Intro to Real Analysis: How to think of real numbers

[@brendanchamberlain9388]

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

[@JoelRosenfeld]

Thank you! I really appreciate you saying so

[@jameslai6879]

I used baby Rubin 30 years ago for learning topology. Can’t believe it is still the gold standard up to this day.

[@timduncankobebryant]

Awesome vid! Music is a bit too much though.

[@JoelRosenfeld]

Thanks! Yeah I think the last song got to be too much in retrospect. I keep working on it

[@kaychimav]

Yeah I was so relieved at 4:04, but it didn't last long... But great vid, it makes accepting/explaining that 0.999=1 so much easier.

[@2minutestomammoth]

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

[@JoelRosenfeld]

Thanks for the input! I’ll try to keep that in mind as I move forward with the series. I’m really glad you like it!

[@saptarshisahoo5075]

Drop the background music .please!

[@coreyevans5734]

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

[@JoelRosenfeld]

The classic textbook for this sort of thing is Polya’s How to Solve it.

[@riadbsdr9821]

😮best real analyse video i saw

[@JoelRosenfeld]

Thank you! I’m glad you liked it!

[@maxpercer7119]

nice sophisticated treatment of the real numbers

[@JoelRosenfeld]

thanks!

[@valentinlishkov9540]

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

[@AJ-et3vf]

Awesome video! Thank you!

[@JoelRosenfeld]

I'm glad you liked it!

[@ronpearson1912]

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

[@JoelRosenfeld]

Wow that’s nuts! I didn’t realize I was sitting on a treasure trove lol. You can find cheaper copies on Amazon www.amazon.com/Mathematical-Analysis-Second-Tom-Apostol/dp/0201002884 I think I picked it up from a retiring professor’s give away pile. Or a used bookstore. It’s been more than a decade now that I’ve had it. If I remember right, I think the big difference is measure theory was added to the second edition. I actually have both editions, but I’ll have to sit down with them side by side to be sure.

[@astraea-paradoxesreasoning6498]

Great explanation as usual. I think of the real numbers as any (R, 1, 0, +, *,

[@nowstronglife3866]

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.

[@JoelRosenfeld]

I appreciate your support! I’ve cut down on the music a great deal in my more recent videos. My plan when I do a couple more in this series is to make a large compilation video, where I’ll go back and trim out the music at more critical parts.

[@RSLT]

Amazing Video!

[@JoelRosenfeld]

Thanks! I’m glad you like it!

[@valentinlishkov9540]

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

[@JoelRosenfeld]

What is dA here? What are you trying to demonstrate?

[@valentinlishkov9540]

@@JoelRosenfeld can there be a length commensurate with all lengths (differential of length)

[@valentinlishkov9540]

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

[@JoelRosenfeld]

@@valentinlishkov9540 no there is no such length and hence, not every number is rational

[@valentinlishkov9540]

@@JoelRosenfeld and the differential of length dl

[@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!

[@JoelRosenfeld]

When you make the rationals from integers, you really only require two integers to define one. For the reals, we use an infinite sequence of rational numbers to define them. If you look at all possible sequences with just two values, 0 and 1, then it is straightforward to see that this gives you uncountable infinity. This isn’t the only proof of uncountability, by the way. Later in the series, I prove the uncountability of the reals buy considering perfect sets. As for their actual appearance in nature, that’s just a matter of philosophy. Numbers in general are not actual objects that appear in nature, but rather are abstractions we use to understand nature. The reals exist no more than a rational number and no more than the number 1. It’s all in our heads. Infinity is a logical construct that we use to interpret things like cardinality and sequences.

[@douglasstrother6584]

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?").

[@JoelRosenfeld]

I think the mode of proof changes discontinuously from many other subjects. Linear Algebra, Abstract Algebra, Combinatorics, etc often have a decently clear path forward, even if that path is hard. The definitions in real analysis are really abstract which violates our intuition about real numbers that we bring to the party. When the definitions seem so otherworldly, proofs with those definitions can seem a bit more blurry. Of course, after some time, you get used to it, and things get much simpler. But that takes time.

[@douglasstrother6584]

@@JoelRosenfeld The definitions are very distilled! You're right about drawing pictures; visualization is an enormous aid to understanding.

[@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

[@JoelRosenfeld]

Yeah I could do that. Probably as a short. This is indeed a proof. It is true simply by definition. We defined real numbers as equivalence classes of Cauchy sequences of rational numbers. If the distance between two sequences goes to zero, then they both represent the same real number, by definition. Hence, 0.9, 0.99, 0.999, … is a sequence that converges to 1 since the difference between it and the constant sequence 1 goes to zero.

