “inspired by Wolfram Language built-in functions”, and implemented in Java.



Scientific Calculator 300 Plus,

Graphing calculator plus 84 83,

Scientific calculator 36 Plus,

Scientific calculator 30 34,

Calc 991 Scientific Calculator

These apps clearly share much code with the Plus 991, even if different developer names are used: Essentially all UI elements other than the keyboard are the same in each of them, and the Symja CAS library is also used (see above).

Thus, it can be expected that they share most of the advantages and disadvantages with the Plus 991.

The main difference seems to be that they are modeled and named after different Casio and TI calculators, and some have a more limited functionality.

Scientific Calculator 300 Plus seems to be essentially the same as the Plus 991.

Graphing calculator plus 84 83 has a significantly different layout, resembling the TI-84 often very closely. It may be better for statistics. Also, it has an alphabetical function catalog.

Below you see the first three.

 

[Skyica]

Scientific Calculator (by Philip Stephens)​

This app stands out with very powerful 3D graphing and some important user interface features that are missing from most or all other calculator apps (see the feature list).

Numeric computation seems to be good, and the statistics and probability functionality is extensive.

Symbolic computation is very limited however.

There are great graphing features (e.g. implicit surfaces), but also problems with graphing.

While the user interface has some great features, it also has significant inconveniences. Documentation is missing, and many things are not obvious.

Numeric and symbolic calculation​

• Symbolic computation essentials:
  • Bad Calculation with exact values mostly does not work:
    E.g, pi²/pi gives pi, but (pi²+pi)/pi already gives a decimal or is not simplified at all. Using the sym() function from sym-menu helps sometimes but not for more complicated expressions
  • Bad Similarly, expressions with variables are only simplified in very simple cases
• Equation solving:
  • Bad The expected problems with simplification
  • Bad Equations with parameter variables cannot be solved
  • Feature Equation systems (up to 3 equations), but maximal one solution is shown
  • Feature Specification of variable range, but only for equations in one variable
• Differentiation and integration
  • Bad Only very simple integrals can be solved symbolically. Integrating e.g. Ln(x)², or 1/(1+x)^2 returns “undefined”
  • Specific multiple / nested operations, but special templates must be selected
• Other calculus and analysis
  • Missing No min / max finding,
  • Bad Sums and products of sequences only numeric
  • Bad Limits only numeric
  • Feature Taylor series
• Matrices and vectors
  • Feature Vector calculus
• Statistics and probability
  (I did not test this much)
  • Missing No plots
  • Feature Confidence interval calculations
  • Feature Many statistical tests
  • Feature Many probability distribution calculations
  • Feature Random number generators for normal, Poisson, binomial distributions

2D and 3D graphing​

• Graph types:
  • Missing No parametric surfaces
  • Feature Implicit plots with high quality
  • Feature Spherical, and cylindrical graphs
  • Feature “Inverse” and implicit plots also for polar, spherical, and cylindrical coordinates
  • Feature Inequality plots
  • Feature Complex plots where real and imaginary parts are shown in two separate 3d graphs.
    In these plots it is however difficult to relate real and imaginary part, especially since rotation is not linked.
  • Feature Many special functions like Error and Gamma function can be used for graphing
• Display:
  • Good Both 2D and 3D graphs are high quality
  • Good 2D: Grid with major and minor lines that adapts to zoom by fading lines in and out
  • Missing No landscape orientation
• Navigation:
  • Feature Independent scale of axes
  • Good Individual translation in coordinate directions
  • Good Individual scale of coordinate axes by using two-finger-pinch on the axis-controls, this is not documented!
  • Bad No translation perpendicular to view-direction
  • Missing 3D: No equal scale of axes. Two-finger zoom does not work to change the drawing area.
• Analysis:
  • Bad Tracing works very poorly

Environment and user interface​

• Variables and functions
  • Only x, y, z can be used as variables. Additionally, there are 5 memories, and 5 constants can be defined.
  • Feature User-defined functions
• Input and output:
  • This is the only app where parts of expressions can be selected and copied in textbook notation. This generally works as expected but could be improved. Pasting at a specific location is inconvenient or impossible because long-tap triggers selection.
  • Missing No undo/redo
  • Feature Automatic calculation without tapping enter key
  • Good Keyboard is well designed for numeric calculations
• Bad History limited to 10 calculations
• Missing No import / export
• Adaption and customization:
  • Good Multi-touch display zoom
  • Feature Scalable keyboard
  • Feature Custom colors for all UI elements

Documentation and source​

• Missing No help document, only a few hints when using some features for the first time.

