Learn German: Ich glaube, dass Sie sich verrechnen, Herr Professor! - I believe that you are miscalculating, Professor!

The base verb is rechnen (to calculate). The inseparable prefix ver- often adds the meaning of doing something incorrectly or making a mistake. By adding it, the verb becomes verrechnen, meaning to miscalculate or to make an error in math.

Why is the reflexive pronoun sich used with verrechnen?

When verrechnen means to miscalculate, it is always a reflexive verb (sich verrechnen). It is very common for German verbs that use the ver- prefix to describe a mistake to also be reflexive. Another example you might know is sich verlaufen (to get lost / to walk the wrong way).

Why is the verb verrechnen at the end of its clause, right before the comma?

The word dass (that) is a subordinating conjunction. It introduces a dependent clause, which means the conjugated verb must be pushed to the very end of that specific clause. The main clause Ich glaube follows standard word order.

Why do we say Herr Professor instead of just addressing him as Professor like we would in English?

In German, it is the standard and polite practice to put Herr (Mr.) or Frau (Ms./Mrs.) in front of an academic or professional title when speaking directly to someone. You will also hear this with doctors, such as Frau Doktor.