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## Visualizing Complex Numbers: Instructor's Guide

### Main Ideas

- Different forms of complex numbers ($x+iy$ and $re^{i\phi}$)
- Use of Argand diagrams

### Students' Task

*Estimated Time: 5 minutes*

Students use their left arm to represent numbers on an Argand diagram.

### Prerequisite Knowledge

- Argand diagrams
- Forms of complex numbers

### Props/Equipment

- Individual students

### Activity: Introduction

Typically this activity follows a short review of complex numbers and various calculations which are commonly used in physics. Students may require an introduction to Argand diagrams and forms of complex numbers. Various forms, representations, and calculations involving complex numbers may be new to many middle division physics students. Alternatively, for students who have previously learned complex algebra, this activity can be used as a review and a way for the instructor to determine students' geometric fluency with complex numbers.

Students are introduced to the idea of using their left arm as an Argand diagram where the left shoulder is the origin and practice using this representation to show various complex numbers. When the arm is parallel to the ground and in front of the student, the number represented is pure real and positive. When the arm is perpendicular to the ground and above the head of the student, the number represented is pure imaginary. Complex numbers can be given in both rectangular ($x+iy$) and exponential ($re^{i\phi}$) forms.

### Activity: Student Conversations

- Individual representation of complex numbers
- 1, $i$, $-i$: These three examples help to orient the students to the representation and very few have issues.
- $-3i$: This brings up the issue of representing length and the challenge when one's arm is only so long.
- $e^{i\pi/4}$: This is when some students begin to have trouble and it is a good idea to remind them at this point of the exponential ($re^{i\phi}$) form of complex numbers where $r$ represents the length and $\phi$ the rotation in the complex plane.
- Multiply by $e^{i\pi/2}$: This allows students to recognize multiplication by a phase results in a rotation in the complex plane.

### Activity: Wrap-up

Short class discussions are encouraged following each example before moving onto another complex number or operation. Further examples can be used to show the geometry of other calculations such as the complex conjugate and the norm. This can lead into introducing when to use a particular form of a complex number and complex algebra calculations which often occur in physics.

### Extensions

This activity is the beginning of the Visualizing Complex Numbers sequence which presents an engaging articulation of complex numbers and their varied representations.

- Follow-up activities:
- Visualizing Complex Two-Component Vectors: In this kinesthetic activity, students work in pairs to represent complex two-component vectors and transformations on those vectors.
- Visualizing Complex Time Dependence for Spin 1/2 Systems: This kinesthetic activity is designed to help students visualize complex time dependence of the spin-1/2 system by using their arms, in pairs, to represent the different time dependencies of the eigenstates and the superposition states.