Is i a Number? An Examination of Advanced Undergraduate Students’ Definitions of Number

Abstract

The purpose of this study was to examine the ways advanced mathematics students define number and the degree to which their definitions extend to different number domains. Of particular interest for this study are learners’ fundamental conceptions of number and the implications for learners’ interpretations of complex numbers (a + bi). Using the theoretical framework of number worlds and qualitative data from interviews, we examined students’ foundational conceptions of number. We identified two distinct categories of participants’ core definition of number, Physical and Symbolic. We compare these core definitions and the extent to which participants could stretch their definition of number to include other types of numbers. This study contributes an initial exploration into the number worlds of college students with substantial mathematical backgrounds.

Appendix A

Thank you for agreeing to help us out with this interview. I just want to remind you that none of what you say or do here will affect your grade. Please keep in mind that we are not looking for particular answers, we really just want to understand your thinking.

• 1. The first question deals with absolute value. In your thinking, what is the meaning of absolute value? [student responds – follow up with “can you say a little bit more of that (if it seems useful)] Ok, thanks. Now how about an algebraic or function definition of absolute value? Can you give me an algebraic or function definition of absolute value [Give student page 1]

  If necessary, follow up writing $|x| =$ on their paper and concurrently asking,

  So how would you complete $|x| = ?$

  Let’s take a look at the second part of the question, which asks for a geometric definition of absolute value. Can you give me a geometric definition of absolute value [student responds]

  Follow up with, “So how would you think about absolute value on a number line?” and draw an empty number line on their paper like the following:

  

  How do you think about the meaning of $|2|$ on a number line?

  How do you think about the meaning of $|-6|$ on a number line?

  When you thought about $|-6|$ on number line did you think about any movement or motion?

• 2. Thanks, let’s take a look at the next problem. How can you use a number line to demonstrate why $3+4=7$?

  [Give student page 2]

  How can you use a number line to demonstrate why $7*(-1) = -7$?

  [Student may do -1 seven times. If so, then ask them “What happens to 7 when you multiply by (-1)?” and concurrently draw number line with 0 and 7 on it].

• 3. The next question is kind of a philosophical question. It deals with what, from your perspective, makes a number a number?

  [Give student page 3]

  [If necessary, follow up with In other words, what characteristics or properties do numbers have?]

  How does zero fit in with that?

  How does -5 fit in with that?

  How does square root of two fit in with that?

  What about i? How does that fit in with this?

• 4. Ok, the next problem is more concrete. [Give student page 4]

  How did you figure that out?

  You mentioned the word root [Or, Other students who solved this problem mentioned the word root] What does it mean for a number to be a root of a quadratic? [If student does not know the root use the term zero instead]

  Do you have any geometric ways to think about the roots [zeros] to each of the quadratic equations?

  What would you get if you plug $2+i\sqrt{2}$ or $2-i\sqrt{2}$ into the third quadratic equation? [Or use whatever root they calculated and if they start doing the computations stop them and ask them if they can anticipate what the result of their computation is going to be]

• 5. Ok, thanks. Now in the next problem you get to do some calculations. [Give student page 5]. Please go ahead and do these problems.

  Do you have a way of thinking about part a) geometrically?

  [If a student says no, then ask the following: Is there some way you could use the complex plane to show how that works? [Draw Real and Imaginary axes for them and plot the point $3+i$ as going over 3 units and up 1 unit if necessary – do not label the point, but do put hash marks on the axes]

  [If a student has their own unique way of thinking about this geometrically then do not introduce the complex plane]

  Do you have a way to think about part b) geometrically?

  Do you have a way to think about $i * (3 + i)$ geometrically?

  How about $2i * 3+i$ ?

• 6. Let’s go back to the first problem where you thought about absolute value. How do you think about absolute value of a complex number, say $3 + i$?

  Do you have a way to think about this geometrically?

• 7. [Go back to page 2] Let’s go back to the second problem where you thought about $7*(-1)$ with a number line.

  One student told me that when they think about $7*(-1)$ on number line she thinks of a 180-degree rotation about zero [demo this in the plane of the paper]. A different student told me that he thinks about going from 7 to -7 through zero on the number line [demo]. Do you usually think about it like the first student, like the second student, or not like either one?

  [After student responds ask the following] So what do you think of the first student’s way of thinking about a 180-degree rotation? Are there advantages or disadvantages between this way of thinking and the second student’s way (or your way of thinking).