 

[Skyica]

Casio fx-CG500 / ClassPad Emulator​

This emulator of Casio’s flagship CAS calculator is clearly one of the most comprehensive and powerful calculator apps.

However, it greatly suffer from emulation related problems, e.g.

• Graphs cannot be navigated with multitouch and are low-resolution.
• Input of expressions is very inconvenient since the physical keyboard is not emulated, and one has to switch between different virtual keyboards often, which requires much tapping, and the keys are very small on phones.

HP Prime Emulator​

This emulator of HP’s flagship CAS calculator is not quite as comprehensive and powerful as Casio’s.

It shares with it most of the emulation-related problems like low resolution. While it supports multi-touch and the hardware keyboard is emulated, the display area is much smaller than Casio’s.

 

[Skyica]

Graphing Tools

GeoGebra​

This app stands out with exceptional 3D graphing (however there are problems with implicit surfaces).

2D graphing is also good, but tracing is missing.

It may also be used as a “general-purpose calculator”, however the keys are quite small on phones for quick calculations.

Numeric and symbolic calculation seems to be very well in general, but there are problems with simplification, and statistics, scientific constants, and unit conversion are missing.

The user interface is sophisticated and intuitive to use. However, a complete user guide is missing, and some things are not obvious.

Note that GeoGebra consists of several apps (mainly Graphing Calculator, 3D Calculator, CAS Calculator, and Geometry), that are available as a suite or singly.

Numeric and symbolic calculation​

• Symbolic computation essentials:
  • Bad Some simple expressions are not simplified automatically, e.g., e^pi/e^(2pi), or e^x/e^(2x). The Simplify function does simplify them however.
  • Feature Factorization, expansion, substitution
  • Feature Domain constraints
• Equation solving
  • Equation solving seems to be quite powerful in general, but there are some problems, e.g., sin(x)=x can only be solved numerically
  • Feature Equation systems
  • Feature Inequalities
• Differentiation and integration
  • Good Symbolic integration is powerful, quite complicated elementary and nonelementary Integrals can be solved.
    However there are sometimes problems with the simplification. (E.g., for the integral of x/(1+sin(x)), a very complicated expression is shown while the solution can be written much simpler. When differentiating the retuned solution, an even more complicated expression is shown, not the originally integrated expression)
• Other calculus and analysis
  • Feature Taylor expansion
  • Feature First and second order ordinary differential equations solving, numeric and symbolic.
• Matrices and vectors
  • Bad Matrices must be entered in linear form
• Statistics and probability
  • Missing No statistics
  • Missing No regression
  • Feature Probability distribution calculator with graphical UI
• Missing No base-n calculation
• Missing No scientific constants and unit conversion

2D and 3D Graphing​

• Graph types
  • Feature Implicit plots, however it does not draw more complicated implicit surfaces like 4*(x^2+y^2+z^2)+16*x*y*z = 1 (Cayley surface)
  • Feature Slope field plots for differential equations
• Display
  • Feature 2D: Customizable grid with major and minor lines
  • Feature 3D: optional semitransparent x-y-plane and x-y-grid
  • Feature 3D: variable transparency and wireframe surfaces
  • Feature 3D: Orthographic and perspective projection
  • Feature Animation of parameter variables with sliders
• Navigation
  • Feature 3D: Automatic (continued) rotation, parameters can be changed while the graph is rotating
  • Feature 3D: Augmented reality navigation
• Analysis
  • Missing Tracing

Environment and user Interface​

• Variables and functions:
  • Feature Custom-named variables for storing values and for use in symbolic computation
  • Feature User-defined functions that can be used in numeric and symbolic calculations
  • Feature Piecewise functions
• Input and output:
  • Missing Parts of expressions cannot be selected and copied, only the whole input or result can be copied.
  • Feature Automatic calculation without tapping enter key
• Feature GeoGebra is also available for iOS, Windows, macOS, Linux, and as web interface. Documents from one device can be saved to the cloud and opened on another.
• Adaption and customization:
  • Bad The keys are quite small for phones
  • Good Quite optimal for tablets

Documentation and source​

• Missing No complete user guide. This is a real problem since some things are not obvious.
• Feature Function / command reference with search (but not full-text) and examples
• Bad Help is fetched from the internet on demand and takes sometimes long to load (several seconds)

Open Source GeoGebra is essentially open source and uses the open-source CAS Giac/Xcas.

 

[Skyica]

Desmos Graphing Calculator​

This app stands out with very good 2D graphing and “flexible” regression.

However, it has no 3D graphing and no advanced calculation functionality (Desmos Scientific Calculator also has no advanced calculation functionality).

