I just installed this app called “μMath+” or “microMathematics plus v2.18.0” from F-Droid on my hand computer, and it’s pretty cool. It’s the first calculator for Android that’s good enough to criticize, so I will.
You have “worksheets” which consist of sequences of “elements” or “objects”, stacked vertically and horizontally; the five types of elements are text fragments, images, equations (really assignment statements and function definitions), result views, and plots.
Text fragments are just paragraphs; the five available styles are “Chapter”, “Section”, “Subsection”, “Subsubsection”, and “Text body”, all with or without numbering. There’s a per-document flag to word-wrap them at some line length.
The expressions in the assignment statements are displayed using conventional math notation, and include Σ, Π, ∫, and derivatives. Values include complex floating-point numbers, Octave-like ranges of them (called “intervals”), multidimensional arrays of them (I think up to three dimensions), and versions of these dimensioned with units. Functions are defined in the assignment syntax by assigning to a function with formal parameters, as “f(x) = x² - 4”. “Result views” pair a user-entered expression, on the left, with its computed result, on the right. Plots come in 2-D, 3-D, and heatmap versions; normally you evaluate a function on an “interval” to get an array result which you can plot on the y-axis. (“Intervals” work very similarly to arrays in APL and related languages, and even more so the independent variables I talked about in Relational modeling and APL and A principled rethinking of array languages like APL. They are not the interval objects used in interval arithmetic, though they have some things in common.)
Moreover, you can export your whole document as LaTeX, though the plots are PNGs — high-resolution PNGs with transparency, but PNGs.
All in all, it can do all kinds of cool stuff.
So you can, for example, write
E := 3 V
I := 0.1 A
E/I =
and it will tell you “30.0 Ω”. You can change these values and recalculate. You can write
x := [-5, -4.9 .. 5]
y := x³ + -2·x² + x
and then plot x versus y. In fact, you don’t even need to define a variable for y; you can enter the polynomial directly on the y-axis of the plot, though because of the horizontal layout of the formula there, it takes up a lot of space. By long-pressing on the plot, you can get a button to view the values of x and y in a table.
You can intersperse your formulas and plots with explanatory text, as in Jupyter, but, at least in theory, somewhat more intimately — you aren’t constrained to vertical stacking — but I haven’t found a way to make that work well yet.
Given that it does all this array and interval stuff, I was sort of hoping it would handle matrix arithmetic (matrix products and inner products and least-squares solutions of systems of differential equations and whatnot) but it doesn’t.
Unfortunately I’m not very happy using it. It feels very clumsy.
It’s hard to figure out how to do anything, and it takes a lot of clicks, because the equation editor is a structure editor, but there’s no visualization of the structure to work on, and you suffer the usual gulfs of execution. For example, x² is an exponentiation node, which you can create by scrolling the toolbar to the right to the create-exponentiation-node button and then pressing it, or by using the “^” key on the keyboard, and then typing “x”. At times, to reach the exponent from there, you can press the soft keyboard’s Enter key (when it displays as “⇥”) but at times you have to tap the spot on the display where you want to put the “2”. It is possible to replace an existing node with a new node that has it as a child; for example, you can place the cursor before the “x” and insert “--”, which makes the “x” node the second child of a new subtraction node, so you can transform it into (1 - x)². So far I haven’t found a way to reverse this transformation (other than using undo — fortunately it does have multiple undo) or to put the focus on the exponentiation node itself so I can transform it into x² + 1.
Analogously, I haven’t figured out how to change the layout, for example changing the order of elements or their stacking direction.
The use of long-press to activate many functions exacerbates the discoverability problem.
The only way to see the value of anything is to add a result view or plot to the document, and typically then to long-press on the result to bring up a dialog with a table of all of its values. This is, however, pretty awesome in that it allows you to bring up a table of all the numeric values being plotted in a plot.
You can’t define an array by listing the items, as R = [1, 2.2, 4.7, 10, 22, 47]. You have to assign the individual items one at a time: R[1] := 1, R[2] := 2.2, and so on.
Error handling leaves a lot to be desired. If there’s an error anywhere in a worksheet, nothing in the worksheet will evaluate. Usually it highlights the error with a red border, which you can tap on to get a transient notification telling you what the error was. I’ve had times where I couldn’t find the error, though, so I opened up a new worksheet. Really good error handling might include suggesting similarly spelled variables when one is not defined, or offering to create a definition for it, or offering to change it the same way you changed the definition that used to exist.
The “New document” menu item discards your current worksheet without confirmation, apparently irretrievably — though only if you’re editing the default autosave.mmt.
