When should one capitalize adjectives/numerals which are used as noun?

Question

In which cases is the noun capitalized and why?

1) Der eine kam, der andere nicht.

2) Peter ist der Beste.

3) Ich denke, wir haben das Wichtigste gesehen.

I have spelled the sentences like I feel they are grammatically correct. Please correct if necessary. Can you give me a general rule? Would be thankful for links.

Answer 1

Your feeling is correct.

General rule for adjectives: Regel 72

1. If used as substantives, adjectives are written upper-case: das Gute, das Angesprochene, Altes und Neues, ...
2. Words like „alles", „etwas", „nichts", „viel", „wenig" indicate the use as substantive: alles Gute, nichts Wichtiges, ...
3. Upper case is also used in special phrases: im Dunkeln tappen, im Trüben fischen, auf dem Laufenden sein, zum Besten geben, ein Programm für Jung und Alt
4. Lower case is used in common adverbial phrases composed of prepositions and non-declined adjectives without article. If the adjective is declined, you can use lower- or upper case: durch dick und dünn, über kurz oder lang - von Nahem oder nahem, bis auf Weiteres oder weiteres

^{(I really do not know how to translate the grammatical expressions correctly, so take a look at the provided link.)}

General rule for numerals:

• Both, upper- and lower case, are correct.

Answer 2

As a general rule of thumb you can capitalize adjectives, numerals, prepositions and adverbs, if those are used as a substantive.

There are some exceptions however.

Here is a short overview of them:

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The first exception is of course if the Noun (here "der Schüler") is given in the same sentence.

Peter ist der beste Schüler.

as opposed to

Peter ist der Beste.

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The second exception is, if it is used in an coolocation such as

durch dick und dünn gehen

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The third exception is, if it is used as a superlative and answers the question "how?". In this example it would be "how cold is it" "the coldest".

Am Südpol ist es am kältesten.

Don't confuse those with those that answer the question "what?".

Wir nehmen nur das Nötigste mit.

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Numerals can be written with a capital or not. Duden says, that you would capitalize it to highlight the Character of the Word as substantive.

Der eine kam, der andere nicht.

Der Eine kam, der Andere nicht.

Also see Duden for the details on "ein/andere".

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Numbers are written in lowercase, numerals are spelled with a capital.

die verhängnisvolle Dreizehn eine Sechs würfeln

as opposed to

Er ist sechzig Jahre alt. die ersten zehn

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For further Details refer to Duden (rule 72 ff).

The above list may not be complete, but should include the most important ones.