Then the book introduces the commutative law, that states that in a multiplication, the order of the factors doesn't matter: a x b is equal tothe same as b x a.

Since you're talking about multiplication in general, no need for the "a" before multiplication

And this is eventuallycan be extended to three or more numbers.

Also, we have to take into account that these numbers must be integers.

Your original sentence makes it sound like you're talking to the reader that they have to "take (this) into account," which is fine in a textbook which is directly teaching the reader something, but this is an explanation of the content in the book.

They could be negative numbers, positive numbers, or zero, but they have to be integers, because only then we arare we able to define multiplication in the way we are doing it, which is the previously mentioned abbreviation of addition.

After we are introduced to the commutative law and after it is extended to three or more terms or numbers, then we are then introduced to the associative law, which states that the multiplication of, say, three quantities, is equal to the multiplication of the product of the first two quantities and the third one.

Not wrong, but "we are then" makes the phrases seem slightly more individual and discrete compared to "then we are".

This allows us to write a given multiplication equation of three or more terms/factors in various ways and combined with the commutative law, describes multiplication almost completely.

Generally the numbers to be multiplied are called factors.

They could be negative numbers, positive numbers or zero, but they have to be integers, because only then we arare we able to define multiplication in the way we are doing it, which is the previously mentioned abbreviation of addition.

After we are introduced to the commutative law and after it is extended to three or more terms or numbers, then we are introduced to the associative law, which. This states that the multiplication of, say, three quantities, is equal to the multiplication of the product of the first two quantites and the third one.

I suggest you break up a very long sentence

Or, alternatively, the multiplication of the first quantity and the product of the two following quantities.

To completely describe multiplication, we need to define the distributive property of multiplication, which links multiplication to addition.