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--- tanszek:oktatas:techcomm:mathematical_expressions_in_tex_language [2025/09/04 12:45] – [5. Aligning Equations] kissa
+++ tanszek:oktatas:techcomm:mathematical_expressions_in_tex_language [2026/09/02 09:14] (current) – [7. Exercise] kissa
@@ Line 19: / Line 19: @@
 **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.
+  - **Papeeria**: We will use Papeeria, a browser-based LaTeX editor that runs document compilation in the cloud: [[https://www.papeeria.com/]]
-    - Sign up at [[https://www.overleaf.com/|Overleaf]].
-    - Overleaf offers collaborative features, version control, and a vast library of LaTeX templates.
 **Basic Document Structure:**
@@ Line 215: / Line 213: @@
 To refer to this equation later, use ''Equation \ref{eq:quadratic}''.
+==== 6. Lists ====
-==== 6. Exercise ====
+LaTeX provides environments for creating both unordered and ordered lists.
+=== Unordered Lists ===
+Use the ''itemize'' environment to create a bulleted list. Each item starts with the \item command.
+<code LaTeX>
+\begin{itemize}
+  \item apple
+  \item banana
+  \item raspberry
+\end{itemize}
+</code>
+{{:tanszek:oktatas:techcomm:pasted:20260902-085819.png}}
+=== Ordered Lists ===
+Use the ''enumerate'' environment to create a numbered list.
+<code LaTeX>
+\begin{enumerate}
+  \item Denver Broncos
+  \item Buffalo Bills
+  \item Dallas Cowboys
+\end{enumerate}
+</code>
+{{:tanszek:oktatas:techcomm:pasted:20260902-085831.png}}
+==== 7. 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}}
+{{:tanszek:oktatas:techcomm:pasted:20260902-091119.png}}
@@ Line 226: / Line 255: @@
 <code>
-Proof: √2 is Irrational
+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.
+Assumptions:
+Assume, for contradiction, that 2 is rational.
+It can therefore be expressed as a fraction a/b.
+a and b are coprime integers.
-Then:
+We can write:
-HERE COMES AN EQUATION.
+EQUATION
 Squaring both sides:
-HERE COMES AN EQUATION.
+EQUATION
 Multiplying both sides by b2:
-HERE COMES AN EQUATION.
+EQUATION
 This implies that a2 is even, so a must also be even. Let a = 2k for some integer k.
-Substituting into the equation:
+Substituting a = 2k into the equation:
-HERE COMES AN EQUATION.
+EQUATION
 Dividing by 2:
-HERE COMES AN EQUATION.
+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.
+Conclusion: If both a and b are even, then they are not coprime.
+This contradicts our original assumption. Therefore, 2 is irrational.
 </code>