هذه مكتبة عامة.

In careful Modern Standard Arabic, it is:

hādhihi maktabatun ʿāmmatun

A few pronunciation notes:

• هذه = hādhihi
  • dh sounds like the th in this
• مكتبة = maktaba / maktabatun
• عامة = ʿāmma / ʿāmmatun
  • the ʿ represents ع, a sound that does not really exist in English
  • the doubled mm shows that the م is stressed or held slightly longer

In normal unvowelled writing, the short endings are usually not written, so learners often hear something closer to:

hādhihi maktaba ʿāmma

Yes. It is a complete sentence.

Arabic often uses this pattern for This is X:

• هذه = this
• مكتبة عامة = a public library

So the whole structure is:

• subject: هذه
• predicate: مكتبة عامة

In Arabic grammar, this is a nominal sentence: a sentence built without a verb in the present tense.

Because in Arabic, the verb to be is usually omitted in the present tense when you are simply equating one thing with another.

So English says:

• This is a public library

But Arabic says more literally:

• This — public library

This is completely normal in Arabic.

If you want the past or future, Arabic does use other forms, for example with كان for was.

Because مكتبة is a feminine singular noun, and the demonstrative has to match it.

• هذا = this for masculine singular
• هذه = this for feminine singular

Since library here is مكتبة, which is feminine, Arabic uses هذه.

English does not show this kind of gender agreement in this, but Arabic does.

Because it has the ending ة, called tāʾ marbūṭa.

This ending often marks a noun as feminine. So:

• مكتبة is grammatically feminine

That does not mean the library is biologically female, of course. It just means the word belongs to the feminine grammatical category, and other words around it must agree with that.

The ة in مكتبة and عامة is usually pronounced like -a when you stop on the word:

• maktaba
• ʿāmma

But when grammar endings are added, or in certain connected forms, it is pronounced as -t:

• maktabatun
• ʿāmmatun

So the ة is special:

• in pause: often sounds like a
• when connected grammatically: often becomes t

Because in Arabic, adjectives normally come after the noun they describe.

So:

• مكتبة عامة = literally library public

This is the normal Arabic order.

English usually puts adjectives before nouns:

• public library

Arabic usually puts adjectives after nouns:

• مكتبة عامة

Because the adjective must agree with the noun it describes.

Since مكتبة is:

• feminine
• singular
• indefinite

the adjective must also be:

• feminine
• singular
• indefinite

So we get:

• مكتبة عامة

If the noun were masculine, you would expect عام instead of عامة.

Because this sentence means This is a public library, not this public library.

In other words, مكتبة عامة here is an indefinite predicate:

• a public library

That is why there is no ال on مكتبة or عامة.

Compare:

• هذه مكتبة عامة = This is a public library
• هذه المكتبة العامة = This is the public library or this public library, depending on context

So the lack of ال is important for the meaning.

With full case endings, the sentence is:

هذه مكتبةٌ عامةٌ

More fully transliterated:

hādhihi maktabatun ʿāmmatun

The -un endings are called tanwīn. They show that the words are:

• indefinite
• in the nominative form here

In everyday Arabic writing, these short vowels are usually not written, which is why you normally just see:

هذه مكتبة عامة

Here it means public.

More literally, عام / عامة can have senses like:

• general
• public
• common

In this context, with مكتبة, the natural English meaning is public:

• مكتبة عامة = public library

So this is a very common adjective-noun combination.

They are different in definiteness and meaning.

• هذه مكتبة عامة = This is a public library

  • indefinite
  • introducing or identifying what something is

• هذه المكتبة العامة = This is the public library or this public library

  • definite
  • referring to a specific library

A useful way to think about it:

• indefinite predicate: هذه مكتبة عامة
• definite noun phrase: هذه المكتبة العامة

So the presence or absence of ال changes the meaning quite a lot.