Thinking Real, Doing Complex: The Case of Exponents and Logarithms

Abstract

The operation of exponentiation stands out compared to other arithmetic operations like addition and multiplication. It is not commutative, associative, or even defined for every pair of real numbers. Perhaps unsurprisingly, the differences between addition, multiplication, and exponentiation of complex numbers are even more stark. Students tend to first encounter expressions of the form in an undergraduate course in complex analysis. This investigation employed two theoretical frameworks, APOS Theory and Toulmin argumentation, to analyze how students reason about complex exponents and logarithms and how they navigate these mathematical differences with specific attention to how they invoked properties of real-valued exponents and logarithms. This invocation has been referred to as Thinking Real, Doing Complex, or TℝDℂ. The first framework, APOS Theory, was used to analyze two interviews with two college students, resulting in four individual interviews. The first individual interviews served to investigate how students understand real-valued exponents and logarithms and complex numbers, and the second interview served to investigate how they understand complex-valued exponents and logarithms. To analyze these interviews, I drew on earlier studies on students' conceptions of exponents or logarithms, which employed an APOS framework, and analogized their findings to synthesize a more general codebook. The second framework, Toulmin argumentation, was used to analyze interviews with a pair of college students collaborating on a series of tasks. These paired interviews helped me investigate how students collaborated in solving the tasks. To analyze these interviews, I modeled their discussion with Toulmin diagrams and tracked which mathematical ideas functioned as shared between the two participants. My findings from the individual interviews suggest that, while students tend to conceptualize of real exponents and logarithms within a single exponent/logarithm schema, they conceptualize complex exponents and logarithms within a distinct complex-valued exponent/logarithm schema. That is, there does not appear to be a general "exponent/logarithm" schema that contains all relevant conceptions. However, while a student's real-valued exponent/logarithm schema might be distinct from their complex-valued exponent/logarithm schema, the organization of the two schemas may be similar. As such, an APOS characterization of Thinking Real, Doing Complex phenomenon can be characterized in terms of similarities of schemas. My findings from the paired interviews suggest that students may also engage with questions about complex analysis by transforming them into equivalent statements about real numbers. In addition to thinking about complex exponents algebraically or geometrically, they may also reason about them numerically. These findings suggest that another manifestation of the Thinking Real, Doing Complex phenomenon can be extended by considering that students may not strictly analogize properties about real exponents and logarithms to complex exponents and logarithms. They might conceptualize complex exponents and logarithms as mathematical objects or operations distinct from their real counterparts, or they might transform statements about complex numbers into ones about real numbers. More broadly, the findings suggest that instruction about complex exponents and logarithms and their relationship to the branch cut can be emphasized more, and perhaps not necessarily confined to undergraduate complex analysis.