Good Navigation is intuitive and very smooth.

Feature Animation of parameter variables with sliders.

Feature Regression is not limited to fixed functions as in most other calculators. E.g., for a logarithmic model, not only the function y = a+b*ln(x) can be used, but additional parameters can be used, e.g., y = a+b*ln(x-c).

Feature Also available for iOS and as web interface. Documents from one device can be saved to the cloud and opened on another.

It is very easy to use and has a good user guide.

Grapher – Equation Plotter​

This is a good 2D and 3D graphing app, however it has important disadvantages against, for example, GeoGebra and the app by Philip Stephens, e.g.

• No “textbook notation”, and no keys for functions like sin(), they must be typed letter by letter
• No implicit surfaces
• When navigating in 3D, the graphs are updated only after the fingers are lifted

Feature It can draw complex functions using domain-coloring. Also, it can draw both real and complex parts for functions of a real variable.

Visual Math 4D​

This app can produce beautiful 3D graphs, however it has important disadvantages against GeoGebra and the app by Philip Stephens, e.g.

• Variable ranges cannot be changed by touch-navigation
• Implicit surfaces are not reliably drawn, the algorithm seems not to adapt to the selected variable ranges

Feature It can draw 3D vector fields.

Graphing Calculator 3D​

This is a 3D graphing app under development. It cannot currently be recommended because e.g.

• Variable ranges cannot be changed by touch-navigation
• Only one graph can be drawn at a time
• Problems with Implicit surfaces
• For complex plots, only one component is drawn at a time
• No “textbook notation”, and no keys for functions like sin(), they must be typed letter by letter

 

[Skyica]

CAS Answer Engines

WolframAlpha​

This app stands out with exceptional calculation capabilities and the possibility to show in-depth information on mathematical (and other) definitions, concepts, and data.

Intermediate calculation steps can often be shown.

Graphing is very limited but still useful.

The user interface is unique in that natural language can often be used to input problems.

Numeric and symbolic calculation​

• Symbolic computation essentials
  • Feature Factorization and expansion
  • Feature Assumptions
• Equation solving:
  • Feature Equation systems
  • Feature Specification of variable ranges
  • Feature Inequalities
• Differentiation and integration
  • Good Very difficult integrals can be calculated. If the result involves special functions, these can be defined
• Other calculus and analysis:
  • BUG Finding minima and maxima: For some (even easy) problems, it returns not all solutions, e.g., for minimize sin(1/x), it returns x = -2/(277pi)
  • Feature Taylor expansion
  • Feature Differential equations solving
• Matrices and vectors:
  • Feature Vector calculus
• Statistics and probability:
  • Feature Histograms and box plots
  • Feature Confidence intervals calculations
  • Feature Many statistical tests
  • Feature Many probability distribution calculations

2D and 3D graphing​

• Graph types:
  • Implicit plots, but more complicated implicit surfaces like 4*(x^2+y^2+z^2)+16*x*y*z = 1 (Cayley's cubic surface) cannot be drawn
  • Feature Complex plots, but not with domain-coloring
• Display
  • Bad Plots are quite small and cannot be enlarged
• Missing Navigation
• Missing Analysis

Environment and user interface​

• Bad Internet connection always required
• Good Nevertheless, results appear very fast
• Missing Values cannot be saved to variables
• Feature Intermediate calculation steps
• Input and output:
  • Missing No input in textbook notation (however the web interface at wolframalpha.com has this possibility).
  • Missing No undo/redo
• Feature Quick info
• Adaption and customization:
  • Missing No dark theme

Documentation and source​

• User guide seems not necessary
• Feature Very many useful examples

 

[Skyica]

Advanced Learning Calculators

Please note that as I said in the introduction, I evaluate also these apps only for their ability to efficiently solve mathematical problems, not for their pedagogical values.

Symbolab​

This app stands out with powerful symbolic computation and an easy-to-use and quite efficient interface.

Equation solving seems to work well, and many difficult integral can be calculated.

Intermediate calculation steps can often be shown.

3D graphing is missing and 2D graphing very limited and buggy.

