Ich rechne im Kopf.

Because the subject is ich (I), and the verb has to match it.

The infinitive is rechnen. In the present tense:

• ich rechne
• du rechnest
• er/sie/es rechnet
• wir rechnen
• ihr rechnet
• sie/Sie rechnen

So Ich rechne is the correct 1st person singular form.

Here, rechnen means to calculate or to do arithmetic.

In German, rechnen often refers to working with numbers, especially math. In this sentence, it suggests mental calculation rather than just general thinking.

It is different from:

• denken = to think
• zählen = to count

So Ich rechne im Kopf is specifically about doing calculation mentally, not just thinking about something.

Im is a contraction of in dem:

• in dem → im

So grammatically, im Kopf literally means in the head.

But in this sentence it is an idiomatic expression meaning in one’s head / mentally.

This kind of contraction is very common in German:

• am = an dem
• im = in dem
• zum = zu dem
• zur = zu der

Because this sentence expresses location, not movement.

German prepositions like in can take either:

• dative for location: wo? = where?
• accusative for movement/direction: wohin? = to where?

Here, the idea is that the calculation happens in the head, so German uses the dative:

• im Kopf = in dem Kopf → dative

If it were about movement into something, then accusative would be used:

• in den Kopf

But that would not fit the meaning here.

Yes, you can say in meinem Kopf, and it is grammatically correct. But im Kopf is the more natural and idiomatic expression in this context.

Compare:

• Ich rechne im Kopf. = natural, standard way to say you calculate mentally
• Ich rechne in meinem Kopf. = understandable, but more literal and less idiomatic

German often uses the article in expressions where English would prefer a possessive.

Yes, it is a complete sentence.

The verb rechnen does not always need a direct object. It can stand on its own when the meaning is clear:

• Ich rechne. = I’m calculating / doing arithmetic.
• Ich rechne im Kopf. = I’m calculating mentally.

If you want, you can also add an object:

• Ich rechne die Aufgabe im Kopf. = I do the problem in my head.
• Ich rechne das im Kopf aus. = I figure that out in my head.

So the original sentence is perfectly fine without an object.

Because this is a normal main clause, and in German main clauses the conjugated verb usually comes in second position.

The structure is:

• Ich = position 1
• rechne = position 2
• im Kopf = rest of the sentence

This is one of the most important German word order rules: the finite verb goes second in a main clause.

You can also move another element to the front, but then the verb still stays second:

• Im Kopf rechne ich.

That version is possible, but the neutral word order is:

• Ich rechne im Kopf.

It is in the present tense.

German present tense can cover several ideas that English sometimes separates:

• I calculate in my head.
• I’m calculating in my head.
• I do mental math.

Which one sounds best depends on context. German often relies on context rather than using a special continuous form like English I am calculating.

Yes, very often.

That sentence means I’m good at mental arithmetic or I can calculate well in my head.

This is a very common pattern:

• im Kopf rechnen = to do mental arithmetic
• gut im Kopf rechnen können = to be good at mental arithmetic

So Ich rechne im Kopf describes the action, while Ich kann gut im Kopf rechnen describes the ability.

Rechnen is the more general verb: to calculate / do arithmetic.

Ausrechnen often means to calculate something out or work something out completely.

For example:

• Ich rechne im Kopf. = I calculate mentally.
• Ich rechne das im Kopf aus. = I work that out in my head.

So ausrechnen is more likely when there is a specific result or problem being solved.

Usually it suggests math or numerical calculation.

If a German speaker says Ich rechne im Kopf, most people will understand it as I do the calculation mentally rather than using paper or a calculator.

In other contexts, rechnen can have broader meanings, but with im Kopf the math-related meaning is the most natural one.