[@SupremeSkeptic]

@@JoelRosenfeld I see... So, if they are defined as the same number according to the real number system because the difference between them approaches 0. Then, there probably is no need to show proofs like the one you are showing because no one would argue that 0.999... approaches 1 and the difference between them approaches 0 right? I am sorry if I sounded rude, I have absolutely no such intention... I am genuinely curious why many mathematicians in youtube even bother with giving step-by-step proofs that 0.999... = 1, if by invoking the real number system, then they are by definition the exact same number... Wouldn't just doing this make 0.999...=1 much less controversial?

[@JoelRosenfeld]

@@SupremeSkeptic you didn’t come off as rude. It’s important to ask questions like that. It’s ultimately a trivial proof, where it’s a single step to verify it fits the definition. Of course, there are other equivalent ways to define real numbers, and that changed the starting point and verification. Real numbers were very mysterious for a long time. Up until Cantor and Dedekind came around.

[@SupremeSkeptic]

@@JoelRosenfeld I see... so it is to verify the definition. I have no problem with 0.999... and 1 be defined as the same number in the real number system But I find that every step-by-step proof showing 0.999...=1 is flawed. The 10x proof, 1/3 proof, 1/9 proof, geometric sum proof, mathologer's proof, they are all flawed. If you don't mind, you can watch Karma Peny's videos about them. I don't agree 100% with Karma Peny but I believe that he is mostly right. Perhaps, after watching Peny's arguments, you can come up with a proof that is better than the ones he has debunked?

[@Morbius907]

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.

[@JoelRosenfeld]

What time in the video are we talking about?

[@Morbius907]

@@JoelRosenfeld The section on convergent sequences. You are basically taking the limit of the sum of 9/(10^x) for the values from 1 to infinity. Basically the limit of a function as x approaches n shows where the function is heading towards at n. It does not mean that the function at x=n equals the limit as x approaches n.

[@JoelRosenfeld]

Are you talking about the proof at 7:48 ? Sorry just trying to make sure we are on the same page, because I technically demonstrate it twice in the video

[@JoelRosenfeld]

Oh, ok. I see what you mean now. This is where the confusion really sets in. What I did here is show that the constant sequence 1 (which is a Cauchy sequence) and the Cauchy sequence that you get when you add 9/10^n for n going from 1 to infinity approach the same value. That is their difference goes to zero. Since we say that a real number is an equivalence class of Cauchy sequences of rational numbers, this means that the Cauchy sequence { 1 } and the Cauchy sequence {0.9,0.99,…} reside in the same equivalence class. By definition, this means they both represent the same real number.

[@masonholcombe3327]

Recently found this channel, big fan!

[@JoelRosenfeld]

@@masonholcombe3327 thank you! Happy to have you here!

[@safeyyaalyahia7198]

Great

[@JoelRosenfeld]

Thanks!

[@valentinlishkov9540]

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

[@tobykelsey4459]

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.

[@JoelRosenfeld]

This isn’t my approach, it is the approach taken by cantor in the 1800s. This is a very standard way of constructing real numbers, and I am very careful in showing how to put them together. The order of operations actually follow from that of rational numbers directly using limits.

[@JoelRosenfeld]

I think you are getting carried away with how you are interpreting what’s going on here. Think back to ordinary calculus 2. If you have two sequences that converge, the limit of the sequence you get from adding these two sequences together is the sum of the two limits. So there is really no ambiguity here. Now when we are looking at real numbers, we think of them as an infinite collection of sequences that are all going to the same limit. That means it doesn’t change anything if we select any one of those sequences to add to another convergent sequence. The limit is the same. Morally, this limit is our real number. Same works for multiplication and division, where with division you just need to be wary of division by zero.