Numeric and symbolic calculation​

• Missing Polar coordinates for complex numbers seem not to be supported
• Symbolic computation essentials:
  • Feature Factorization and expansion
• Equation solving:
  • Bad Some simple equations cannot be solved, e.g. sin(x) = x/2.
  • Good Apart from some cases (see above), equation solving seems to work very well, also for periodic functions
  • Feature Equation systems
  • Feature Inequalities
• Differentiation and integration:
  • Good Symbolic integration seems to be powerful, quite complicated elementary and nonelementary Integrals can be solved.
    (the complicated integral of x/(1+sin(x)) is not well simplified)
• Other calculus and analysis:
  • Bad Minima and maxima often cannot be calculated, e.g. for sin(1/x)
  • Missing For infinite sums, it only returns if it converges, not the value, e.g. for 1/n².
  • Feature Taylor expansion
  • Feature Differential equation solving
• Missing No random numbers and prime factorization
• Statistics and probability:
  • Missing Only one-variable statistics, no data editor
  • Missing Regression
• Missing No Scientific constants and unit conversion

2D and 3D graphing​

Missing No 3D graphing.

Missing Most important 2D graphing features are missing. Graphs are not always correctly drawn (try sin(1/x)). Navigation is juddery and inconvenient. Input of functions in the list does not work well.
There is also a separate app called Symbolab Graphing Calculator, but it does not work better.

Environment and user interface​

• Bad Internet connection always required
• Good Nevertheless, results appear very fast
• Missing Values cannot be saved to variables
• Feature Intermediate calculation steps
• Input and output:
  • The keyboard is quite well designed for phones and leaves a large area for the display. The fly-out keys work well.
  • Missing No copy and paste
  • Missing No undo/redo
• Feature There is also an iOS app and a web interface. Problems can be synced across devices.
• Adaption and customization:
  • Missing Display text size not adjustable

Documentation and source​

• Missing No complete user guide
• Feature Very many useful examples

 

[Skyica]

Maple Calculator​

This app stands out with powerful symbolic computation and an easy-to-use and quite efficient interface.

Complicated integrals can be calculated, but there are sometimes problems with simplification. There are significant bugs and problems with equation solving. No calculation of minima and maxima.

Intermediate calculation steps can often be shown.

Graphing is very limited but still useful.

Numeric and symbolic calculation​

• Symbolic computation essentials
  • Bad Expressions are not always displayed in the simplest form, e.g., for 2x^2=pi, it shows the positive solution as √(2)√(pi)/2 instead of √(pi/2) (although in the solution steps, it shows ±√(pi/2) as “intermediate step”)
• Equation solving:
  • BUG Equations involving periodic functions are not correctly solved, e.g., for sin(x)=1/2, only one solutions is shown (i.e., not even all solutions between 0 and pi are shown), for sin(1/x)=0, it shows the empty set as solution
  • Feature Equation systems
  • Feature Inequalities
• Differentiation and integration:
  • Good Powerful symbolic integration, nonelementary integrals can be solved with special functions, but no further information is given and no graph is shown. There are sometimes problems with simplification, e.g. for x/(1+sin(x)).
• Other calculus and analysis:
  • Missing No minima and maxima finding
  • Feature Differential equation solving
• Missing No base-n calculation
• Missing No random numbers and prime factorization
• Statistics and probability:
  • Missing No data editor
  • Missing No regression
• Missing No scientific constants and unit conversion

2D and 3D Graphing​

• Missing Only one 3D graph can be drawn at a time (in 2D, multiple functions must be entered in a vector)
• Graph types:
  • Missing No polar plots, not parametric plots
• Display
  • Missing No custom colors
• Navigation
  • Missing Ratio of axes scale is set by the app and cannot be changed
  • Missing 3D: No complete navigation, graphs are drawn in a fixed variable range that cannot be adjusted
• Missing No analysis features

Environment and user interface​

• Bad Internet connection required for most calculations. Even for the simplest problems, it takes about 1 second until the results are shown, sometimes a bit longer.
• Missing Values cannot be saved to variables
• Feature Intermediate calculation steps
• Input and output:
  • Good The keyboard is well designed for phones. It is quite efficient to use while leaving a large area for the display.
  • Missing No copy and paste (although there is a special mechanism to select parts of expressions)
  • Feature Automatic calculation without enter, even graphs are shown and updated automatically.
• Feature Quick Info
• Missing No history
• Bad When the app is closed, previous calculation is not restored
• Feature Calculations can be copied to desktop Maple (did not test this)
• Adaption and customization:
  • Missing Text size cannot be adjusted
  • Missing No dark mode
• Bad Sometimes the app enters fullscreen mode for no reason

Documentation and source​

• Missing No user guide, just a FAQ that cannot be searched

 

[Skyica]

Graphing Calculator + Math​

This app stands out with very good (although not optimal) 2D graphing, quite good 3D graphing, and some useful user interface features that are missing from most or all other apps.

Its calculation capabilities are quite limited however (e.g., simplifications are often not done, more complicated equations cannot be solved, no symbolic integration).

While the user interface has very useful features, it also has significant inconveniences.

The user manual is comprehensive and excellent for people just learning math.

