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--- tanszek:oktatas:techcomm:mathematical_expressions_in_tex_language [2024/09/02 14:28] – [Aligning Equations] kissa
+++ tanszek:oktatas:techcomm:mathematical_expressions_in_tex_language [2024/09/10 08:55] (current) – [5. Exercise] kissa
@@ Line 1: / Line 1: @@
 ===== Introduction to LaTeX for Mathematical Expressions =====
-The goal of the lesson is to become familiar with the TeX language, specifically for the purpose of writing 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 ====
@@ Line 7: / Line 7: @@
 **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.
+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?**
-  - **Precision and Control**: LaTeX allows precise formatting of documents and mathematical expressions.
+  * **Handling Complex Documents**: It is ideal for mathematical formulas, citations, and cross-referencing in technical writing.
-  - **Consistency**: LaTeX automatically manages references, labels, and numbering, ensuring consistency throughout your document.
+  * **Consistent Layout**: Automatically ensures a uniform, professional design by separating content from formatting.
-  - **Professional Quality**: Documents created in LaTeX look professional and are publication-ready.
+  * **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:**
@@ Line 23: / Line 25: @@
 **Basic Document Structure:**
-<code>
+<code LaTeX>
 \documentclass{article}  % Specifies the document class (article, report, book, etc.)
 \begin{document}          % Begins the content of the document
@@ Line 44: / Line 46: @@
 == Example ==
-<code>
+<code LaTeX>
 \documentclass{article}
 \begin{document}
@@ Line 68: / Line 70: @@
 == Examples ==
-<code>
+<code LaTeX>
 \documentclass{article}
 \begin{document}
@@ Line 88: / Line 90: @@
 {{:tanszek:oktatas:techcomm:pasted:20240902-141644.png}}
-==== Aligning Equations ====
+==== 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 ===
@@ Line 96: / Line 142: @@
 == Example ==
-<code>
+<code LaTeX>
 \documentclass{article}
 \usepackage{amsmath}
@@ Line 121: / Line 167: @@
   - Example:
-<code>
+<code LaTeX>
 \begin{equation} \label{eq:quadratic}
 x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
@@ Line 128: / Line 174: @@
 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>