Fall 2025 : Math 100 Introduction to the Profession

Math 100 Introduction to the Profession

Instructor: Hemanshu Kaul
Office: 125C, Rettaliata Engg Center.
E-mail: kaul [at] iit.edu

Class Time: 1:50pm-3:05pm, Tuesday and Thursday
Place: 222, Alumni Memorial Hall (Previously, 106, Rettaliata Engg Center).

Office Hours: Tuesday 12:45-1:15pm and Thursday 3:05-4pm. And by appointment in-person or through Zoom (send email to setup appointment).
Questions through Canvas Discussion Forums are strongly encouraged.

TA Office Hours: Minjung Michelle Kang, Mondays 11:30am-1pm & Thursdays 11:30am-1pm, at Kaplan Institute Cafe till RE remains closed (previously, RE 129 or at Math Tutoring Center). At this link you can find the latest TA office hours schedule for all TAs.

Course Information:

This course is an introduction to the nitty-gritty of work and study in the mathematical sciences. Fresh applied math majors are introduced to undergraduate studies and careers in applied math; to the effect of ethics, diversity, and related social issues to work and study in mathematics; to the study of mathematical statements, logic, and proofs; to experimentation and computation in mathematics; and they work on a semester-long group project that investigates a mathematical topic.

The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, as well as other relevant information. Read it carefully!

Advice for students:

Here is my Welcome Letter to students like you.

Excellent advice by Francis Su on good mathematical writing.

Why do we have to learn proofs? by Josh Cooper.
On a more abstract note, here is a discussion by Tim Gowers on Language and Grammar of Mathematics - which is what you are learning in a course like this.

Excellent advice for math majors by Terry Tao, 2006 Fields medallist. Required reading.

Read this book on a variety of experiences in the journey to learn mathematics: Living Proof

Some of the primary sources of information/discussion for careers in Mathematical Sciences:
MAA - Careers
SIAM - Careers
INFORMS - Careers
AMS - Careers

Class Announcements:

Monday, 8/25 : See above and in the Course Information Handout below for the updated TA office hours information.

Tuesday, 8/19 : Check this webpage regularly for homework assignments, announcements, etc.

Course Topics:

Undergraduate studies in Applied Math - During and Afterwards : Careers - Documentary, discussions, presentations by instructor and by Career Management Center. Managing Courses, how to study, and making the jump from High School Math to Undergrad Math.
Ethics in Mathematical Sciences : How ethics, diversity, and related social issues affect work and study in Mathematics and its applications.
Mathematical Statements and Logic : Second order logic and simple mathematical statements, Comparison of Existential and Universal statements, the conjunction, disjunction, and their negations. Comparing Mathematical statements to real-life statements. Understanding Mathematical definitions and Statements of Theorems.
Mathematical Proofs : Classical examples of proofs (and non-proofs) from Geometry, Number Theory, Set Theory. Different types of proofs and how to find them and how to write them - Direct implications, Contradiction, Contrapositive, Mathematical Induction.
MATLAB/Python : How to use MATLAB/Python to learn experimentation and computation in Mathematics - Fibonacci Sequence, Matrices, etc.
Project : A semester-long group project on investigation of a mathematical topic.
Scavenger Hunt : Meeting faculty and learning through discussion.
Special Lectures : Lectures by faculty in Applied Math introducing different fields/topics within Applied Math and their connection to real-world, further courses in the curriculum.

Special Lectures:

Thursday, 10/16: Professor Michael Pelsmajer (Discrete Math)
Thursday, 10/23: Professor Nathan Kirk (Computational Math)
Thursday, 10/30: Professor Chun Liu (Dynamical Systems)
Thursday, 11/13: Professor Jacek Dzikowski (AI & LLM)
Tuesday, 11/25: Professor Lane Yeung (Randomness)

Projects:

The project requires you to work in teams of 2-3 students with the Professor offering the project. The project adviser will specify the details of what his/her project entails, the outlines will be given in the descriptions below.
The successful completion of the project entails:

Before the project Starts -

1) Send me an email with your choice of project(s) on Thursday, 10/2 after class. Projects will be assigned in first-come first-serve order.
2) After I confirm your project and team members by Friday, 10/3, write an email to your chosen project adviser by Sunday, 10/5 to set up an appointment with him/her.