Numeric and symbolic calculation​

• Symbolic computation essentials:
  • Bad Limited calculation with exact values (e.g., logarithms are always evaluated numerically)
  • Bad Limited simplification, e.g. e^(x+1)/e is not simplified
    (however, it simplifies 2x/(x^2-1)-1/(x-1) which Hiper does not simplify)
  • Feature Polynomials are displayed in both expanded and factored form
• Equation solving:
  • Bad Equations involving periodic functions cannot be solved symbolically
  • Bad Equations with parameter variables cannot be solved
  • Feature Inequalities
• Differentiation and integration
  • Bad For symbolic differentiation the simplification problems are an issue, e.g., the derivative of ln(√x) is given as 1/(2x^(1/2)* √x) instead of 1/(2x).
  • Missing No symbolic integration
• Other calculus and analysis:
  • Bad Sums and products of sequences only numeric
  • Bad Limits only numeric
• Statistics and probability
  • Missing Only 1 variable statistics, no data editor
  • Missing Only linear regression
• Missing No bit-operations

2D and 3D graphing​

• Missing In 3D, only one graph can be shown at a time
• Feature Multiple workspaces
• Graph types:
  • Feature Implicit plots in 2D and 3D.
  • Good Also more complicated implicit surfaces can be drawn, e.g. 4*(x^2+y^2+z^2)+16*x*y*z = 1 (Cayley's surface), which GeoGebra does not draw. However, Scientific Calculator by Philip Stephens draws implicit surfaces in much better quality.
• Display
  • 3D: Perspective projection only
  • 3D surfaces display: Wireframe only
• Navigation
  • Feature Independent scale of axes in 2D, well implemented.
  • Missing No simultaneous pan and zoom with 2 fingers
  • 3D: Complete navigation, but not very efficient (and unusual)
• Analysis
  • Good Tracing is well implemented, also “reverse” tracing with a horizontal line

Environment and user interface​

• Variables and functions:
  • Feature Custom-named variables
  • Feature User-defined functions that can be used in numeric and symbolic calculations
• Input and output
  • Bad Much functionality is hidden behind unlabeled keys (e.g., for integrals and derivatives, long-tap the x^n key!)
  • Missing Input of expressions in linear notation (simultaneously, they are displayed in textbook notation, which can also be tapped to position the cursor). The advantage over most other calculators is that parts of expressions can be selected and copied at least in this way.
  • Bad Very limited undo
• Feature It has a limited “notebook interface” where elements can be rearranged and comments can be added.
• Feature Multiple workspaces, however all variables and functions are shared
• Adaption and customization:
  • The keys are quite small for phones (the keyboard can “resized” to fewer, larger keys, but then important functionality is missing).
  • Bad User interface does not adapt to different screen sized and orientations

Documentation and source​

• Good The user manual is really comprehensive and excellent for people just learning math, available online and as PDF.

 

[Skyica]

Photomath, ​

Mathway, ​

Microsoft Math Solver​

These apps are similar to Symbolab and Maple Calculator (they also recognize problems from photos and show step-by-step solutions), however their calculation capabilities seem to be more limited, as the following comparison shows.

	Symbolab	Maple	Photomath	Mathway	Microsoft
solve sin(1/x) = 0	correct not simplified	wrong (empty set)	x	x	correct
solve ln(x) = -x	correct	correct	x	correct	x
solve sin(x) = x/2	x	only 1 solution	x	correct	x
solve ln(x*y) = y*ln( y) for x	correct	correct not simplified	x	correct not simplified	correct
integrate (e^x+1)/(e^x-1)	correct	correct not simplified	correct	x	x
integrate x/ln(x)	correct	correct	x	x	x
integrate 1/(1+sin(x))	correct	correct	correct	x	correct
integrate x/(1+sin(x))	correct not simplified	correct not simplified	x	x	x

HiEdu Scientific Calculator,​

Scientific Calculator He-580​

These apps may be good for learning math, however their capabilities are very limited, e.g.:

• No symbolic computation
• The equation solver can only give one solution
• Graphing is limited to simple polynomials

Also, they hide status bar and action bar, and there seems no way to show them permanently.

 

[Skyica]

Advanced calculators of other types

MATLAB Mobile​

This app allows to create and execute MATLAB scripts in the cloud.

Feature MATLAB is a full-featured programming language, extensive numeric and symbolic calculation functionality is available, and 2D and 3D graphs can be created (but not navigated).

The standard editor seems to work well and provides a custom keyboard, autocompletion, and syntax highlighting.

A command-line is also available.

