MATHEMATICAL DEMONSTRATIONS IN LATEX

A repository for documenting and sharing mathematical proofs, written in LaTeX with clarity, rigor, and professional style.

---

Description

This project is a growing collection of fundamental and classic mathematical demonstrations. It aims to provide reference material, support for self-study, and practice in mathematical writing.

Whether you are a student, educator, or self-taught learner, this document offers step-by-step explanations of mathematical proofs that build foundational understanding and abstract thinking.

---

Demonstrations Table

Category	Demonstration
Arithmetic	$1 = 0.999\ldots$
Arithmetic	$1 > 0$
Arithmetic	$0 \cdot a = 0$
Number Theory	$\sqrt{2}$ is irrational
Arithmetic	Sum of two even numbers is even
Arithmetic	Sum of two odd numbers is even
Arithmetic	Square of an odd number is odd
Arithmetic	If $ab=0$, then $a=0$ or $b=0$
Summation	Sum of first $n$ natural numbers: $1+2+\cdots+n = \frac{n(n+1)}{2}$
Number Theory	There is no maximum natural number
Number Theory	$\sqrt{3}$ is irrational
Arithmetic	Product of two odd numbers is odd
Algebra	$(-a)(-b) = ab$
Inequalities	AM-GM Inequality for two numbers
Inequalities	$\sqrt{ab} \leq \frac{a+b}{2}$ for $a, b \geq 0$
Algebra	$n^2 \geq n$, $\forall n \geq 1$
Algebra	$2^n > n$, $\forall n \in \mathbb{N}$
Prime Numbers	There exist infinitely many primes
Arithmetic	Product of an even number by any integer is even
Summation	Sum of first $n$ odd numbers: $1 + 3 + \cdots + (2n-1) = n^2$
Arithmetic	Difference of two even numbers is even
Arithmetic	Square of an even number is even
Algebra	$(a+b)^2 = a^2 + 2ab + b^2$
Algebra	$a^2-b^2=(a-b)(a+b)$
Algebra	Quadratic Formula ($ax^2+bx+c=0$)
Algebra	$a^3 + b^3 = (a+b)(a^2-ab+b^2)$
Geometry	Pythagorean Theorem
Geometry	Area of a triangle
Geometry	Sum of interior angles of a triangle
Physics	$d = vt$ (relationship between distance, velocity, and time)
Physics	Newton's Second Law ($F = ma$)

---

Quick Compilation

1. Clone this repository:

   git clone https://github.com/Alvaro-Manzo/DEMOSTRACIONES.git
   cd DEMOSTRACIONES

2. Compile the main file (main.tex):

   pdflatex main.tex

   Or use online platforms like Overleaf.

---

Project Structure

DEMOSTRACIONES/
├── main.tex      # Main LaTeX document
├── main.pdf      # Generated PDF after compilation
├── LICENSE       # License and terms of use
└── README.md     # This guide

---

Why This Project?

• Mathematical Clarity: Promotes precise language, clarity, and mathematical rigor.
• Step-by-Step Proofs: All arguments and demonstrations include detailed, line-by-line explanations.
• Accessibility: Ideal for students, educators, and self-taught learners.
• Reference Material: Includes a mathematical and physical glossary for quick reference.
• Open Source: Collaborative project encouraging contributions and feedback.

---

Key Features

✓ Well-formatted proofs using LaTeX's theorem environments
✓ Logical progression from basic to advanced concepts
✓ Complete explanations with no steps skipped
✓ Professional typography ensuring readability and elegance
✓ Free and open-source under Creative Commons license

---

Contributing

We welcome contributions! If you would like to:

• Add a new demonstration
• Report errors or typos
• Improve existing proofs
• Enhance documentation

Please feel free to:

1. Open an issue
2. Fork the repository and submit a pull request

---

Author

Álvaro Yamil Manzo Ayora
GitHub @Alvaro-Manzo
Email

---

License

This work is licensed under the
Creative Commons Attribution 4.0 International (CC BY 4.0) Please check it and do the same with CONTRIBUTING if you are interested to make an edit. :p
See LICENSE for details.

---

Getting Started with LaTeX

If you're new to LaTeX, here are some helpful resources:

• Overleaf Tutorials
• The Not So Short Introduction to LaTeX
• LaTeX Project

---

Recommended Reading Order

For beginners, we suggest studying demonstrations in this order:

1. Basic arithmetic proofs ($0 \cdot a = 0$, $1 > 0$)
2. Parity and divisibility proofs
3. Proof by induction
4. Proof by contradiction
5. Classical theorems (Pythagorean, prime infinitude, etc.)

---

We encourage you to recommend, share, and use this material in your studies!

"The beauty of mathematics reveals itself only to its most patient followers." — Maryam Mirzakhani