One of the goals of this column is to present mathematical concepts in a fun, inclusive way. I know that for some, math can be a scary or intimidating subject, and I think a lot of this has to do with some of the terminology used by mathematicians. As you will see below, some mathematical words sound truly spooky. This article will try to describe what these words mean for a general audience, and hopefully make them seem (at least slightly) less monstrous.
Imaginary numbers: Despite the name, originally coined by René Descartes—who is also the namesake of the Cartesian plane—these numbers do exist! The fundamental imaginary number is typically labeled i, and it formally indicates the square root of negative one. But I think it helps to visualize i using Cartesian coordinates. Imagine a graph where the x-axis is labeled with real numbers, and the y-axis with imaginary numbers: i, 2i, -3i, and so on. Multiplying a real number by i corresponds to making a 90-degree rotation, bringing, say, the coordinate (5,0) on the x-axis to (0, 5i) on the y-axis! These types of visualizations help make complex analysis—the study of real and imaginary numbers—a little less complex.
Stochastic: Some of you may see this term in classes on stochastic calculus or probability. It is essentially a synonym for “random,” although I remember when I first learned about this word, I found it to be a little shocking-sounding. Its origin is from a Greek word meaning “to aim at a mark,” or more simply, “to guess.” This is also how it was originally used in more general English, appearing first in the 1600s to describe anything related to “conjecturing” (another fancy word meaning to guess). I really like the “aim at a mark” definition though, because it invokes in my mind the image of throwing darts. While it is impossible to predict exactly where the object will land on a dartboard, there is still some structure to the problem, based on the skill of the thrower or other conditions in the game. I think this is why “stochastic” is used in place of “random” to indicate this combination of structure (that comes from probability distributions) and unpredictability (from other aspects of the problem without a deterministic property).
Mollifiers: This term may not show up specifically in an undergraduate course, but I find it to be a cool mathematical idea that nicely shows the benefits of limits. Although the name has an anxious connotation, mollification in mathematics refers to a kind of smoothing. In particular, mollifiers are a sequence of smooth functions (i.e., functions whose derivatives are continuous) that approximate nonsmooth ones in the limit. Nonsmooth “functions” include the delta distribution, which is zero everywhere except for a spike at one point, and they appear often in mathematical modeling. Mollifiers help ensure that the mathematical theory of smooth functions can in some ways be extended into the realm of these nonsmooth, “generalized” functions, which are harder to describe with traditional calculus.
The above are only three examples, and I could have picked many other bits of mathematical jargon that sound wild to someone first hearing them. (Another favorite of mine, which I’ll leave for another article, is “functor” in category theory.) But I find that learning what the words mean, and having one’s own intuitive definition, can go a long way in demystifying mathematical concepts, or concepts in almost any other field more generally. While I don’t expect you to share these definitions or explain complicated mathematical terms as a party trick (though there’s always someone who is polite enough to listen in my experience!) I hope you can all embrace and enjoy some spookiness this Halloween.