Feature For tablets, a “notebook interface” is also available, where the code is placed on an interactive document together with the calculation results, graphics, and formatted text.

While this concept is optimal for longer complicated calculations, the app has serious issues with this currently. E.g., a stable internet connection seems to be always required while editing, the cursor cannot be moved with the arrow keys, and when text is selected, the keyboard disappears.

Already for working with the standard editor, the dependence on the cloud may be a disadvantage of course. Even with Wi-Fi, the connection gets lost from time to time, and one has to wait until the program is executed, or until one can continue editing.

Maxima on Android​

This is the long-established open-source computer algebra system Maxima (with roots going back to the 1960s) with a command-line interface.

Sadly, it was not updated since 2018, while Maxima is under continuous further development.

Features include

• Autocompletion of commands
• Results in LaTeX
• 2D and 3D graphs using Gnuplot (which cannot be navigated however )
• User guide for Android and complete Maxima manual

Xcas Pad​

This is the open-source computer algebra system Xcas/Giac with a command-line interface.

Features include autocompletion of commands, results in LaTeX, and a command reference with examples.

Note that GeoGebra is based on Xcas/Giac.

micro Mathematics​

This is a calculator app with a “notebook interface”:

Feature Definitions of variables and functions, calculations, 2D and 3D graphs, text, and images can be placed on an interactive document.

While this concept is optimal for longer complicated calculations, the interface of the app is quite inefficient. For example, to enter sin(x)^2+1, you have to

• Type sin(x
• Long-tap sin
• Type ^2
• Long-tap sin
• Tap the “expand selection” button
• Type +1

(This could admittedly be done more efficiently by starting with the outermost operator, i.e. + in this case. But note that when modifying an expression, e.g. adding a summand, this will not work.)

The calculation and graphing capabilities are quite limited:

• No symbolic calculation.
• The equation solver cannot find more than one solution and does not work reliably.
• Graphs cannot be navigated by touch, the variable ranges, and view angles for 3D, must be typed.
• The z-range for 3D plot cannot be defined.
• No implicit plots.

A good user manual is available.

Open Source, see here.

DYsolve Math Solver​

This is another calculator app with a “notebook interface”.

While expressions are typed in standard linear notation, which is faster than the method of micro Mathematics, there are serious problems currently, e.g.

• Selecting, editing, placing and moving elements does not work well, and sometimes not at all
• A bug on newer devices, the whole screen becomes blue sometimes and working on is not possible
• The keys are very small on phones

The calculation and graphing capabilities are quite limited as well. I has a numeric solver for equation systems and ordinary differential equations (I did not test this), however

• No symbolic calculation
• 3D graphs can be navigated using sliders only, which is inefficient, and view changes are not persisted
• Z-range for 3D plot cannot be defined.
• No implicit plots.

It is stated that it supports if/else blocks and for and while loops, however this is not described in the help and I did not find examples.

The app is aimed at engineering students, and there are worksheets for example problems available online.

I will include screenshots when the user interface is improved.

SMath Studio​

This is a further calculator app with a “notebook interface”.

It seems to support differential equation solving and programming, however the user interface is very limited, and I could not find out how to enter more complicated expressions.

TechCalc Scientific Calculator​

This is a uniquely comprehensive “multi-function calculator” with many specialized calculation modes for mathematical, scientific, engineering, and financial applications.

Feature It includes many important mathematical, mechanical, electrical, chemical, financial, and other formulas, ready for calculation. Further custom formulas can be added.

Feature It also has reference sections, e.g. differentiation & integration rules, statistics formulas, the periodic table, etc.

Feature There are several further useful features, it is appropriately called a "Swiss Army Knife".

While it has some advanced functionality like symbolic integration and 3D graphing, it is in general unsuitable for advanced mathematics:

The calculation functionality is distributed over many modes and sub-modes, e.g. for symbolic derivatives, numeric derivatives, symbolic integrals, numeric integrals, limits, etc.

Expressions involving derivatives, integrals etc. are not supported, so that for example the divergence or curl of a vector field cannot be calculated in one step.

Similarly, the graphing functionality is distributed so that for example a function graph and an implicit graph cannot be plotted together.

Further, textbook-style input is not supported, and editing expressions is inconvenient: To make changes in places other than the end, one must switch to an “edit mode” by long-tap, and switch back again before the result can be shown.

The interface is idiosyncratic, but well-organized comprehensive help is included.

Scientific Calculator Scalar​

This is a calculator app with a command-line interface, some advanced calculation features, some scripting functionality, and limited 2D graphing.

