# Guidelines for translation – Serlo

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• ↳ Project "Serlo"
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Contents "Real Analysis"
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• Introduction • Complex numbers • Supremum and infimum • Sequences • Convergence and divergence • Subsequences, Accumulation points and Cauchy sequences • Series • Convergence criteria for series • Exponential and Logarithm functions • Trigonometric and Hyperbolic functions • Continuity • Differential Calculus ## Guidelines for translation

### Translation style

• Capital letters in titles: Like on Wikipedia, we suggest to only capitalize the first word of a page or section title (like "Real numbers" instead of "Real Numbers" or "real numbers")
To-Do:

Collect some more guidelines

### Content

• If you come up with new explanations or solutions to exercises, you are invited to add them in the English version.
• Some German articles contain German media content (e.g. videos). you may just leave them in place and add a note like (in German) in the captions. Those videos will be replaced later.
• There are German articles (A) which embed parts of other German articles (B) - so if you click in (A) on "edit" to translate, you will just find a link to (B), but no translatable text. The trick is to follow the link to (B), where you can copy the source code (containing the translatable text) by clicking on "edit" in (B). Insert the source code in the English version of (A) and you can start translating.
• German expressions in formulas can and should be translated (like $\quad \delta >0$ für alle $\varepsilon >0\quad$ is changed to $\quad \delta >0$ for all $\varepsilon >0\quad$ )

## Mathematical conventions

The following mathematical conventions are a translation of this German article. New conventions and changes are usually published there first and it might take some time for them to be translated, so that article should be treated as the master document.

### Notation of sequences

For sequences we use the notation $\left(a_{n}\right)_{n\in \mathbb {N} }$ . The source code to produce this is \left(a_n\right)_{n\in\N}. For sequence elements we write $a_{n}$ , not $a(n)$ (cf. this survey (in German))

### Natural numbers

We understand $\mathbb {N} =\{1,2,3,\ldots \}$ (so zero is not regarded as a natural number). If you want to refer to the set $\{0,1,2,3\ldots \}$ , you should use $\mathbb {N} _{0}$ . (cf. this survey (in German))

### Imaginäre Einheit

The imaginary unit is written as \mathrm{i}. Appearance: $\mathrm {i}$ ### Subsets

We denote subsets with \subseteq, for example $A\subseteq B$ . For proper subsets we use $\subsetneq$ , e.g. $A\subsetneq B$ . We do not use the notation $A\subset B$ . There is an extended discussion of this convention (in German).

### Disjoint unions

We denote the disjoint union of sets as $A_{1}\uplus A_{2},\quad \biguplus _{i=1}^{n}A_{i}$ .

### Column vectors

In running text we write $(1,2,3)^{T}$ for column vectors (LaTeX code: (1,2,3)^T). In an equation environment we use the common notation ${\begin{pmatrix}1\\2\\3\end{pmatrix}}$ (LaTeX code: \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}). The reason is that actual column vectors used in running text exceed line height by far and mess up the formating (take this text as an example). Therefore we try to avoid them in running text. Cf. this survey (in German) for the decision.

Beispiel:

This is running text, so I write $(1,2,3)^T$. In an equation environment I instead use

{{Formel|$2 \cdot \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}$}}


Ergebnis:

This is running text, so I write $(1,2,3)^{T}$ . In an equation environment I instead use

$2\cdot {\begin{pmatrix}1\\2\\3\end{pmatrix}}={\begin{pmatrix}2\\4\\6\end{pmatrix}}$ ### Linear span / hull of vectors

The linear span of a set $M$ of vectors is denoted $\operatorname {span} (M)$ . Source code: \operatorname{span}(M) (link to survey (in German)).

### Transformation matrix

The transformation matrix of a linear map $L$ is written $M_{C}^{B}(L)$ , where $B$ is the basis of the source vector space, while $C$ is the basis of the target vector space. Source code: M_C^B(L)

### Ordered basis

We do not have any special notation for ordered bases. You should explain in running text, whether a basis is ordered or not.