1 – Always about the Real Numbers. Why?

Every textbook in Real Analysis has The Real Numbers as its first chapter. But why?

Well, it’s Real Analysis, so we need to talk about real numbers, but why do we need to understand how to construct the set of real numbers by starting from the set of natural numbers and then going through the integers, rationals and irrationals?

Also, during this process, they talk about some terms such as fields and order properties. They don’t focus on such subjects, but as with Discrete Mathematics, if you knew about it previously to real analysis, it would probably help a lot.

Thus, let’s at least give a big picture of what is a Field and what are Order Properties. Honestly, I didn’t need to know them in detail to understand real analysis, so…

1.1. Field (Definitely not a soccer field)

Wikipedia: a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

Okay, didn’t help much for me.

It’s interesting how textbooks and even wikipedia forget one important thing about teaching: whenever you are explaining something, you need to start from the beginning and not just explain what it is. It just doesn’t work. You need to put some effort into teaching. Let’s see.

Extracted and Edited from the textbook Fundamentals of Abstract Analysis (Andrew Gleason):

i) Mathematical Configuration: Every mathematical concept is described by sets and all mathematical relationships are represented by the interlocking membership relations between the various sets of what we shall call configurations.

Let A be any set. Let’s form some power sets: …

ii) Definition of Binary Operations: A binary operation in the set S is basically a function from to . The domain has elements of the type ordered pairs. Thus, .

Let’s condider the addition operation: 3 + 4. How do I relate this with the definition of binary operations?

Well, , right? What about the result? Well, 3+4 = 7 and the result is the image under the function, i.e., under this binary operation. Thus, .

ii) Binary Operation Properties: Binary operation can be associative, commutative , blah blah blah.

iii)

Addition and multiplication in the real numbers are the best known examples of binary operations.