Can you explain the difference between "complex" and a "compound" sentences to a mathematician?

It seems to me as if a dependent clause is simply an independent clause and conjunction lumped together.

Be careful. This may be true in some cases, but not in all cases.

Now, the distinction between a complex and a compound sentence lies with the conjunction. A compound sentence uses a coordinating conjunction, because the two clauses - both independent - are of equal importance. A complex sentence uses a subordinating conjunction, because the two clauses are of different importance: one is independent and the other is dependent.

Most of your sentences are compound sentences. They can't be changed into complex sentences by "drawing the boundaries" wherever you like. This is because they utilize coordinating conjunctions, and this doesn't change by including or excluding them.

If a computer were tasked with differentiating between compound sentences and complex sentences, searching for conjunctions would be a good way to start (the other way would be to check for independent/dependent clauses, which I think requires intuition that a computer cannot possess).

In the previous answer, note the important distinction between "sub"ordinating and "co"ordinating. "Sub" means "under" or "secondary to"; "co" means "equal to" or "of equal weight or value". In a complex sentence, the writer can use different kinds of clauses to indicate which information is the more important: the information in a subordinate clause is of secondary weight to the information in the main clause. On the other hand, in a compound sentence, the information in each of the two clauses is of equal weight or value. One finer point of syntax: in a compound sentence, you can decide which clause is first, which second, depending on which you think should come to the reader's attention first. Mathematically, I think of the equal sign = versus the other signs (greater than >, less than <, etc.).

As a mathematician, I would feel very, very suspicious about an explanation made to “appeal to a mathematical mind”. Especially since I know that English is complicated and trying to press it into simple rules is going to fail.

The rules in KINDS OF SENTENCES AND THEIR PUNCTUATION might help (examples omitted):

A sentence may be one of four kinds, depending upon the number and type(s) of clauses it contains.

• An independent clause contains a subject, a verb, and a complete thought.
• A dependent clause contains a subject and a verb, but no complete thought.

1. A SIMPLE SENTENCE has one independent clause.

Punctuation note: NO commas separate two compound elements (subject, verb, direct object, indirect object, subjective complement, etc.) in a simple sentence.

2. A COMPOUND SENTENCE has two independent clauses joined by

• A. a coordinating conjunction (for, and, nor, but, or, yet, so),
• B. a conjunctive adverb (e.g. however, therefore), or
• C. a semicolon alone.

Punctuation patterns (to match A, B, and C above):

• A. Independent clause, coordinating conjunction independent clause.
• B. Independent clause; conjunctive adverb, independent clause.
• C. Independent clause; independent clause.

3. A COMPLEX SENTENCE has one dependent clause (headed by a subordinating conjunction or a relative pronoun ) joined to an independent clause.

Punctuation patterns:

• A. Dependent clause, independent clause
• B. Independent clause dependent clause
• C. Independent, nonessential dependent clause, clause.
• D. Independent essential dependent clause clause.

4. A COMPOUND-COMPLEX SENTENCE has two independent clauses joined to one or more dependent clauses.

Punctuation patterns:

Follow the rules given above for compound and complex sentences. A compound-complex sentence is merely a combination of the two.

CONNECTORS--COMPOUND AND COMPLEX SENTENCES

Two independent clauses may be joined by

• 1. Coordinating conjunctions Ic, and ic
• 2. Conjunctive adverbs Ic; therefore, ic.

A dependent (subordinate) clause may be introduced by

• 1. Subordinating conjunctions (ADVERB CLAUSE) Dc, ic. or Ic dc.
• 2. Relative pronouns (ADJECTIVE CLAUSE) I, dc, c. or I dc c.
• 3. Relative pronoun, subordinating conjunctions, or adverbs (NOUN CLAUSE)

Mathematical examples:

x=7 and y=5        (compound: two independent ideas)
x=7 or x=5 or x=3  (compound: three independent ideas)
x=7 therefore x>0  (complex: the second idea depends on the first)
x=7 and y=5 therefore x+y=12 (compound-complex)
x+y=10 and x-y=2 therefore x÷y=1½ (compound-complex)

Or think of it as:

• Compound sentences can be listed in point form (e.g. "all of the following are true:" or "at least one of the following is true:")
• Complex sentences cannot be listed in point form without changing the meaning of the sentence (e.g. the last example above would lose the idea of the 1½ having been deduced from the first two values).