## Activity: Using Arms to Visualize Complex Numbers (MathBits)

Lie Groups and Lie Algebras 23 (4 years)
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
• This activity is used in the following sequences
What students learn
• Geometric representation of complex numbers in rectangular $x+iy$ and exponential $re^{i\phi}$ forms;
• Multiplication of complex numbers in exponential form adds the phases;
• Complex conjugation is a reflection in the real axis.

## Instructor's Guide

### Introduction

Start with a short review of complex numbers in rectangular $x+iy$ and exponential $re^{i\phi}$ forms. Include an introduction to the graph of the complex plane (Argand diagram). Be aware that, while some students may have seen this content as early as high school, other students may only know that $i^2=-1$. Discuss basic complex number algebra (addition and subtraction, multiplication, and division). See for example the content in the first six sections of LinAlg.

Draw the complex plane on a board at the front of the room and have students stand facing the right-hand side of the room. Introduce students to the idea of using their left arm as an Argand diagram where their left shoulder is the origin. Ask them to sweep their left arm around in a circle and show them that their arm aligns with the complex plane on the board. Point out that when their arm is parallel to the ground and in front of the student, the number represented is pure real and positive. When their arm is perpendicular to the ground and above the head of the student, the number represented is pure imaginary. Have students practice by showing some simple complex numbers, e.g. $1$ and $-i$.

Now show the students a written prompt, have them close their eyes and act out the given complex number. Complex numbers can be given in both rectangular $x+iy$ and exponential $re^{i\phi}$ forms. After each prompt, have students open their eyes and compare. Discuss as necessary.

### Student Conversations

This activity is particularly useful in helping the instructor identify which students have only minimal background on complex numbers and may need extra practice.

• Individual representations of complex numbers
• 1, $i$, $-i$: These three examples help to orient the students to the representation and very few have issues.
• $-3i$: This example brings up the problem of representing length when one's arm is only so long.
• $e^{i\pi/4}$: This example is when some students may begin to have trouble. Remind them of the exponential ($re^{i\phi}$) form of complex numbers where $r$ represents the length and $\phi$ the angle in the complex plane.
• Multiply the previous complex number by $e^{i\pi/2}$: Introduce the word "phase" for a complex number of unit norm, i.e. $r=1$. This example is intended to allow students to recognize that multiplication by a phase results in a rotation in the complex plane.
• Multiply by $i$: This example is mathematically the same as the previous one, but conceptually harder. Emphasize that to multiply two complex numbers it may be easiest to first write them in exponential form. Discuss that multiplication by $i$, in particular, is a rotation by $\frac{\pi}{2}$ in the complex plane.
• Find the complex conjugate of the previous complex number: Emphasize that complex conjugation is reflection in the real axis BECAUSE the phase goes to the negative of itself.

### Wrap-up

Short class discussions are encouraged following each example before moving onto another complex number or operation. These examples can lead into a mini-lecture about when to use a particular form of a complex number, i.e. rectangular form for addition and, often, exponential form for multiplication and division.

Use your left arm to represent the complex plane, with your left shoulder representing the origin. For each complex number or operation given to you by your instructor, move your arm so that it points to the appropriate complex number.
The prompts are deliberately not given here, as they are best given one-by-one during class. A summary of the prompts appears in the solution.

• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

##### Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.
• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• accessibility_new Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
• group Fourier Transform of the Delta Function

group Small Group Activity

5 min.

##### Fourier Transform of the Delta Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students calculate the Fourier transform of the Dirac delta function.
• accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

accessibility_new Kinesthetic

30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

group Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
• assignment Circle Trig Complex

assignment Homework

##### Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2023 (2 years)

Find the rectangular coordinates of the point where the angle $\frac{5\pi}{3}$ meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

• group Spin-1 Time Evolution

group Small Group Activity

120 min.

##### Spin-1 Time Evolution
Quantum Fundamentals 2023

Students do calculations for time evolution for spin-1.
• assignment Phase

assignment Homework

##### Phase
Complex Numbers Rectangular Form Exponential Form Square of the Norm Overall Phase Quantum Fundamentals 2023 (3 years)
1. For each of the following complex numbers $z$, find $z^2$, $\vert z\vert^2$, and rewrite $z$ in exponential form, i.e. as a magnitude times a complex exponential phase:
• $z_1=i$,

• $z_2=2+2i$,
• $z_3=3-4i$.
2. In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$
• group Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

Author Information
Corinne Manogue
Learning Outcomes