Essence of Analysis: Real Numbers

Essence of Analysis: Real Numbers. In this overview of analysis, I go through the different number systems like natural, rational, and real numbers. I explain why the real numbers are better than the rational or even the complex numbers. It's because the least upper bound property is true, which has to do with sup and max.
Real number playlist: • Real Numbers
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[@mathkaveli11]

I don't think I have ever appreciated the LUB of the reals so much, when I had Real Analysis, my professor didn't emphasize its importance in a meaningful way like you did here. Thank you for a great explanation.

[@TomSkinner]

It's amazing how much a simple explanation of the significance of an introduced concept can accelerate students' learning. And how few teachers bother to do it. Many thanks to Dr. Peyam.

[@cparks1000000]

Rational analysis is Number Theory (or Algebra, depending on which you prefer).

[@slavinojunepri7648]

This is a terrific piece of video central to the understanding of real analysis. After watching it, one begins to appreciate the concepts of suprimum, infimum and others as the basic building blocks (or holy grail to repeat Dr. Peyam) of real analysis.

[@BlackEyedGhost0]

8:29 You can't include infinity as a least upper bound because infinity isn't a real number. If you can say infinity counts, then you can make the same statement that a least upper bound always exists for the rational numbers as well, which completely eliminates the purpose of the property.

[@leonidasliao5288]

WOW, the comment recited in the end is truly mesmerizing. Kind of reminds me of power sets, so is "the set of all the ways to divide the rational numbers" like the set that contains the sup of all elements of the powerset of rational numbers?

[@drpeyam]

Yes, kind of, it’s the set of all cuts, as defined in section 6 😄

[@General12th]

Hi Dr. Peyam!
I'm really looking forward to getting back to university and doing proper math classes again!

[@GhostyOcean]

Would we have an equivalent definition for supremum if we replace the second requirement with "If L is an upper bound of S, then M ≤ L"?

[@tom13king]

Yes. His definition is "anything smaller is not an upper bound", yours is the contrapositive of that i.e. "all upper bounds must be at least as big".

[@GhostyOcean]

@@tom13king oh! I always forget about the contrapositive. Thanks!

[@dominicellis1867]

Could you make a video on the langrange inequality and Cauchy’s inequality? I’m taking complex analysis and we’re supposed to use algebraic and geometric reasoning to prove the various versions of the triangle inequality for inner and outer product spaces.

[@behzat8489]

if you define real numbers as an ordered field with all axioms (commutativity of both operations, order axioms etc...) but do not include lub property as an axiom, then there is a theorem saying that there are infinitely many ordered fields with any cardinality you like. so, lub property is a characteristic property of real numbers.

[@Will-Ch]

Great, thanks dr Peyam.

[@aravindkr]

this is great, really liked your explanation ! , do you have a video that explains Dedekind cuts ?

[@drpeyam]

It’s on my playlist!!

[@theproofessayist8441]

How can you express the squeeze theorem in terms of infimum and supremum of sets of real numbers?

[@tom13king]

Are you going to talk about how the LUB property is axiomatic i.e. we just assume it's true?

[@iabervon]

We take it as an axiom for the real numbers, but we do prove that various objects (Dedekind cuts, equivalence classes of Cauchy sequences of rational numbers) obey the axiom and are therefore valid models for the real numbers. Really, what we're just assuming is that each author who's working with the real numbers has picked some set that obeys these axioms.

[@almenarab]

Is zero a natural number?

[@hehgendary]

No, zero not a natural number.

[@guydror7297]

Yes

[@pzorba7512]

@@guydror7297 Pareil pour les nombres premiers, quand j'étais au lycée en 1958 1 était premier, depuis ce n'est plus le cas dans les programmes français.

[@jan-willemreens9010]

... In The Netherlands we consider 0 as an element of the set of the natural numbers: N = {0, 1, 2, 3. ... } ...

[@tom13king]

Depends on who you ask or whose course you're taking. When I started uni, the analysis class used the convention that 0 wasn't a real number but the foundations course said it was.

[@popodori]

student are in N, no negative or half student or sqrt(2)*student

[@drpeyam]

I’ll show you half a student 😂

[@u.s.r.00]

Wow

[@purim_sakamoto]

うむ