![]() There’s more about how naive interval arithmetic graphing looks around a typical singularity here:Ĭheck out that program, GrafEQ. The bands start to widen, indicating growing uncertainties, near points where division by zero starts to occur.įrink does interval arithmetic natively, allowing you to play around with this very powerful technique very easily, but the simple examples I’ve posted don’t try any detailed tricks to refine errors around singularities. Especially if it’s losing precision badly. The appearance of the graph that you posted looks exactly what happens when using interval arithmetic techniques around a singularity, though. (Be sure to follow the “view the source” link to see how ridiculously simple this program actually is, yet it can easily plot equations that you or I couldn’t begin to guess the behavior of!) For example, here’s an intentionally simple example of a program written in my programming language Frink ( ) that can plot arbitrary equations using interval arithmetic techniques, *in only a few lines of code!* (Try to teach a computer how to graph an equation that contains arbitrary expressions of x and y using any other technique.) However, it can be done in only a few lines of code using interval arithmetic techniques. ![]() Interval arithmetic is almost magically powerful when it comes to graphing arbitrary equations. As intervals are added or multiplied with each other, those uncertainties are propagated through all your calculations. Interval arithmetic is a relatively new branch of mathematics that represents each number as an interval within a certain range, say,, but you’re not sure where the true value may lie within that range. This looks like it’s quite probably an artifact of using “interval arithmetic” techniques to graph the equations. ![]() ![]() You can find more of my work with Desmos here. I explore this theme in greater depth in my talk “ When Technology Fails“. In doing so, they’ll not only learn some mathematics and some computer science, but they’ll also develop a healthier relationship with technology, by learning to understand how it does what it does, and perhaps more importantly, what it doesn’t do. Lots of interesting questions emerge from such anomalies, and these are great questions for students to explore. It’s the mathematical technology that is behaving strangely, as it tries to represent the function. Now, it’s not the function here that’s behaving strangely: its behavior is well-understood. This graph has a hole (a removable discontinuity) at the point (-2,-1), which I have colored blue.īut look what happens when you zoom in around the hole:Īt a very small scale, some very curious behavior emerges! But some of my favorite mathematical questions arise when technology does something we don’t expect.įor example, here’s the graph of. And like most worthy instructional technologies, it’s really a learning technology: it’s easily accessible to students as well as teachers.Īs far as technology goes, Desmos works very well. I use it almost every day in my classroom: to sketch simple graphs, demonstrate mathematical relationships, and dynamically explore mathematical situations. I am huge fan of Desmos, the free online graphing calculator. Published by MrHonner on NovemNovember 21, 2013 ![]()
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