232 lines
5.3 KiB
Markdown
232 lines
5.3 KiB
Markdown
@slide(layout=chapter-slide)
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@number
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7
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@title
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Typesetting mathematics
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@slide(layout=content-and-preview)
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@title
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Formula environments 🧮
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@content
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``` {.lang-tex .hljs}
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$2 \sqrt{\frac{\pi^2}{3} \cdot c_2}$
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```
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Mathematical formulas are only accepted in the so-called **math mode.**
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In-line formulas must therefor be guarded by two dollar signs.
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We can also use the `equation` block environment:
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``` {.lang-tex .hljs}
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\begin{equation}
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2 \sqrt{\frac{\pi^2}{3} \cdot c_2}
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\end{equation}
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```
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Packages: `amsmath`, `amsthm`, `amssymb`, `mathtools`
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@preview
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@slide(layout=content-only)
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@title
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Examples
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@content
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<table>
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<tr>
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<th>source code</th>
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<th>result</th>
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</tr>
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<tr class="fragment">
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<td>`\sum_{i=1}^{n}x^2`</td>
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<td>$\sum_{i=1}^{n} x^2$</td>
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</tr>
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<tr class="fragment">
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<td>`12 \leq 4 x^2 + 13`</td>
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<td>$12 \leq 4 x^2 + 13$</td>
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</tr>
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<tr class="fragment">
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<td>`{n \choose k}`</td>
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<td>${n \choose k}$</td>
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</tr>
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</table>
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@slide(layout=content-and-preview)
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@title
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Aligning a group of equations
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@content
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The `align` environment permits us to align equations at certain positions like the ‘ = ’ character.
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``` {.lang-tex .hljs}
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\begin{align}
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13 \cdot (4a - 3)^2 &= 13 … \\
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&= 208a^2 - 312a + 117
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\end{align}
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```
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* ++ The equations will be aligned with respect to the ampersands (‘`&`’).
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* ++ We can mark a new line using ‘`\\`’.
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* ++ `align` and `equation` will not be numbered if we add an asterisk after their names (e. g. `\begin{align*}` and `\end{align*}`).
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@preview
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@slide(layout=task-without-number)
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@task-number
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7a
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@title
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Typesetting mathematics
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@content
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Code up the following formulas in the file `exercises/maths/math-formulas.tex`.
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| Meaning | Result |
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|---------|--------|
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| Gravitational acceleration | $9.81\,\frac{m}{s^2}$ |
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| One formula to solve quadratic equations | $x_{1,2} = - \frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}$ |
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| The other one | $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ |
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| Catalan numbers | $C_n = \frac{1}{n+1} {2n \choose n} = \frac{(2n)!}{(n+1)!n!}$ |
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@slide(layout=task-without-number)
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@task-number
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7b
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@title
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Typesetting mathematics
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@content
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| Meaning | Result |
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|---------|--------|
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| Definition of factorial | $n! = \prod_{i=1}^{n} i$ |
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| Set of all odd natural numbers | $\{ x \mid x \in \mathbb{N}, \text{odd}(x) \}$ |
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| Elimination $\neg\exists x$ | $\neg\exists x . p(x) \Leftrightarrow \forall x . \neg p(x)$ |
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Find further examples on the [Chair of Algorithms and Complexity Theory's LaTeX exercise page](https://www.algok.uni-bamberg.de/practice/KaTeX%20Formula%20Practice%20Page.html).
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@slide(layout=extra-content-only)
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@title
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More examples
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@content
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<table>
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<tr>
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<th>source code</th>
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<th>result</th>
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</tr>
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<tr>
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<td>`(x)`, `[x]`, `\lbrace x \rbrace`, `\lvert x \rvert`</td>
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<td>$(x), [x], \lbrace x\rbrace, \lvert x\rvert$</td>
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</tr>
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<tr>
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<td>`\exists`, `\forall`, `\in`, <br />`\notin`, `\infty`</td>
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<td>$\exists,\forall,\in,\notin,\infty$</td>
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</tr>
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<tr>
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<td>`\alpha`, `\beta`, `\Gamma`, <br />`\Delta`, `\varepsilon`, `\pi`</td>
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<td>$\alpha, \beta, \Gamma, \Delta, \varepsilon, \pi$</td>
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</tr>
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<tr>
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<td>`\rightarrow`, `\leftarrow`, `\Rightarrow`, `\Leftarrow`, `\Leftrightarrow`</td>
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<td>$\rightarrow, \leftarrow, \Rightarrow, \Leftarrow, \Leftrightarrow$</td>
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</tr>
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<tr>
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<td>`(A \cup B) \cap C`</td>
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<td>$(A \cup B) \cap C$</td>
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</tr>
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<tr>
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<td>`(A \lor B) \land C`</td>
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<td>$(A \lor B) \land C$</td>
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</tr>
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<tr>
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<td>`(A \cdot B) \times C`</td>
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<td>$(A \cdot B) \times C$</td>
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</tr>
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</table>
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@slide(layout=extra-content-only)
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@title
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Height-adapting braces
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@content
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<table>
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<tr>
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<th>source code</th>
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<th>result</th>
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</tr>
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<tr>
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<td>`\left( \frac{1}{2} \right)`</td>
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<td>$\left( \frac{1}{2} \right)$</td>
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</tr>
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<tr>
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<td>`\left[ \frac{1}{2} \right]`</td>
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<td>$\left[ \frac{1}{2} \right]$</td>
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</tr>
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<tr>
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<td>`\left\lbrace \frac{1}{2} \right\rbrace`</td>
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<td>$\left\lbrace \frac{1}{2} \right\rbrace$</td>
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</tr>
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</table>
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``` {.lang-tex .hljs}
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$4 \cdot \left( \frac{1}{2} % …
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```
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<p style="text-align: center;">$4 \cdot \left(\frac{1}{2} +\frac{3}{ 12 \cdot \left( 2 + \frac{1}{86 \cdot \left(\frac{1}{2} + 24 \right)} \right)} \right)$</p>
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@slide(layout=extra-content-and-preview)
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@title
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Depicting boundaries
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@content
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The bounds of an integral can be enforced to appear above and below the integral symbol using the `\limits` command.
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This is the standard behaviour for sums, products and limits.
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``` {.lang-tex .hljs}
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\sum_{i=1}^{n^2}(x+2)
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\prod_{j=1}^{100}(3 \cdot x)
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\lim_{x \rightarrow \infty}(14x^3 - 12)
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\int\limits_{-12}^{4}(14x^3 - 12)
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```
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<p data-category="Caution!">Don't use `\limits` inline.</p>
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@preview
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@slide(layout=extra-content-only)
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@title
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Set-builder notation
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@content
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In certain situations, it is more adequate to use textual predicates or long function names within the set builder notation.
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This is where `\text{}` comes into play.
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<p style="text-align: center;">$\left\lbrace x \mid frequency(x) \geq 20\right\rbrace$</p>
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<p style="text-align: center;">$\left\lbrace x \mid \text{frequency}(x) \geq 20\right\rbrace$</p>
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``` {.lang-tex .hljs .fragment}
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$\left\lbrace x \mid \text{frequency} …
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``` |