After the project starts -

3) Regular meetings with the project adviser and following his/her instructions.
4) At the end of each week before Friday 11:59pm, each student will also write a short 2-3 line summary of what she/he did related to the project over the past 7 days. This will be directly written in the MS Doc file for each project I will share with you.
5) A midterm report on the status of the project before Saturday, 11/1 .
6) A final meeting with project adviser before Wednesday, 12/3 to confirm the completion of the project, and discuss the outcome of the project.
7) Submission of project report by email to me before the start of the Final exam week on Saturday, 12/6, 11:59pm.
8) Submission of project presentation file (as a PDF file) by email to me before the start of all presentations on TBA.
9) Final project presentation in front of your classmates during the Final exam week: TBA

Project Topics:

Project #1: Advisor: Prof. Hemanshu Kaul.
Project #2: Advisor: Prof. Hemanshu Kaul.
Project #3: Advisor: Prof. Hemanshu Kaul.
Project #4: Advisor: Prof. Robert Ellis.
Project #5: Advisor: Prof. Robert Ellis.

Weekly Class Log with Discussion Questions and HW:

Week #1 :
- Topics and Readings:
  Introduction to the course and each other; President Garfield's proof of Pythagoras's Theorem; a foundational discussion of President Garfield's proof, and the geometric lemmas and axiom(s) underlying it.
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 3.1 to 3.7 of Alcock, How to Study as Math Major; Read Sections 2.1 and 2.2 of Devlin, Intro to Mathematical Thinking.
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#1 for Submission [PDF]. Due Thursday, 8/28, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Week #2 :
- Topics and Readings:
  - Presentations by Student Organizations: SIAM Student chapter; Machine Learning club; Women in STEM.
  - Dissection of a square into smaller n squares; Axioms, Definitions, and related concepts; Truth table and comparing logical statements; Proof by contradiction - There are infinitely many primes; Universal vs Existential statements - how to prove and disprove such statements; All real numbers are irrational vs. There is a real number that is irrational; Proof of sqrt(2) is irrational.
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 4.1, 4.2, 4.5, 4.6, 4.7, 4.8 of Alcock, How to Study as Math Major; Read Sections 2.3 and 2.4 of Devlin, Intro to Mathematical Thinking.
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#2 for Submission [PDF]. Due Thursday, 9/4, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Week #3 :
- Topics and Readings:
  - Presentations by Office of Student Affairs/ Career Services/ Elevate program: Experiential Learning; Career Services; and Student support resources at Illinois Tech.
  - Proof of sqrt(2) is irrational. Discussion of proof and disproof of two statements with multiple quantifiers and interchanged "for all" and "there exists" quantifiers from Alcock. Discussion of HW problems from Devlin.
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 4.9-4.10, 5.1-5.6 of Alcock, How to Study as Math Major; Read Sections 3.1-3.3 of Devlin, Intro to Mathematical Thinking.
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#3 for Submission [PDF]. Due Thursday, 9/11, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Week #4 :
- Topics and Readings:
  Discussion of Mathematical Statements, implication, truth tables, contrapositive, logically equivalent statements, quantifiers, and HW problems from Devlin. Discussion of how to understand "product of three consecutive integers is divisible by 6" from Alcock through examples and non-examples, a non-proof, a case-by-case proof, and finally a short proof that captures the mathematical truth underlying the case-by-case analysis.
- Homework:
  Reading HW (Be ready to discuss in class next week): Read Sections 6.1-6.4 of Alcock, How to Study as Math Major; Read Sections 3.4-3.5 of Devlin, Intro to Mathematical Thinking.
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#4 for Submission [PDF]. Due Thursday, 9/18, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Week #5 :
- Topics and Readings:
  - Discussion of 5 types of proofs with examples: Direct argument, Case-by case analysis, Proof by Contrapositive, Proof by Contradiction, Principle of Mathematical Induction. Discussion with example of how sometimes we can discover a short argument (the central truth) underlying case-by-case argument. Discussion of "characterization"/"if and only if" through examples. Discussion of some old and new HW problems.
  - Discussion of "Pythagoras Triple Square Day"; Two types of Pi Day - which is one is better?.
  - Discussion of "Learning how to learn" as an educational principle; utility and dangers of computational technology/ Google/ AI. Also look at Are AI hallucinations unavoidable?.
- Homework:
  Reading HW (Be ready to discuss in class next week):
  We will begin our study of MATLAB as a tool for mathematical experimentation from next week. We will access MATLAB through the IIT's Virtual Computing Lab (VCL) which is accessible through your Illinois Tech portal. Try using the VCL on your own before classes next week and email me/OTS if you face any issues. See FAQs regarding VCL. Refer to VCL Instructions here.
  Also, create a Mathworks account using your official hawk.illinoistech.edu email address.
  Watch/ Read the following and be ready to try examples in class next week:
  - MATLAB Overview
  - Getting Started with MATLAB
  - MATLAB Development Environment
  - Preface of Moler's Textbook
  - Chapter 1 of Moler's Textbook (We will start with these examples next week.)
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading.
  HW#5 for Submission [PDF]. Due Thursday, 9/25, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Weeks #6 and #7 :
- Topics and Readings:
  - Further discussion and examples of proofs by Induction. Proof of formula for sum of first n natural numbers. Strong idunction and proof of Fundamental Theorem of Arithmetic (every positive integer can be factored into prime numbers).
  - Discussion of HW problems including how to show sqrt(3) is not rational.
  - What is iteration? Exploring experimentally the formula for sum of first n natural numbers and verifying its correctness using a computer. Writing pseudocode for each part and comparing the results. Iteration in MATLAB using chapter 1 of Moler. Approximating Golden ratio.
  - (9/30 R.Ellis) Special lecture on Machine learning and related algorithms, etc.
  - Discussion of projects and their structure and expectations.
  - Discussion of recursive definitions, like sum for first n natural numbers or infinite tower of sqrt(2), in math and in MATLAB. - Further discussion of induction. Discussion of how to study an infinite tower of sqrt(2)^sqrt(2)^ ....: mathematical expression and its properties. Experimentation using MATLAB/Python.
- Homework:
  Reading HW (Be ready to discuss in class next week):
  - Review the MATLAB reading HW from last week.
  - Read and try everything in Chapter 1 of Moler's Textbook as we did in class.
Weeks #8 and #9 : Fall Break on 10/14
- Topics and Readings:
  - (10/7 R.Ellis) Special lecture on Machine learning and related algorithms, etc.
  - Discussion of how to study an infinite tower of sqrt(2)^sqrt(2)^ ....: mathematical expression, proof of its properties using mathematical induction, and how to find the value of this infinite expression(!).
  - Discussion of MATLAB HW and using MATLAB copilot.
  - (10/14) Fall Break
  - (10/16) Special Lecture Prof Michael Pelsmajer on Discrete Math.
- Homework:
  Reading HW (for next two weeks; due to holiday on 10/14 and special lecture on 10/16):
  Work on your projects and send me weekly updates.
  Read the following and be ready to try examples:
  - Chapter 2 of Moler's Textbook (We will start with these examples and problems in the week after next in class. See below for the corresponding problems you will have to submit.)
  
