The copula verb in Japanese. While it does correspond to "be" and "equal" in English, it is as usual a completely different concept.

Desu doesn't at all behave like other Japanese verbs. It's so irregular as to be different part of speech altogether, which it is, as far as Japanese grammar is concerned. It has its own "conjugations" and is used all over the place. It does not form stems, bases, or have endings, in any regular sense of the word.

The general pattern for a sentence involving desu is [X] wa [Y] desu, where A is the topic of discussion and B is a definition, description, location, or somesuch. Different forms of desu are as follows:

Note that dewa is often contracted to ja in ordinary conversation. Furthermore, desu (in whatever form) can often be omitted; however, this renders the sentence slightly more casual/less polite. Politeness in Japanese is complicated, but among friends and family you can usually leave the desu out, whereas with strangers, superiors, and casual acquaintances you had better leave it in. It's probably better to err on the side of politeness rather than rudeness, anyways, in any language or culture.

Kore wa totemo oishii desu yo.
This is very tasty!
(literally: As for this, (it) is very tasty.)

Sore wa nan desu ka.
What is that?

Watashi wa ichiban muzukashii kurasu wa Nihon-go deshita.
My hardest class was Japanese.
(literally: As for me, as for number one hard class, (it) was Japanese.)

Kuruma ga hitsuyoo ja arimasen.
I don't need a car.
(literally: Car, needed, is not.)

Imooto wa aisu kuriimu ga suki ja arimasen deshita.
As for my younger sister, she didn't like ice cream.

You get the idea. You may have noticed that the verb "to be" does not appear in some of the English translations. This is because definitions and descriptions are expressed very differently in Japanese than in English, as you can see from the semi-literal translations included with some of the examples.

Desu also has some informal forms. These are da and datta, for present and past respectively. These are quite casual and should only be used among friends or certain family members, or (more likely for foreigners just visiting) not at all, except in a few fixed expressions that require them. Likewise, dewa arimasen and dewa arimasen deshita take the informal forms ja nai and ja nakatta, respectively. Da and datta are slightly more common/acceptable among groups of men, but as usual this gender difference is overrated.

Deshoo should probably be considered a form of desu, but it's a bit different and more limited, as it means "probably is".

In many cases where English uses "to be", the Japanese use one of aru, iru, or (rarely) oru, which actually correspond to "to exist" in English. This is especially true of phrases involving location, posession, or (naturally) existence, rather than definition or equality.

Desu is always written in hiragana.

As an additional note to the writeup by Jeeves, desu is in fact, more regular than it appears.

The conventional form "desu" is a contraction of the form "de arimasu" or "de aru". This form is often considered more literary, and is commonly found in books, dissertations, theses, etc.

"Aru" however, has an irregular negative conjugation, that is, "nai" (in the plain form). The form "dewa arimasen" which is used for the negative of "desu", is therefore a slightly irregular negative conjugation of "de aru" with the particle "wa" inserted.

Ergo: de aru -> de wa nai

The form "deshou" has the plain form "darou". This form is sometimes called the suggestive, because it is variously translated as "you see", "isn't it", "probably". Because this form expresses more than just "desu", it is used to soften sentences, sometimes in effeminate speech.

The particle "no" is also often prefixed to desu to form something called the demonstrative. Used in this way, "no" is sometimes shortened to just "n". This form expresses something of an explanation.

Jikan ga nai n desu yo
I don't have time, you see.

Jikan ga arimasen
I don't have time.

As you can see, these various forms are hard to translate, and sometimes are ignored, leaving very wooden translations.

Also, fans of samurai/Edo-jidai cartoons may be interested to note that "de aru" is the origin of the form "de gozaru"/"de gozaimasu". Gozaru is the irregular honourific that is used in place of aru.

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