Essence of Analysis: Real Numbers

Рет қаралды 8,015

Жыл бұрын

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.
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Пікірлер: 34

@mathkaveli11 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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

@slavinojunepri7648 Жыл бұрын

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 Жыл бұрын

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.

@Will-Ch Жыл бұрын

Great, thanks dr Peyam.

@leonidasliao5288 4 жыл бұрын

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 4 жыл бұрын

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

@General12th Жыл бұрын

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

@aravindkr 6 ай бұрын

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

@drpeyam 6 ай бұрын

It’s on my playlist!!

@behzat8489 Жыл бұрын

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.

@dominicellis1867 6 ай бұрын

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.

@GhostyOcean Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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

@theproofessayist8441 Жыл бұрын

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

@u.s.r.00 Жыл бұрын

Wow

@tom13king Жыл бұрын

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

@iabervon Жыл бұрын

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.

@popodori Жыл бұрын

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

@drpeyam Жыл бұрын

I’ll show you half a student 😂

@almenarab Жыл бұрын

Is zero a natural number?

@hehgendary Жыл бұрын

No, zero not a natural number.

@guydror7297 Жыл бұрын

Yes

@pzorba7512 Жыл бұрын

@@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 Жыл бұрын

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

@tom13king Жыл бұрын

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.