When I ask anyone who is not a mathematician (and even some who are) what a “number” is, they never define what a “number” is, but say something along the lines of “well a thing you add to get another thing, like 2+2=4”. But here we had defined “+”, which I do not know of, have not defined and do not have much care for; I asked a different question. We see numbers as “left and right” on some big “interval”, this is also wrong. We will see this by the end of the first lecture.
The motivation behind this course is to build some proper rigour into understanding what, how and why analysis is/operates/works.
(When probability is taught wrong, it leads to bad knowledge of quantum physics, statistics(bizarrely), stochastic models, differential equations. Anything with randomness.)
When real analysis is taught wrong, it leads to bad knowledge of everything taught later on in maths: probability, calculus (vector calculus too), geometry, complex analysis, Fourier analysis, group theory, just a few to name, really cannot be taught without some specific and excellent knowledge of real analysis.
It makes sense (which through out this blog I will be saying… a lot) to begin with my notes for the course “An Introduction To Real Analysis”.
But… What if you have no mathematical knowledge? No worries. The prerequisites are nothing. Especially not calculus (integration or differentiation). But you should have some mathematical ability, what do we assume? That you know that a “real number” is. That is all. You do not even need to know what addition or multiplication are. To be honest; I hope you do not.
If you do have some mathematical knowledge, the notes will not be so interesting but the excercises should keep (some are very difficult, some very easy) you awake. And if you have a lot of mathematical knowledge, you may wish to simply download these notes for your own need; to teach others as a lecturer, as a prerequisite to your course, or just to have some real analysis notes, which is always handy.
Arbias Hashani 
In how much detail do you have to know what a real number is?
I confess when I try explaining them to people who swear they can’t do mathematics, I realize I can do very nicely starting from the counting numbers, to the integers, to the rationals, and then getting into exactly what real numbers are I realize I can’t do much beyond say they include roots of rational numbers (square roots and sometimes cube roots are comfortable to people), and then some more stuff.
First, thank you for commenting and you ask a terrific question, one that we think is limited to real numbers but then applies to complex numbers and then we think, why limit ourselves to numbers? We go from normed spaces to inner product spaces to metric spaces to topological spaces. In all areas of mathematics this is apparent; “the next level” as so to speak.
The real numbers are the classical tool to finally beginning to analyse things. But how do we define them? First, how do we define numbers? Whenever I ask that, usually the person responding, if they had any colour in their face to begin with whilst discussing numbers, promently disappears and are stumped.
It is relatively accepted how Von Neumann defined numbers, as sets and then progressions of sets, all beginning with the empty set. But this requires knowledge of set theory and some of Cantor’s earlier arguments which are relatively trivial when explained, but take time to appreciate. So we rule out this approach. And even if we do use this approach, all we get are the natural numbers.
So for me, the way to introduce real numbers is to teach sequences – Cauchy and convergent ones specifically, and apply them to the rationals. We see that for some (in fact, many, or infinitely many) sequences in the rationals, we get a “limit” that is not in the rationals, despite only working with the rationals. This is the supremum and of course, does not have to be in the set it’s assigned to.
Another similar way is to show that the rationals are incomplete and use a completion method (usually Cauchy) to complete the rationals to get the reals.
But even that is too much work, in my experience give the person the equation x^2=2 and ask if the solution is in the integers, rationals, natural numbers, etc. it’s not, of course, so we define a new set – just conceivably called real numbers.
Rationals and real are both dense. I always found that if you grasped how the rationals were dense, you’d get the reals on your own.