Math on iPad #1
Apple’s iPad hasn’t been available for very long but there is already a wealth of mathematical apps available for it and I expect the current crop to only be the tip of the iceberg. So, this is the beginning of a new series of articles on Walking Randomly where I’ll explore the options for doing mathematics on this new platform.
Update: Part 2 is now available
The Rolls Royce of mobile mathematical applications and one that I have been using since my days as a Windows Mobile user. The iPad version was one of the first apps I bought when I received my device and it is just beautiful! Symbolic algebra and calculus, 2 and 3D interactive plotting, scripting, fractals linear algebra…the list of functions just goes on and on. I would have loved to have access to this app when I was in high school or early university.
If you want to get an idea of the quality of SpaceTime’s graphical capabilities then check out the free demo, Graphbook, but be aware that there is a lot more to SpaceTime than just graphics.
Regular readers of Walking Randomly will know that I am a big fan of Wolfram’s Demonstration project which is made possible by Mathematica’s Manipulate function. Well, SpaceTime has a similar, albeit simplified, version of Manipulate – a function called Scroll. Interactive Fourier Series on the iPad anyone?
Something else that I like about SpaceTime is the fact that it is cross-platform with versions for Linux and Windows available in addition to iPhone, iPad and Windows Mobile. So, students could use it in a classroom setting on PCs and use what they have learned on their own iPad/iPhone version.
If you only buy one mathematical application for iPad then this should be it. It’s relatively expensive for an iPad app at £11.99 (at the time of writing) but is worth every penny and I bought it without hesitation – so should you!
PocketCAS Pro is a computer algebra system that started out life as a Windows Mobile app and is now available for iPhone and iPad. I haven’t had chance to try it out yet so I can’t comment on its quality but it has a lot of features including symbolic algebra and calculus, 2D plotting, numerical solution of equations and more.
At the time of writing, it is the same price as SpaceTime mathematics – £11.99 – and yet my first impression is that it has less functionality. No 3d plotting for example. I’ll know more when I buy a copy next month.
There is a free lite version available which includes some of the functionality of the main product to allow you to try it out.
- PocketCAS Pro on iTunes.
- PocketCAS lite – free, cut down version of PocketCAS Pro
- PocketCAS main website
My favourite operating system is Linux where there is a philosophy of “Write programs that do one thing and do it well”. fxIntegrator does one thing -the numerical solution of 1d integrals – but does it do it well?
Well, it’s not bad. You enter the function you want to integrate using the nice, specially designed keyboard, then you enter the limits and press the = button to get the result. Couldn’t get any easier and I like it. The equation editor is very nice resulting in well formatted integrands but I did manage to confuse it once or twice. FxIntegrator is also very cheap at only 59p – a real bargain!
I tried a few straightforward integrals on it and it gave the correct answer in all cases. Then I got nasty and tried the following which has an algebraic-logarithmic singularity at the origin (original source for this integral).
I wasn’t expecting fxIntegrator to cope and it didn’t. Rather than giving the answer I just got an unhappy face indicating that it couldn’t compute the solution. This isn’t a criticism though! I like the fact that rather than giving numerical garbage, fxIntegator simply said ‘I can’t do that’.
There are some niggles, however. First of all, the list of elementary functions available is rather limited as it only includes square roots, powers,the trigonometric functions sin,cos and tan, the natural logarithm function ln and basic arithmetic. Even when I was in high school I would have wanted more such as inverse trig functions. (update: There are now a lot more functions including inverse trig)
Another problem with it is that although you can use the customised keyboard to enter the integrand, when you try to enter limits the standard iPad keyboard pops up.
These niggles aside, however, this is a nice little app for 59p and I hope the author continues to develop it. If he does then here are some suggestions for functionality I’d like to see.
- Add a few more functions. Inverse trig for a start. If possible then maybe things such as Bessel functions. (update: Inverse trig has now been added)
- Help turn this into a better teaching and learning tool by implementing a range of numerical methods for computing the integrand and allow the user to choose between them. Methods such as the rectangle rule, trapezoidal rule and simpson’s rule along with the ability to change the sub-divison size. The more methods the better :) (update: There are now three integration methods)
- Perhaps add some tutorial notes on each numerical method.
- Give the calculation time for the result in seconds along with the number of evaluations of the integrand. This will help students compare the trade off in speed/accuracy of each method. (update: This has now been done and looks great)
- Add the ability to plot the integrand along with the limits. Allow the user to change limits by moving them on the graph as well as by direct input. Once the calculation is performed, show the points on the curve where the algorithm sampled the function.
This good little app could be turned into a great little app.
Update (December 2010)
fxIntegrator has been updated several times since this article was written and it has improved even further. There are now three integration methods (Simpson, Trapeze and Rectangle) and several new functions have been implemented including inverse trig. Without doubt, this is now a great little app.
Update (February 2011)
More info on fxIntegrator. Of the version I originally reviewed I said “although you can use the customised keyboard to enter the integrand, when you try to enter limits the standard iPad keyboard pops up”. This is no longer the case in the current release. In fact, when entering the limits, not only you don’t see the standard iPad keyboard anymore, but you can freely write full-blown formulas (which can be just as complex as the integrands) to be used as integration limits. Further more, infinite limits are now possible too!