Der Taschenrechner, den ich im Kurs benutze, liegt auf dem Tisch.

Because they do two different jobs in the sentence.

• Der Taschenrechner is the subject of the main clause, so it is in the nominative case.
• den ich im Kurs benutze is a relative clause, and inside that clause the calculator is the direct object of benutze, so it has to be in the accusative case.

Since Taschenrechner is masculine singular:

• nominative: der Taschenrechner
• accusative: den Taschenrechner

So der and den both refer to the calculator, but each reflects its role in its own clause.

It is a relative clause. A relative clause gives extra information about a noun.

Here, it tells you which calculator is meant: the one that I use in class.

So the structure is:

• Der Taschenrechner = the noun being described
• den = the relative pronoun, meaning that/which/whom depending on context
• ich im Kurs benutze = the rest of the relative clause

In English, this works like The calculator that I use in class...

Because in German, a conjugated verb goes to the end of a subordinate clause, and a relative clause is a type of subordinate clause.

Main clause:

• Der Taschenrechner liegt auf dem Tisch.
• The verb liegt is in the normal second position.

Relative clause:

• den ich im Kurs benutze
• The verb benutze moves to the end.

This is one of the most important word-order rules in German.

Because German always separates relative clauses with commas.

The relative clause is inserted into the middle of the main clause:

• main clause without the insertion: Der Taschenrechner liegt auf dem Tisch.
• inserted relative clause: den ich im Kurs benutze

So the commas mark the start and end of that extra descriptive part.

English sometimes uses commas differently, but in German this punctuation is required.

Because the relative pronoun must match the noun it refers to in gender and number, but its case depends on its role inside the relative clause.

The noun is Taschenrechner, which is:

• masculine
• singular

So the relative pronoun must also be masculine singular. Then we choose the case based on its function in the relative clause.

Inside den ich im Kurs benutze, the pronoun is the thing being used, so it is the direct object of benutze. That means accusative masculine singular, which is den.

A quick comparison:

• nominative masculine singular: der
• accusative masculine singular: den
• dative masculine singular: dem
• genitive masculine singular: dessen

Im is a contraction of in dem.

So:

• in dem Kurs → im Kurs

This is very common in German.

Here im Kurs means something like:

• in class
• in the course

The exact English translation depends on context, but grammatically it is simply in + dative article.

Because this sentence describes a location or setting, not movement into somewhere.

German often uses:

• dative for location: where something happens
• accusative for direction/movement: where something is going

Here, ich benutze den Taschenrechner im Kurs means I use the calculator in class / during the course, so it is a location or situation, not movement. That is why German uses in dem / im, which is dative.

Compare:

• Ich bin im Kurs. = I am in the course/class.
• Ich gehe in den Kurs. = I go into the course/classroom.
  (This is movement toward it.)

Because auf dem Tisch describes location, not motion.

With two-way prepositions like auf, German uses:

• dative for location: where?
• accusative for movement toward a place: where to?

So:

• Der Taschenrechner liegt auf dem Tisch. = The calculator is lying on the table.
  Location → dative
• Ich lege den Taschenrechner auf den Tisch. = I put the calculator onto the table.
  Movement/destination → accusative

That is why dem is correct here.

German often prefers specific verbs for physical position.

• liegen = to lie, to be lying
• stehen = to stand, to be standing
• sitzen = to sit, to be sitting

A calculator is normally thought of as lying flat on a table, so German uses liegen:

• Der Taschenrechner liegt auf dem Tisch.

You could sometimes say ist auf dem Tisch, and people would understand it, but liegt sounds more natural because it describes the object's position more precisely.

Yes, in this sentence it is clearly masculine, because it takes der in the nominative:

• der Taschenrechner

German noun gender has to be learned with the noun, so it is best to memorize them together:

• der Taschenrechner = calculator

Then the other forms follow from that:

• nominative: der Taschenrechner
• accusative: den Taschenrechner
• dative: dem Taschenrechner
• genitive: des Taschenrechners

No, not in standard German.

In English, you can say:

• The calculator that I use in class...
• The calculator I use in class...

German normally cannot omit the relative pronoun. You need it:

• Der Taschenrechner, den ich im Kurs benutze, ...

Leaving out den would be ungrammatical in standard German.

The core sentence is:

• Der Taschenrechner liegt auf dem Tisch.

Then German inserts a relative clause right after Taschenrechner:

• Der Taschenrechner, den ich im Kurs benutze, liegt auf dem Tisch.

So the pattern is:

• noun
• relative clause giving more information about that noun
• rest of the main clause

This is a very common German sentence pattern. If you remove the relative clause, the main sentence still works perfectly on its own.

Yes, welchen is possible in very formal or old-fashioned style, but den is much more normal in everyday German.

So:

• natural, standard: Der Taschenrechner, den ich im Kurs benutze, ...
• more formal/literary: Der Taschenrechner, welchen ich im Kurs benutze, ...

For learners, den is definitely the form to use.