Nobody teaches the same topic in the exactly the same way.
There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include.
Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers:
#1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers.
#2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number.
You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with.
Lastly, there is always a better way to explain the same material.
There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include.
Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers:
#1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers.
#2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number.
You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with.
Lastly, there is always a better way to explain the same material.