Feature Advanced calculation features:

• Expressions can be saved to variables (“arguments”) and reference each other
• Functions can be defined and can reference each other, also recursively
• Variadic and piecewise functions can be defined
• Sums and products with conditionals
• “rep” statement for repetition with a fixed range of variable values
• “if” function for simple alternatives
• Very many special functions, like gamma, log-gamma, and digamma function, error function, etc.
• Prime-counting function using the sieve of Eratosthenes, and approximations
• Arbitrary-base numbers
• Random variables for different ranges and distributions

Feature Scripts can be created, and the editor works well. However, the possibilities are quite limited since there are no flexible loops, no GOTO, and control structures cannot be nested.

Feature 2D graphing is well suited for discrete functions.

Otherwise, the calculation and graphing capabilities are quite limited:

• No symbolic calculation
• The equation solver cannot find more than one solution
• Graphs cannot be navigated by touch, the variable ranges must be typed
  (they can be zoomed and then moved by touch, but are not recalculated by this)
• Only 3 functions can be drawn at a time
• No parametric and implicit plots
• No 3D graphing

The keyboard is quite well designed, with scrollable key rows. It is efficient for phones while leaving a large display area. However, it does not adapt to tablets.

The app is easy to use, and there are many in-app hints, examples, video tutorials, and a long user manual, which is quite repetitive however.

Source The expression parser is open source, see here.

 

[Skyica]

Intermediate scientific and graphing calculators

I use the feature list for basic scientific calculators as baseline here (scroll down this post).

Graphing Calculator – Algeo​

This app has a simple user interface but some advanced features like 2D graphing.

The textbook-notation input and the always visible calculation history are very convenient.

• Calculation:
  • Missing Random numbers, prime factorization, lcm
  • Feature Fractions
  • Feature Complex numbers
  • Feature Very limited equation solver (returns NaN for sin(x) = 1 and 1/x = -1)
  • Feature Symbolic differentiation, Numeric integration, Taylor series
  • Feature One variable statistics
  • Feature Calculation with units and unit conversion
  • Feature scientific constants
• Graphing:
  • Feature 2D cartesian, polar, and parametric plots with tracing
• Memories/variables, functions:
  • Bad Only 4 memories, saving is inconvenient
• Input and output:
  • Feature Input in textbook notation
  • Feature Automatic calculation without tapping enter key
• Number formatting:
  • Missing No custom number of decimal places

Jade Scientific Calculator​

This is another app with a simple user interface but some advanced features.

The textbook-notation input and the always visible calculation history are very convenient.

It has some useful customization options.

• Calculation:
  • Missing Random numbers, prime factorization, gcd, lcm
  • Feature Calculation with exact expressions like fraction, roots, pi
  • Feature Simplification of expressions with variables
  • Feature Numeric and symbolic differentiation and integration,
    1/(1+sin(x)), but not others
  • but no equation solving
  • Feature Complex numbers (rectangular)
  • Feature Scientific constants
• Input and output:
  • Feature Input in textbook notation
• Number formatting
  • Missing No custom number of decimal places
• Adaption and customization
  • Feature Different keyboard sizes
• Missing No user guide, and some things are not obvious

Graphing Calculator (X84)​

This app is designed to closely resemble the TI-84, while making use of the touch-interface of Android devices.

It seems to do this really well, but graphing could be implemented better.

Also, it seems to include virtually all functionality of the TI-84 calculators, including programming in TI-BASIC (but not Python).

• Calculation:
  • Feature Complex numbers
  • Feature Calculation with fractions
  • Feature Base-n calculation
  • Feature Numeric equation solving (1 real variable)
  • Feature Numeric differentiation and integration
  • Feature Numeric minima and maxima finding
  • Feature Sums and products of sequences
  • Feature Matrix calculation
  • Feature Base-n calculation: conversion and bitwise operations for bases 2 to 16
  • Feature One and two variable statistics with histograms
  • Feature Statistical tests
  • Feature Probability distributions
  • Feature Confidence intervals
  • Feature Regression with plots
  • Feature Unit conversion
  • Feature Scientific constants
• Graphing:
  • Feature 2D cartesian, polar, and parametric plots with tracing
  • Bad No equal-axes-zoom, no simultaneous pan and zoom with 2 fingers
  • Bad Quality is unnecessarily low (visible for functions like sin(1/x))
  • Bad Grid not adapting well to zoom
• Programming
  • Feature TI-BASIC
• Memories/variables, functions
  • Feature Saving expressions
• Input and output:
  • Feature Input in textbook notation
  • Feature Undo & Redo
• Bad Status bar is hidden and cannot be shown permanently
• Feature Function / command reference

Class Calc Graphing & Statistics​

This app has a quite sophisticated interface, and statistics and regression seem to be implemented well.

