Understanding Math Vs Understanding Math

One of the things hard about engineering — electrical engineering, in particular — is that you can’t really visualize what’s important. Sure, you can see a resistor and an LED in your hands, but the real stuff that we care about — electron flow, space charge, and all that — is totally abstract. If you just tinker, you might avoid a lot of the inherent math (or maths for our UK friends), but if you decide to get serious, you’ll quickly find yourself in a numerical quicksand. The problem is, there’s mechanically understanding math, and intuitively understanding math. We recently came across a simple site that tries to help with the latter that deserves a look.

If you don’t know what we mean by that, consider a simple example. You can teach a kid that 5×3 is 15. But, hopefully, a teacher at some point in your academic career pointed out to you what the meaning of it was. That if you had five packages of three items, you have 15 items total. Or that if you have a room that is five feet on one side and three feet on the other, the square footage is 15 square feet.

That’s an easy example. But as you get higher into math it is more common to just use a result and not really understand the why of it.  For example, you’ve probably heard of convolution and you know it has applications in digital signal processing among other areas. But could you explain it to a 9th grader? Using a combination of animations and graphics, the author takes you through not only what convolution does, but how you arrived at it and some applications. There aren’t many topics on the site, but the detail on each that is there is great. There are topics on algebra, calculus, complex numbers, and a great pair of articles on the Fourier transform. There is even an article on a different way to think about multiplication that lets you factor things graphically!

We love sites that make math more understandable. We enjoyed solving equations with colors, for example. And we couldn’t stop watching the visual representation of the creation of square waves with harmonically-related sine waves which effectively demonstrates the Fourier transform.

26 thoughts on “Understanding Math Vs Understanding Math

  1. Thanks Al nice reminder.
    FWIW, the key math one seems to need as substantive foundation is calculus, easiest being integration re summation over areas which I understand can be fairly easily taught to primary school kids. Then when approaching differentials allows higher level math(s) to be more easily appreciated on a intuitive basis once some hard work done eg Gauss as foundation for fields, volumes, etc towards tensors such as in Einstein. In the digital set theory domains of sampled data control systems such as in Kalman filters offer interesting potential but, still variations from the central theme.
    It’s interesting and some would argue somewhat troubling that the universe and fundaments of relationships are rather well allied as a whole being well describable and predictable through relatively minor extensions of; add, subtract, multiply and divide as the central theme – which all practical maths is ultimately predicated upon, implicitly itself just lots of it with immense permutations :-)

      1. Hmm, for most part one could draw it down to those two. I’d thought this way as diluting to the most simplest in school then came across phase operators ah lah the famous sqrt(-1) ie ‘i’ or ‘j’ in traditional electronics filter theory issue. Although one might then use addition & subtraction re infinite series (Taylor & McLauren) sort of – ugh, the multiplication and division issue by i, j etc somehow is rather outside addition and subtraction as phase operators pull towards abstractions in set theory. I guess that’s what I had in the back of my mind when I touched on sets as its an odd but useful abstract with little direct connect to addition and subtraction that I can see…

  2. This is convolution:


    Each guitar is playing the same sequence of notes.. a function of time. The three instances of the function are offset from each other in time.

    The second guitar is a bit quieter than the first one, and the third is a bit quieter than the second.. or each instance of the function is scaled by a constant related to its offset from the others. We can also call the scaling constants values of a second function calculated at t={0,1,2}.

    The song is the offset instances of one function, scaled by a function of the offsets, added together.

    That’s a convolution.

    Jump to 6:50 for the end product.

    1. This is what I would call “doing it the hard way”. I mean, why even HAVE three guitarists? Using proper keying, you can even have three copies of the first guy, playing it at the different delays, on the video.

  3. Oh! Is Maths a valid UK-English thing? Do they use it everywhere that in the US we would say Math or is it specific to a certain shade of meaning? I always just assumed it was something some people like to say because they are trying to be cute but aren’t like when someone talks about ‘da Internets’.

    1. Grossly simplifying, maths seems to be more popular here as it doesn’t depluralise “mathematics”, and mathematics feels more proper as it covers a wide range of fields (e.g. there is one history, but good luck describing what the “mathematic” is). I always imagined American English dropped the ‘s’ to fit it in with other singular school subjects.

      1. How do y’all across the pond abbreviate economics? Econ or econs?

        I’m fine with either form, I always thought that trying to get English to be perfectly self-consistent was a fool’s errand. Never gonna happen, it’s a Frankenstein language. Especially with little things like informal abbreviations.

          1. No, it’s not. Every language I’ve ever looked at has exceptions to exceptions to exceptions. This is the way of natural languages, since they evolve in multiple ways, simultaneously.

  4. Much of the mystery of math is in the notation. Sure, for people who do this every day (and by that I mean mathemeticians, not engineers or scientists), it’s nice to have a shorthand that makes the writing of equations conscise. But while I once knew, quite well, all four of what are commonly referred to as Maxwell’s equations, as in, what each one was called, and how each could be stated in English, and how they applied to everyday situations, twenty-five years later, I find the notation of those symbolic equations to be nearly opaque. Which is very sad, because all four of these express very basic and elegant principles that are useful anywhere electromagnetic fields are found. They’re the Newton’s laws of the electromagnetic universe, but they elude my grasp because of those damned slippery symbols.

    1. Sure, if we don’t use it… we lose it. Especially, with the quality of water, agriculture, food and drugs going from medication to more like poison if not in fact poison(ed). I just a few days back had a…, wow… I can’t even comprehend French anymore and don’t recall most of the words I had learned at one time. Spanish isn’t so great either… though I did use more Spanish over the years for work.

      I usually struggle more with the units of the variables, though admit. 20+ years later and not, using the math or, really solidly learning outside of how to plug and chug in a scientific calculator or mathematics software application… I forgot a lot. Head injury and extra lead didn’t help either.

  5. I use JavaScript in a browser to render visual models of maths for different things to do with electronics, mechanics etc.

    For those who remember a hatred for JavaScript – The browser wars are over and you can easily write JavaScript that is cross browser compatible.

    What I would really like to see here at HAD is some articles on the maths of forward and inverse parallel kinematics or even non-linear sequential kinematics.

    I would also like to see articles on what higher level languages are best suited to the more complex math of non-linear and parallel kinematics. I was looking at LUA and while it’s up to the task but LUA has quirky data objects that that make it hard for a newcomer to read and write the code without some prior experience with LUA.

    Even an article on what hardware platforms exist for the more complex maths of non-linear kinematics. 8-Bit micro-controllers are not suited to the maths needed for more complex kinematic systems.

    These sorts of articles may leads to newer and cheaper CNC devices like 3D printers, Pick and Place etc.

    Arc Delta is much cheaper than Linear Delta to construct and micro-controllers are dirt cheap now. Parallel kinematics will also cut construction costs.

    Sequential linear kinematics (XYZ) is probably the most expensive to construct and yet it’s the most common.

    Another important maths consideration is error or tolerance. For linear CNC is not a lot more than steps per inch plus play. It is far more complex for arc CNC and that a very important part of the design metrics of the mechanical construction.

    PS: It’s great to see an article of interest that isn’t just a write up on a video.

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