There is one website that I both constantly recommend to students and regularly use myself to help with almost any problem in all areas of mathematics. Impossible, you say? Not at all. The website is Wolfram Alpha.
Wolfram Alpha is, (at least according to Business Insider):
“probably the most useful site on the internet. It’s not a search engine, it’s not an encyclopedia, and it’s not a calculator, but it’s a little bit of all of that.“
For a Math or Engineering student however it’s real strength is in providing solutions to just about any algebraic problem.
With an extremely flexible interpreter, you can ask it mathematical questions (in the same way you would google just about anything else) and it will give a solution, list assumptions made and provide a report of other relevant information such as alternative written forms, graphs, intersects, roots, limits or turning points.
This article provides a brief introduction to the interface, as well as some examples of the most useful features.
Introducing the Wolfram Alpha Search Engine
1. Search Bar
This is where you tell Wolfram what you need to know. Some examples of input are:
- “expand (x^2 + 1)(x^2 – 1)(x+1)^3”
- “simplify ((x-1)^2(x-2)^3(x^2-1))/(x^3-x^2-4x+4)”
- “factor x^12 – y^12”
- “vector from (0,3,4) to (6,-2,9)”
- “second derivative of x^4”
You can also solve simultaneous equations:
Or draw graphs with multiple variables:
These example queries only scratch the surface of the types of questions you can ask Wolfram. Aside from a full suite of mathematical analysis tools, Wolfram can answer questions about hundreds of topics including Statistics, Finance, Chemistry, Physics and Engineering (Mechanical, Electrical, Acoustic, Control Systems, Materials) and that just scratches the surface.
2. Step-By-Step Worked Solutions!
Step-by-step solutions are available to paid subscribers (or for free if you use the Android App), and as you would imagine this is a massively useful feature – the biggest complaint my students have is that their study notes are written by professors who show inadequate or no working because they expect those “trivial” details to be obvious or because “it is pre-requisite knowledge”.
A dropdown menu also allows users to select which solution method they would like to follow. The excerpt below shows part of the worked solution for “x^2 + 3x = 3” via the “Quadratic formula Method”, but there is also an option to view the steps for the “Complete the Square” method.
There are other sites that provide step-by-step worked solutions for free (I recommend checking out Symbolab), however the overall ease-of-use, powerful functionality and relatively low price Wolfram Alpha subscriptions (last time I checked they were going for 2.99 USD/month for students after a free one-month trial) has led to myself and a number of my students signing up.
(Edit: One of my students has informed me that the Wolfram Alpha Android App now provides free step-by-step worked solutions without requiring a paid membership).
3. Copyable Plaintext
The ability to copy the plaintext version of any of the output equations makes it very quick to modify the input, copy across to other websites/software packages for cross-checking/further analysis, or even copying directly into the Microsoft Word equation editor (hint: alt + “=”) to make those tedious design or lab reports slightly less painful.
4. “More” (Show Alternate Forms)
A common issue for students is that when checking answers against those provided in a textbook or by a course instructor, the answers provided don’t quite seem to match yours, and even though you may have verified your solution using a self-check technique, you really want to know whether or not you have been approaching the question correctly. It is likely that the instructor didn’t simplify the terms in the same way you did, or perhaps they re-arranged the equations in a different way.
To view the solution output in various rearranged forms, try tapping the “Show More” button:
When to Use Wolfram Alpha (or other Online Equation Solvers)
Let’s imagine your professor has given you an assignment question which conceptually wasn’t really that difficult… except for the five pages of rearranging, expanding, re-differentiating and simplifying required to reach the solution. After carefully working through the problem step by step, you can’t shake the feeling that you could have made a silly mistake somewhere which has likely propagated throughout the rest of the assignment. After a quick scan through your working, you can’t see any mistakes, but you can’t shake the feeling that something isn’t quite right.
Back in the day, there wasn’t much to do about this other than:
- re-doing the problem (after telling your friends you probably won’t have time to join them doing that fun thing that was happening on the weekend),
- comparing your solution with a friend’s (if they are still around that is – see point above), or
- muttering something suitably nihilistic white hitting the “submit assignment” button and just seeing what happens (probably nothing until you get your grades back with no feedback 6 weeks later).
Thankfully, now, all you need to do is ask Wolfram. Just as “have you tried Googling it?” is now a standard question to be asking, “have you tried asking Wolfram?” is a regular question I now ask my students along with “what did the class notes say” or “what was in the textbook”?
Online equation solvers aren’t a silver bullet for all of your math problems, however they are a formidable addition to your regular study toolkit. Which leads us to…
The Ethics of Using Online Equation Solvers
The use of online equations solvers only came into prominence when I was about halfway through completing my degree, and I distinctly recall having mixed feelings about using these powerful tools.
One one hand, I could clearly see that when used properly, they could greatly improve the efficiency of my study, sparing me from undertaking hours of re-working for questions that, while I already had a good conceptual understanding of the techniques involved, even a small algebraic error would have cost me a disproportionate amount of assignment marks. They also provided significant value when completing practice exams for which lecturers had not provided solutions, and therefore made it very difficult to actually close the feedback loop of trying, checking, learning and adjusting without having to wait several days until the next on-campus tutorial session, or worse, never knowing at all whether or not my methods were correct.
On the other hand, there were at least a couple of lecturers who were vocal in discouraging the use of such websites. While it may not have been so clear back them, it is now quite obvious that these were the instructors who had been re-using the same assignment questions for years, and a lack of creativity in formulating the questions meant that often they could be directly solved by a number of online solvers. This meant more work was required by them to keep up to date with evolving technologies and learning methods, and they were just trying to avoid the extra work.
We live in a world where instant feedback to any question is increasingly in demand, and questions that cannot be quickly and effortlessly answered are increasingly being put aside and forgotten about in favour of those that can. Course instructors need to embrace technology that facilitates this kind of instant feedback, and adapt their assessment methods to allow for these new information channels.
Now go forth and enjoy the increased learning efficiency!
Do you have any favourite websites, apps or other tools that you use for your Maths and Engineering studies? Have a question but still aren’t quite sure how or what to ask an online solver? Drop me a message, or if you are keen to go over a question in depth as soon as possible, book a session now.