However, it has serious bugs and problems, e.g.:

• For the derivative of e^x, it returns something like (1196457)^-x*(32535443)^x (a fraction-approximation of e^x)
• For the integral of 1/(1+sin(x)) it incorrectly returns ln(1+sin(X))
• It seems not possible to solve equations
• No complex numbers
• Zooming of graphs does not work well

The web interface has the same problems.

NumWorks Emulator​

This is an emulator of a good intermediate scientific graphing calculator, however it is quite inconvenient since it supports input over the emulated keyboard only.

Scientific Calculator Advanced​

This app has some advanced features like 2D graphing, matrix calculations, and equation solving. However, the user interface is inconvenient for this functionality. Also the status bar is hidden and cannot be shown permanently.

Taculator Graphing Calculator​

This is another app designed to closely resemble the TI-84.

Serious problems are immediately evident:

• Lag between tapping a key and the reaction, getting worse with using the app (restarting helps)
• Entering 2π gives 23.1415…
• 2D graphing: when zooming, the graphs are not recalculated, so that they get jaggy.

(The iPhone version has fewer problems.)

Also, the lifetime Pro version costs $64 (against $2 / $4 for the best apps).

 

[Skyica]

Basic scientific calculators​

I use the feature list for basic scientific calculators as baseline here (scroll down this post).

ChampCalc Scientific Calculator​

This is a very sophisticated and customizable basic scientific calculator.

Memory/variable management is better than in most advanced calculators. Also, it has advanced base-n calculation and a large and well-organized library of units and scientific constants. There is a comprehensive user guide.

• Calculation:
  • Missing gcd, lcm
  • Feature Fractions
  • Feature Advanced base-n calculation for different integer sizes
  • Feature Complex numbers
  • Feature One variable statistics
  • Feature Many scientific constants
  • Feature Many unit conversions
  • Feature Large numbers and high precision with up to 100 decimals of significand
• Memories/variables, functions:
  • Feature Memory and variables manager
  • Feature Storage of expressions
• Input and output:
  • Feature Syntax highlighting and highlight of matching brackets
  • Feature Automatic calculation without tapping enter key
• Number formatting
  • Feature Repeated decimals
• Feature Quick info with prime factorization and value as fraction, mixed, DMS, Roman, BIN, OCT, HEX
• Adaption and customization:
  • Feature Many useful settings like long-tap time
  • Feature Many high-quality themes and custom colors

Panecal Scientific Calculator​

This app may be very efficient for longer simple calculations.

In contrast to most other apps, parts of expressions can be selected and copied.

• Calculation:
  • Missing Random numbers, prime factorization, gcd, lcm
  • Feature Base-n calculation
  • Feature Many scientific constants
  • Feature Many unit conversions
• Input and output
  • Feature Parts of expressions can be selected and copied
  • Feature Large multi-line display

Scientific Calculator by CalcTastic​

This is a quite sophisticated basic scientific calculator with advanced base-n calculation.

The large display with always visible history is convenient.

• Calculation
  • Missing Prime factorization, gcd, lcm
  • Feature Advanced base-n bitwise and bit operations for different integer sizes
  • Feature Calculation with fractions
  • Feature Complex numbers
  • Feature One variable statistics
  • Feature Unit conversion
  • Feature Scientific constants
• Memories/variables, functions
  • Missing Variables cannot be used in expressions (only value inserted)
• Input and output
  • Feature Syntax highlighting and highlight of matching brackets
  • Feature Automatic calculation without tapping enter key
• Adaption and customization
  • Feature Special keyboard layout for tablets
  • Feature Many high-quality themes

MathsApp Graphing / Scientific Calculator​

This is a quite sophisticated scientific calculator with some advanced features.

The always visible history is convenient, however the keyboard seems to be unnecessarily small.

The price for the full version is quite high comparably.

• Calculation
  • Feature Advanced base-n calculation for different integer sizes
  • Feature Complex numbers
  • Feature One variable statistics and Probability distributions
  • Feature Matrix calculation
  • Feature Scientific constants
• Graphing:
  • Feature 2D cartesian and polar plots with display of coordinates of intersections, zeros, extrema, but no tracing
• Feature Import / export of history
• Adaption and customization
  • Missing No light theme

RealCalc Scientific Calculator​

This is a basic old-style one-line scientific calculator. Its more “advanced” features include result history, 10 memories, unit conversion and physical constants, and fraction calculation.

 

[Skyica]

reserved

 

[Skyica]

If this review was helpful to you, I would appreciate it if you checked out App Finder.

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