Intro to Real Analysis: How to think of real numbers

Mathematicians didn't feel that the decimal expansion of a number was quite rigorous enough in the 19th century, so they came up with some new ways to build them. Two prominent methods are Dedekind Cuts and Equivalence Classes of Cauchy sequences of rational numbers (or in other words, real numbers are convergent sequences of rationals).

//Books

//Exercises
- Finish showing that the Cauchy sequence definition of real numbers provides a field.

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0:00 Introduction
0:23 What are real numbers?
1:49 1 is 0.9999
2:32 Two approaches to defining real numbers
4:00 An example through pi
6:32 Convergent sequences
9:35 Cauchy Sequences
13:21 Sums of Cauchy Sequences
16:40 Inequalities

[brendanchamberlain]

Your channel is such a gem! Thank you so much for making these videos.

[ronpearson]

Hey where did you get that second edition, thats like $1000 on amazon. How significant are the differences between apostol first and second edition?

[minutestomammoth]

I think this video is really great! I like the hand-written proofs over the video. I think this could be improved by using more diagrams and talking about proof ideas/sketches/morals before diving into the symbols/epsilons/equations

[coreyevans]

Great video.
Do you have any recommendations for materials to help understand and improve writing proofs?

[maxpercer]

nice sophisticated treatment of the real numbers

[timduncankobebryant]

Awesome vid! Music is a bit too much though.

[nowstronglife]

I know it's been commented on below, but I would add that your stuff is so good, it doesn't need any music or stock videos. I'm sure this will cut down on your editing time and hopefully allow you to make more amazing content.

[valentinlishkov]

can there be a length commensurate with all lengths (differential of length)?then all numbers would be rational

[astraea-paradoxesreasoning]

Great explanation as usual. I think of the real numbers as any (R, 1, 0, +, *, <) that satisfies the axioms of a Dedekind complete ordered field. Intuitively, any set of elements in which +, -, *, / work as usual, play nicely with a linear order <, and have no "gaps". Such an (R, 1, 0, +, *, <) with R a set of elements, 1 and 0 being constants, + and * being 2-ary functions, and < being a 2-ary relation (all with respect to R) is a model satisfying the real number axioms. I think of the real numbers in a Platonic sense as The Dedekind complete ordered field of which we construct versions. I don't mean the reals necessarily exist in a Platonic sense, just that I imagine them that way. Basically everything we construct in set theory involves lots of arbitrary choices to get to the kinds of structures we are interested in. The way I think about is is that we are interested in satisfying certain axioms with our constructions, and the construction is just there to prove that we can create something to satisfy those axioms given our original set theory axioms (and maybe it can teach us some insights along the way).

[WorldsBestGuys]

So real numbers are built upon rational numbers which are built upon integers but yet the real numbers are uncountable. I don’t understand. I also think without time the real numbers would not exist because it’s dependent upon a process; it’s not an actual thing that is fixed. this is one reason I have trouble believing that diagonalization is correct and not hand wavy and reliant on infinity which I don’t even know if that exists either.Can you please try and convince me otherwise? Thanks for your content!

[Morbius]

You confused the limit of a function with the actual value of the function. For example the function f(x) = x / x. What is the value when x = 0? It is undefined. The limit as x approaches 0 is 1 but that does not mean that when x=0 the value is 1.

[douglasstrother]

Question for the Math Geeks: "What makes Real Analysis so difficult?"
My exposure to Real Analysis is limited to what is covered in Calculus to demonstrate that limits, sequences & series, derivatives & integrals are basically legal. I found Abstract Algebra much more difficult than Calculus, ODEs, PDEs, etc.


Of the Four Food Groups of Physics (Classical Mechanics, Electromagnetism, Thermodynamics/Statistical Mechanics and Quantum Mechanics), their order in ascending difficulty is: Classical Mechanics (This is the foundation, which is hard enough.), Quantum Mechanics (This applies the mathematics of Classical Mechanics, waves and Linear Algebra to put a new spin on Classical Mechanics.), Electromagnetism (The simplest problems grow into difficult boundary value problems which obscures the Physics. "What am I looking for again?"), and Thermodynamics/Statistical Mechanics ("Consider a gazillion harmonic oscillators (Classical & Quantum) in an electromagnetic field." and "What the heck are Enthalpy and Entropy, again?").

[SupremeSkeptic]

Can you do a more in depth video on 0.999...=1 please?
The proof that you gave... (that called a proof?)
I am still not convinced that is true

[valentinlishkov]

Issue:
What is a differential of an irrational argument?
Let a= some rational approximation, and A be the irrational number itself (if that makes sense).
Then A - a > dA and there is no way a + dA > A

[tobykelsey]

These definitions are pretty sloppy. You are throwing together infinite sets of infinite sequences of rationals. You then perform infinite numbers of pairwise operations.on them - you end up with multidimensional infinities and sums, and you don't even consider or define how the order of operations might affect the outcome. I don't know why you think you can blindly assume you can do any arithmetic operation with a real number when it might require an infinite amount of computation to even define it with your approach.

[masonholcombe]

Recently found this channel, big fan!

[valentinlishkov]

dl=sqrt(dx.dx + dy.dy) = probable 90% irrationality

[tobykelsey]

irritating and distracting background "music"/noise.

[Morbius]

Your proof that .999... = 1 proof violates algebraic rules.

[johnlabonte-chul]

I have been on a journey, for less than a year, in YouTube discussions to find why most maths believe that ".99..." is a good representation of 1.
My background is middleschool basics and readings of mathmatical texts, a large number being Calculas texts and other more general books.
I have learned a lot.
The first observation is that in decimal notation ".99..." does not have the form of an integer, since there are nonzero digits after the decimal point.
".99..." does not fit the definition of rational numbers as it is not the ratio of two integers.
I have found a problem on how math treats infinity.
Real numbers are unique and precise. Infinity is incomplete, inconsistent and imprecise. How can arithmetic (and algebra) be complete, unique and precise with infinite numbers without some mathic?
One particular implication, that ".99..." converges to 1. ".99..." has a one way limit to 1 from below with 1 being a upper bound. Further that in number bases greater than 10 there are at least one number in each base that is closer to 1 than ".99..." in base 10. (1/16^n, 1/60^n are closer to 0 than the 1/10^n in base 10.).
How is the sum of infinite positive >0 terms in a series equal to a unique and precise real number value?
In the sense of incomplete long division, ".33..." is equal to 1/3 but it is not equal to the real number value of 1/3.
Are all 1/n^°° the same number 0?
Too many problems to conclude that ".99..." is 1.