Every time the screen orientation changes, the worksheet scrolls back to the top. Zooming is also troublesome; positioning in the document after a zoom is unpredictable, perhaps because the zoom feedback moves the document horizontally as well as vertically, while the actual resulting position is much more horizontally constrained. At times zoom is disabled for no apparent reason, and at times the document moves multiple screen widths diagonally when I’m pinch-zooming in place.
The formula formatting looks ugly; it’s using a sans-serif monoline font without even any obliquing for the variables, except of course that Σ and ∫ are in a serif font with diagonal stress, and Π must be too but it looks like a child’s drawing. The parentheses and brackets do not match the rest of the font, being thinner, and they aren’t properly spaced; “(x + 1)!” will butt the “!” up to the parens. Superscripts and subscripts are smaller than the text they modify, but still too large and too far away; then at four levels of superscript the font has gotten far too small.
It includes export to LaTeX and HTML, but although it seems to use SVG internally (in its documentation, and maybe for its plotting), it doesn’t seem to be able to export SVG.
The 3-D plots are lovely but being able to interactively rotate them to get a good angle would be a big plus. This is a general problem with a lot of things: to change a property you have to bring up a dialog box, but in order to see the result of changing the property, you have to close it. The even more egregious result of this is that there’s no way to resize a 2-D plot by dragging; you have to bring up a dialog box on top of it, type in a new width in pixels, and tap OK. (And the default plot size is pretty small.)
It’s possible to plot multiple functions on the same plot, and it’s possible to plot parametric functions, but doing both at once seems to be impossible.
There doesn’t seem to be a way to label plot axes by anything other than the actual expression that produces them, which would be useful if you have, for example, two plots covering different intervals of the same axis, which you will necessarily have to name with different variables.
You can’t just sum an array or an interval; you have to go the whole nine yards with Σᵢ₌₁⁹ xᵢ instead of just Σx.
You can define n := [0, 1 .. 32] and then ask 2ⁿ = and get a result, but you can’t just inline the interval value in the superscript. In fact, I think that’s true of all operations on intervals — the interval literal can’t occur in any context other than an assignment statement.
Because it uses the regular Android soft keyboard, numeric entry is a pain; you often have to switch modes, and the number keys are always tiny.
The HTML export doesn’t include plots, or, if it does, they aren’t showing up in Chrome.
Although dimensional analysis works for simple quantities, if you do this you get “V/A” instead of Ω for the units, although they’re at least still correct values:
E0 := [2.5, 2.6 .. 3.5]
E := E0 · 1 V
I := 0.1 A
E/I =
However, if you make I also an “interval” without units, then you get the error message on E/I: “There are indirectly referenced intervals: [E0]”. Dividing E0/I directly gives a matrix-like display of results, though with no real way to see which result corresponds to which inputs.
Although you can define an array by enumerating values as described earlier and it displays the same way intervals do, attempting to divide E/R produces the error, “Array is not allowed for this field.”
I mentioned above that four levels of subscript produce text that is too small. Five levels returns to the normal size again, but it’s not the normal size for your zoom level; it’s the normal size for your display. So if you zoom out, your fifth superscript will be huge compared to everything else in the document, while if you zoom in far enough, it will be tiny compared to everything else, because it always stays the same size on the screen.
Interactive calculator and drag-and-drop calculator for touch devices talk a bit about how calculators could take advantage of multitouch screens, but for pure formula entry, I feel like it’s really easy to do better than using the standard onscreen soft keyboard — the built-in calculator app does it already. The exception is when you’re assigning a name to something, when you need to be able to type the name, or when you have so many variables and functions that you need to find one by text search instead of picking from an LRU list.
The standard Android calculator these days displays the result of each formula incrementally as you edit it. This is a big help for simple calculations like 500×46.5. μMath+ doesn’t, for some reason.
Note that the ersatz Casio calculator profiled in Usability of scientific calculators, though very limited in hardware, got a bunch of things right: once you have a formula, you can attach it to a name. Unfortunately this, along with the whole RPN universe, adds an extra gulf of execution to the problem: you have to figure out that the way to get “y = x²” on the screen is to first get “x²” on the screen and only then activate “STO Y”, as the ersatz Casio’s keyboard labels call it.
Going further, you could imagine writing actual generalized equations rather than assignment statements, then submitting the glob of equations to some kind of solver; or, in the opposite direction in some sense, writing explicit algorithms with loops and conditionals.
OCR of handwritten equations might also be a good approach, given how clumsy interacting on the touchscreen is.
Structure editors are hard to make usable, but not impossible. A useful step would be providing visible handles to each of the tree nodes you might be trying to select.
Given the amount of symbolic computation and ad-hoc UI interaction tailoring this problem needs, writing it in Java was probably not a good idea; a development environment that better supports symbolic computation and iterative development, especially code changes without program restarts, would probably work better.