Envisioning the Invisible

To Be the Remainder “What is it like to be the remainder?” I answer this question with a mime sketch in a performance created specifically to motivate thinking about mathematics. I play the part of a clown who, almost compulsively, divides objects into two groups as they emerge from a suitcase. Repeatedly, one object is left: the remainder of the division. The pantomime touches on a universal human experience—being alone. In fact, the dramatic line of the piece relies heavily on a tension between togetherness and separateness. This sketch guides the audience, mathematicians and nonmathematicians, through a cerebral journey of parity. The mathematical claim is quite simple: division by two of any odd number will result in a remainder of 1, whereas two divides evenly into any even number. The clown, in his simpleminded way, struggles to accept this fact. With increasingly large numbers, he is empathetically saddened by a sole remainder and satisfied when groups divide evenly. Yet, almost tempting mathematical fate, he peers into his suitcase even after forming groups of equal size. This daring spirit results in new, larger sets of objects that he must divide. The sketch begins with one, then two, and quickly three objects. Even as the set grows to six and seven items, the audience can mentally group the objects to quickly discern the parity of the set. Later in the sketch, the set grows first by three and then by five. At this stage, it is harder to visually anticipate the resulting remainder. Primary school audiences are often heard counting the number of objects in order to calculate whether such aloneness will result. The clown almost always seems to discover his mathematical destiny more slowly than the audience. Children and adults alike often giggle when they compute sets of odd size. This sketch, like many others that I perform, contextualizes a mathematical idea, enabling an audience to place an abstract concept into a story. This mathematical mime piece presents mathematics in a physical rather than algebraic way. Whether written in formulas or portrayed through the movement of mime, such expressions are pointers to an intellectual experience. This sketch does not offer a proof of the divisibility of odd and even numbers. It does reflect what mathematicians often do prior to writing a proof—analyze simple examples and look for trends in cases of increasing complexity. This sketch is quite popular with audiences of varying ages. Nonmathematicians enjoy anticipating the clown’s ultimate fate with the set of ever-growing size. Mathematicians also enjoy the sketch. The concept it portrays is quite elementary. Mathematicians can be silenced by elegant proofs of simpler concepts and somehow dissatisfied with seemingly clumsy proofs of complex material. The pleasure brought about by a good proof can similarly be evoked through the performing arts—in this case, through a mimetic translation of mathematical thoughts. Why shouldn’t we be silenced by an elegant sketch? Completed proofs uncover some truth; we have all experienced the exhilaration of such discoveries. Mathematicians can be quite excited by the mime sketches I perform as they illustrate their abstract world. Regarding this sketch on remainder, a child from a multi-age thirdand fourth-grade classroom wrote,