Learn Hardcore Greek: Αυτό το βιβλίο είναι σπάνιο.

You can say αυτό το βιβλίο (very natural) or το βιβλίο αυτό (also correct, a bit more “the book, this one”). But το αυτό βιβλίο is not normal modern Greek.

Why is it Αυτό and not Αυτός or Αυτή?

Greek demonstratives agree with the gender, number, and case of the noun.

βιβλίο (book) is neuter in Greek.
The neuter nominative singular form of “this” is αυτό.

Basic singular forms of “this”:

Masculine: αυτός (this man) → αυτός ο άντρας
Feminine: αυτή (this woman) → αυτή η γυναίκα
Neuter: αυτό (this book) → αυτό το βιβλίο

So you must say Αυτό το βιβλίο, because βιβλίο is neuter.

Why is the article το and not ο or η?

Again, it’s about gender agreement.

The definite articles in the nominative singular are:

ο = the (masculine) → ο άντρας (the man)
η = the (feminine) → η γυναίκα (the woman)
το = the (neuter) → το βιβλίο (the book)

Since βιβλίο is neuter, its article is always το in the nominative (subject) form:

Αυτό το βιβλίο = this book
Το βιβλίο = the book

That’s why we don’t say ο βιβλίο or η βιβλίο.

Why does σπάνιο end in -ο and not -ος like a typical adjective?

Adjectives in Greek also agree with the gender, number, and case of the noun they describe.

The adjective σπάνιος (rare) has these basic singular forms:

Masculine: σπάνιος → σπάνιος πίνακας (rare painting, masc.)
Feminine: σπάνια → σπάνια ευκαιρία (rare opportunity, fem.)
Neuter: σπάνιο → σπάνιο βιβλίο (rare book, neut.)

Since βιβλίο is neuter, the adjective must be neuter: σπάνιο.

So we get: Αυτό το βιβλίο είναι σπάνιο.
All three key words are neuter singular nominative:

αυτό – neuter demonstrative
το – neuter article
βιβλίο – neuter noun
σπάνιο – neuter adjective

Can the word order change? For example, can I say Το βιβλίο αυτό είναι σπάνιο?

Yes. Greek word order is fairly flexible, especially with demonstratives and adjectives. All of the following are correct:

Αυτό το βιβλίο είναι σπάνιο.
Neutral, common: this book is rare (pointing out this one).
Το βιβλίο αυτό είναι σπάνιο.
Slight nuance: this book is rare, with a bit more focus on “this book (as opposed to others)”.

Other possible but less neutral patterns (often for emphasis):

Αυτό το βιβλίο σπάνιο είναι. – emphasizes σπάνιο.
Σπάνιο είναι αυτό το βιβλίο. – rare it is, this book, focus on rare.

For a learner, using Αυτό το βιβλίο είναι σπάνιο is the safest, most natural pattern.

Could I leave out Αυτό and just say Το βιβλίο είναι σπάνιο?

Yes, but the meaning changes slightly.

Αυτό το βιβλίο είναι σπάνιο.
→ This book is rare. (You’re pointing to or clearly identifying a specific book.)
Το βιβλίο είναι σπάνιο.
→ The book is rare.
More general or context-dependent (we already know from context which book is being talked about).

So Αυτό adds the demonstrative idea “this (one)”, not just definiteness.

Why do we need είναι? In some languages you can say “This book rare” without “is”.

In modern Greek, the verb είμαι (to be) is normally required in the present tense for sentences like this.

Αυτό το βιβλίο είναι σπάνιο. = This book is rare.

You generally cannot drop είναι here.

Only in very specific, elliptical, or headline-like contexts might it be omitted, but that’s not standard full-sentence grammar. So, as a learner, always include είναι.

How do you pronounce Αυτό το βιβλίο είναι σπάνιο?

Approximate pronunciation (in syllables):

Αυτό → af-TO
το → to (short “o”, like British “got”)
βιβλίο → vee-VLEE-o (the β is like English v)
είναι → EE-neh
σπάνιο → SPA-nee-o

Rough overall: af-TO to vee-VLEE-o EE-neh SPA-nee-o

Key points:

αυ before a voiceless consonant (here τ) is pronounced /af/, not “av”.
β is pronounced /v/, never /b/.
The accent marks (ά, ό, etc.) show stress position, not a different vowel sound.

How would I say “These books are rare” instead?

You need the plural forms:

These (neuter plural): αυτά
the (neuter plural article): τα
books: βιβλία (neuter plural of βιβλίο)
are: είναι (same form for plural)
rare (neuter plural): σπάνια

So:

Αυτά τα βιβλία είναι σπάνια.
These books are rare.

Again, everything agrees in neuter plural:
αυτά – τα – βιβλία – σπάνια.

Is σπάνιο placed correctly? Could it go before βιβλίο, like in English “rare book”?

In Greek, the most neutral pattern is:

article + noun + είμαι + adjective
Το βιβλίο είναι σπάνιο.

If you want “a rare book” as a noun phrase, you usually say:

ένα σπάνιο βιβλίο (a rare book)
→ article + adjective + noun

So:

Αυτό το βιβλίο είναι σπάνιο. = This book is rare.
Αυτό είναι ένα σπάνιο βιβλίο. = This is a rare book.

In your sentence, σπάνιο correctly appears after the verb είναι, because it is the predicate adjective (“is rare”).