===== Introduction to LaTeX for Mathematical Expressions =====

The goal of the lesson is to become familiar with LaTeX, specifically for the purpose of writing mathematical expressions.

==== 1. Introduction to LaTeX ====

**What is LaTeX?**

LaTeX is a high-quality typesetting system, primarily used for technical and scientific documents. It is particularly powerful for formatting complex mathematical equations and formulas, making it a preferred choice in academia and research.

**What are the advantages of LaTeX?**

  * **Handling Complex Documents**: It is ideal for mathematical formulas, citations, and cross-referencing in technical writing.
  * **Consistent Layout**: Automatically ensures a uniform, professional design by separating content from formatting.
  * **Scalability**: Suitable for large projects, allowing version control and modular structure.
  * **Longevity**: LaTeX's plain text format ensures long-term compatibility and durability.
  * **Academic Preference**: Often required in academic and scientific publishing.

**Getting Started:**

  - **Overleaf**: We will use Overleaf, a free online LaTeX editor, which allows you to write and compile LaTeX documents directly in your browser.
    - Sign up at [[https://www.overleaf.com/|Overleaf]].
    - Overleaf offers collaborative features, version control, and a vast library of LaTeX templates.

**Basic Document Structure:**

<code LaTeX>
\documentclass{article}  % Specifies the document class (article, report, book, etc.)
\begin{document}          % Begins the content of the document
% Your content goes here
\end{document}            % Ends the content of the document
</code>

  - **\documentclass{article}**: Defines the overall layout and style of the document.
  - **\begin{document}** and **\end{document}**: Everything between these commands will be included in the output document.

==== 2. Writing Basic Mathematical Expressions ====

=== Inline vs. Display Math ===

  - **Inline Math**: For mathematical expressions that appear within a line of text, use ''$...$''.
    - E.g. $E = mc^2$ is written as ''$E = mc^2$'' in LaTeX.
  - **Display Math**: For standalone equations, use ''$$...$$''.
    - E.g. To display $$E = mc^2$$ on its own line, use ''$$E = mc^2$$''.

== Example ==

<code LaTeX>
\documentclass{article}
\begin{document}

The equation $E = mc^2$ is famous in physics. It is so important that we can highlight $$E = mc^2$$ by putting it to a separate line.

\end{document}
</code>

This code will became:

{{:tanszek:oktatas:techcomm:pasted:20240902-141219.png}}

=== Basic Mathematical Symbols ===

  - **Exponents (superscripts)**: Use ''^'' for superscripts.
    - E.g. $x^2$ is written as ''$x^2$''.
  - **Subscripts**: Use `_` for subscripts.
    - E.g. $a_1$ is written as ''$a_1$''.
  - **Fractions**: Use `\frac{numerator}{denominator}`.
    - E.g. $\frac{a}{b}$ is written as ''$\frac{a}{b}$''.

== Examples ==

<code LaTeX>
\documentclass{article}
\begin{document}

% Exponent and subscript
The formula for the area of a circle is $A = \pi r^2$.

% Fraction
The equation $\frac{a}{b} = c$ represents a fraction.

% Combined
The equation for kinetic energy is $K = \frac{1}{2}mv^2$.

\end{document}
</code>

This code will become:

{{:tanszek:oktatas:techcomm:pasted:20240902-141644.png}}

==== 3. Special Mathematical Symbols in LaTeX ====

LaTeX provides a variety of symbols to accurately represent mathematical expressions.

  - The **plus-minus symbol** is used to denote values that can be either positive or negative and is written as ''\pm'', which displays as $\pm$.
  - To express **square roots**, the square root symbol is used, which is written as ''\sqrt{...}''. For example, ''\sqrt{2}'' produces $\sqrt{2}$ 
  - For **higher-order roots**, such as a cubic root, the syntax is ''\sqrt[3]{...}'', yielding $\sqrt[3]{9}$
  - Another common symbol is the **infinity symbol**, represented as ''\infty'', and it is displayed as $\infty$ 
  - For **greater than or equal to** and **less than or equal to** symbols, use ''\geq'' and ''\leq'', which render as $\geq$ and $\leq$, respectively.

=== Summation ===

The general form of summation in LaTeX is written using the ''\sum'' command. For example, the sum from $i=1$ to $n$ is given by:

$$\sum_{i=1}^{n} i^2$$

This expression sums the squares of integers from 1 to $n$.

=== Derivative ===

The derivative of a function $f(x)$ with respect to $x$ is represented in LaTeX using the ''\frac'' command for fractions. The notation for the derivative of $f(x)$ with respect to $x$ is:

$$\frac{d}{dx} f(x)$$

This gives the rate of change of $f(x)$ with respect to $x$.

=== Partial Derivative ===

For partial derivatives, the ''\partial'' command is used. The partial derivative of a function $f(x, y)$ with respect to $x$ is:

$$\frac{\partial}{\partial x} f(x, y)$$

This expression gives the partial derivative of $f$ with respect to $x$, holding other variables constant.

=== Partial Integration ===

Partial integration, also known as integration by parts, can be expressed in LaTeX. For the specific example of integrating $x \sin(x)$ from $a$ to $b$, the integral is written as:

\[
\int_{a}^{b} x \sin(x) \, dx
\]

This represents the definite integral of $x \sin(x)$ with respect to $x$ from $a$ to $b$.

==== 4. Aligning Equations ====

=== Align Environment ===
  - The `align` environment is used to align multiple equations. Each line of the equation is aligned using the `&` symbol, typically before the equal sign or any other operator.
  - Use `\\` to separate lines.

== Example ==

<code LaTeX>
\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{align}
  3x + 2y + 0z &= 6 \\
  4x - y &= 25
\end{align}

\end{document}
</code>

This code will become:

{{:tanszek:oktatas:techcomm:pasted:20240902-142323.png}}

**Explanation:**
  - **\usepackage{amsmath}**: The `amsmath` package is required for advanced mathematical typesetting features, including the `align` environment.
  - **&**: This symbol is used to align equations at the specified point, usually before an operator like `=`.

**Tips:**
  - You can label equations using the ''\label{}'' command and refer to them later with ''\ref{}''.
  - Example:

<code LaTeX>
\begin{equation} \label{eq:quadratic}
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}
</code>

To refer to this equation later, use ''Equation \ref{eq:quadratic}''.


==== 5. Exercise ====

Reproduce the following mathematical proof in LaTeX. Use inline and display math, as well as basic and special symbols! Save the result in PDF format!

{{:tanszek:oktatas:techcomm:pasted:20240902-144955.png}}


Raw text:

<code>
Proof: √2 is Irrational

Assume, for contradiction, that 2 is rational. Then it can be expressed as a fraction a/b, where a and b are coprime integers.

Then:
HERE COMES AN EQUATION.

Squaring both sides:
HERE COMES AN EQUATION.

Multiplying both sides by b2:
HERE COMES AN EQUATION.

This implies that a2 is even, so a must also be even. Let a = 2k for some integer k.

Substituting into the equation:
HERE COMES AN EQUATION.

Dividing by 2:
HERE COMES AN EQUATION.

This implies that b2 is even, so b must also be even.

But if both a and b are even, they are not coprime, which contradicts our original assumption. Therefore, 2 must be irrational.
</code>