I don't understand why people sometimes use 'du' instead of 'des' when the meaning of both articles is the same. For example, "some pencils" is des crayons whereas "some cheese" is du fromage.
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Use du (masculine) or de la (feminine) followed by a singular noun for uncountable quantities. Use des followed by plural for countable ones.
Sometimes there are nouns that can be both uncountable and countable:
Note: du or de la don't always denote uncountable quantities, de is a widely used preposition in French.
If you would like to use an undefinite quantity following a de preposition simply use de + singular (uncountable) or de + plural (countable), without an additional article. Examples:
Note that using “du bonheur” (the general notion) would mean something different from “de bonheur” (some subset of the general notion). Same for “de l'ami” (the friend, i.e. a specific friend), “des amis” (all of them), or “d'un ami” (one of them), all are different from “d'amis” (some of them).
Use du (masculine) or de la (feminine) followed by a sing. noms for articles indéfinis(uncountable) while des followed by plural noms for article définis(countable).
example of articles indéfinis
example for articles définis
Some of the most difficult words in a language are articles and prepositions. The words du and des can combine a preposition with an article, so it's inevitable that this can get a little complicated.
The word du is always a contraction of de le. The corresponding feminine form is de la, and we use de l' for the masculine when it's followed by a vowel. The plural is des. Then, to complicate matters:
In your example sentence, du is a partitive article. Du fromage is the singular form, and des fromages is the plural form.
The singular form du fromage uses fromage as an uncountable noun. “Je mange du fromage”: “I'm eating cheese” (or: “I'm eating some cheese”; French doesn't normally distinguish between these two). The singular is used because cheese is a substance, and I'm eating an indeterminate amount of that substance. Contrast:
Jez' answers misses a crucial point, I'm afraid: you can say des fromages!
The distinction is simply between mass nouns (which allow the use of de with a singular) and count nouns (which don't). You can say Je veux des crayons (in which des does not mean "of the") because you can use that with almost all French nouns (there are very French words that are not count nouns in at least some meanings), but you can hardly say je veux du crayon because crayon is not a mass noun (vs. la mine du/des crayons, "the pencil's/pencils' lead")
The distinction is only troublesome to English speakers because English drop articles for both mass nouns and count nouns (compare I'm eating cake/cakes and je mange du/des gateaux, which make exactly the same distinction).
Use du when the object of the sentence is singular. du is short for de le, and des is short for de les, so:
In your examples, crayons is plural (so des is used) whereas fromage is singular (so du is used).