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading. When reading a algorithms or programming textbook, have the relevant computer software up and running next to you and try out the various commands, etc. as you read them. Use "help"/"copilot" to further explore each topic.
  
  HW#7 for Submission: Solve and submit the following problems from the Chapter 2 of Moler's Experiments with MATLAB 2.3, 2.4, 2.6, 2.9.
  Due Thursday, 10/23, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.
Weeks #10 and #11 :
- Topics and Readings:
  - Discussion of MATLAB chapter 2 and the related HW problems.
  - Discussion of Matrices and matrix equations; MATLAB chapter 4 and related HW problems
  - Discussion of Research opportunities with professors and through REUs.
  - (10/23) Special Lecture Prof Nathan Kirk on Computational Math/Monte Carlo.
  - (10/30) Special Lecture Prof Chun Liu on Dynamical Systems/PDEs.
- Homework:
  Reading HW:
  Work on your projects and send me weekly updates.
  Read the following and be ready to try examples:
  - Chapter 4 of Moler's Textbook (We will start with these examples and problems in the week after next in class. See below for the corresponding problems you will have to submit.)
  
  Follow the HW instructions and suggestions in the Course Information Handout. When reading a mathematics textbook, you need to have a pencil and paper next to you so you can make note of any thoughts of confusion (or of clarity!) that strike you as you read, and you can scribble the details of examples and non-examples of the concepts you are reading. When reading a algorithms or programming textbook, have the relevant computer software up and running next to you and try out the various commands, etc. as you read them. Use "help"/"copilot" to further explore each topic.
  
  HW#8 for Submission: Solve and submit any 4 out of the 5 following problems from the Chapter 4 of Moler's Experiments with MATLAB 4.2, 4.4, 4.5, 4.6, 4.10ab
  Due Thursday, 11/6, by 11:59pm. Submit a PDF file through Canvas Assignment.
  Questions? Ask on Canvas. Or, send email. Or, stop by office hours.

MATLAB Information:

Our textbook:
Moler's Experiments with MATLAB

MATLAB Access:
We will access MATLAB through the IIT's Virtual Computing Lab (VCL) which is accessible through your my.iit.edu account.
See FAQs regarding VCL.
Refer to VCL Instructions here.

Introduction to MATLAB:

Matlab Links for More Information

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