[@MathwithMing]

Hi. I wrote this Q&A in order to clarify some confusion. 1. Before defining real numbers, what objects are considered well-understood? Ans: the set of rational numbers, that is, the ratio of integers. Denoted as Q. 2. What do we want from real numbers, exactly? Or why can't we just work with Q? Ans: We wish to have a number system suitable for calculus, that is, computing limits, derivatives, integrals, series, etc. This is a task ill-suited for rational numbers, unfortunately. For example, the sequence of rationals 1, 1.4, 1.41, 1.414, 1.4142,... fails to converge to a rational (which, of course, is the square root of 2 that we wish to define). Informally, we say Q has "gaps", i.e., we cannot assign limits to seemingly "convergent" sequences in Q. Of course, in our new number system, we also wish to retain the usual rules for rational numbers: arithmetic rules, and order. 3. Can we write our "wish list" for our new number system more formally? Ans: Our wish list consists of three groups of axioms (In modern mathematics, "axiom" does not mean "self-evident", it means "assumption", or "property". Whether or not a given object meets those axioms needs to be verified). I will use Principles of Mathematical Analysis (3rd edition), by Walter Rudin, as a reference (It is an open secret that this book is available online). For example, p3 means page 3 in Rudin. Axioms (or properties we wish the real numbers satisfy): (1) Least upper bound property (1.7-1.10, p3-4): This property "fills up" the "gaps" in Q. (2) Field axioms: How do arithmetics, i.e., addition and multiplication, work? (1.12, p5) (3) Order axioms: How does "

[@tobykelsey4459]

irritating and distracting background "music"/noise.

[@JoelRosenfeld]

I’m sorry you feel that way. I’m working on improving the selection of music.

[@Morbius907]

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

[@JoelRosenfeld]

Oh yeah? What did I get wrong?

[@Morbius907]

@@JoelRosenfeld The problem is when you multiplied .999... by 10. If you take the number .9 and multiply it by 10 you get 9.0. .99 x 10 = 9.90, .999 x 10 = .9999 x 10 = 9.9990 What you are doing is basically like saying .9 x 10 = 9.9 and .99 x 10 = 9.99 and so on. Actually .999... x 10 = 9.999...990

[@sirius1255-]

@@Morbius907 you are confusing between infinite sum and finite sum.

[@johnlabonte-ch5ul]

I have been on a journey, for less than a year, in KZbin 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.

[@JoelRosenfeld]

@@johnlabonte-ch5ul you are essentially running with 1800s notions of arithmetic. It was very hard to arrive at the right definition of a real number. This was done simultaneously by Dedekind and Cantor. Essentially, we no longer define real numbers by decimals, but rather we define them in another way from which we can extract a decimal representation of that number. The method presented here is one such method of defining real numbers.

[@johnlabonte-ch5ul]

​​@@JoelRosenfeldThanks for the reply. I have found KZbin to be a interesting way to intuitively discuss the topic of ".99..." is 1. Modern media is quick but not decisive. To rigorous discuss requires more verification and usually time to gather resources. That is why I am careful to say I am not well studied in math and I am learning. My idea of arithmetic and algebra are much older than the 1800's. The idea of ".99..." is 1 and the way to handle infinity was not generally accepted until the twentieth century. (according to the Princeton Companion to Mathematics) The basics of Mathematics are axioms. Consider the statement that ".99..." is not an integer due to its form that has nonzero digits to the right of the decimal point. This is extremely weak. Modern texts that I have found do not directly say that. On the other hand, the definition of rational numbers persist such as in Analysis 1 by Tao. (Tao also says that ".99..." is 1 the best I can tell is that Tao uses some kind of super limit. Even converging limits are subject to verification, such as lim as x->1, x²-1/(x-1) equals 2) So it is easier to argue that ".99..." is not rational. 9/9 does not result in ".99...". (Obviously don't agree that 10 times ".99..." equals "9.99...". That 1/n^°°?=?0) I have never felt it is a paradox that to move (change) you first have to get 1/2 way there. The only way to describe motion (change) by real numbers is to consider them as a continuum such as time and space. Due to the idea of infinity you can not determine if you ever stop moving or get to your destination. Infinity is dangerous, incomplete, inconsistent and imprecise. Math seems to have decided that infinitesimals are 0 but there is no largest number. It seems reasonable that ".11..." +".88..." = ".99...". An argument could be made that there are different cardinality involved by how far the infinity was determined (very weak). Then comes ".99..." +".11..." =? What happens to the 0 at the end (in infinite land). Should the real answer be less than "1.11..."? If infinitesimals are 0, is there a smallest number? Is there any real number that can be precisely described next to 1 in human terms? Archimedean property says no.

[@johnlabonte-ch5ul]

The decimal place value system goes back to the 5th century C.E. Digital representation of fractions was introduced in the late 16th century. I was not around then, so I am depending on the Princeton Companion To Mathematics for my info on these two points. [For those "put off" by the size of the book, it is section II.1]