Analysis really is a beautiful subject - I like how analysis is rigorous yet intuitive and pretty. I think that sometimes, math students that get the axiomatic definition of the reals thrown at them during the first lecture (this is worse where I live because we have no such thing as standard calculus courses - analysis and linear algebra are usually the first two college math courses you take) fail to appreciate the ingeniousness of these concepts. Axioms are not so much fundamental rules of nature but basic principles - assumptions, if you want to call them that -, and assuming different things leads you to different conclusions. A really interesting example of this would be hyperreal numbers - alternatively, instead of using the least upper bound property, one may axiomize the reals as an ordered field using the Archimedian axiom and Cauchy completeness, and the former implies that there are no nonzero infinitesimals in the real numbers. This ties into the history of calculus, where Leibniz, Newton, Euler and others made breathtaking discoveries using this notion of infinitesimal numbers they were never able to rigorously justify. And when mathematicians tried to put mathematics as a whole on a more rigorous footing in the 19th and early 20th century, they found that you really don't need infinitesimals to define limit processes - which led to the introduction of the famous epsilon delta proofs, brought about in their modern form by Weierstrass. But in the 1960s, a mathematician named Abraham Robinson found a way to rigorously introduce infinitesimals, creating the hyperreal numbers - and it involved simply postulating the existence of one number greater than zero but smaller than any positive real number and ditching the Archimedian axiom to avoid a contradiction. He also showed that the hyperreal numbers are equally logically consistent as the reals - in other words, if you believe the real numbers are legitimate, you must consider the hyperreals to be an equally legitimate number system. And because of what's called the transfer principle, all basic properties of the reals also hold for the hyperreals (more formally, what's meant are formulas of first order predicate logic - if you haven't heard of first order predicate logic before, though, just ignore this remark). Hyperreals are nice because they, at least in my view, very clearly align with the intuitive picture many math students have, and a lot of things become much cleaner - continuity, for example, is very easy to define using hyperreals. You may define a function to be continous at a point _x_ if and only if any infinitesimal displacement _dx_ on the real number line produces only an infinitesimal change in the value of the function. This becomes particularly vivid if you introduce the operator _st(x),_ which basically "rounds" any hyperreal number to the real number "closest" to it. Then, continuity at a point _x_ simply means that _st(f(x + dx)) = f(x)_ for all infinitesimal _dx._ Compare this with how tricky it is to define continuity using epsilon and delta - and although it may not be as directly apparent, it's also pretty straightforward to generalize this definition to an arbitrary number of dimensions.
I was wondering if you would do a video on finding green's functions to determine solutions to ode/pde problems with boundary conditions. I haven't been able to understand other videos and I feel like it would be clearer coming from you.
Hey Dr.Peyam I'm wondering if you can make a video about summable families in the playlist series? There's like No video on it in TH-cam which is sad :( Thank you!
Y= x ^ 2 is not compact x ^ 2 + y ^ 2 = 1 is compact is there a method to understand when F (x, y) = 0 is compact? perhaps a group is associated and if this group is compact then the design contained in the formula is also compact? Thenks
In general, a set S \subset R^2 (for that matter, R^n) is compact if and only if it is closed and bounded (Heine-Borel Theorem). If F is continuous, (as in F(x,y)=x^2+y^2), then the set S={(x,y) l F(x,y)=0} is already closed, S is compact if and only if S is bounded, that is, all the solutions to F(x,y) is in some large rectangle [-M,M]^2 centered at the origin. Hope that helps.
Analysis really is a beautiful subject - I like how analysis is rigorous yet intuitive and pretty. I think that sometimes, math students that get the axiomatic definition of the reals thrown at them during the first lecture (this is worse where I live because we have no such thing as standard calculus courses - analysis and linear algebra are usually the first two college math courses you take) fail to appreciate the ingeniousness of these concepts. Axioms are not so much fundamental rules of nature but basic principles - assumptions, if you want to call them that -, and assuming different things leads you to different conclusions. A really interesting example of this would be hyperreal numbers - alternatively, instead of using the least upper bound property, one may axiomize the reals as an ordered field using the Archimedian axiom and Cauchy completeness, and the former implies that there are no nonzero infinitesimals in the real numbers. This ties into the history of calculus, where Leibniz, Newton, Euler and others made breathtaking discoveries using this notion of infinitesimal numbers they were never able to rigorously justify. And when mathematicians tried to put mathematics as a whole on a more rigorous footing in the 19th and early 20th century, they found that you really don't need infinitesimals to define limit processes - which led to the introduction of the famous epsilon delta proofs, brought about in their modern form by Weierstrass. But in the 1960s, a mathematician named Abraham Robinson found a way to rigorously introduce infinitesimals, creating the hyperreal numbers - and it involved simply postulating the existence of one number greater than zero but smaller than any positive real number and ditching the Archimedian axiom to avoid a contradiction. He also showed that the hyperreal numbers are equally logically consistent as the reals - in other words, if you believe the real numbers are legitimate, you must consider the hyperreals to be an equally legitimate number system. And because of what's called the transfer principle, all basic properties of the reals also hold for the hyperreals (more formally, what's meant are formulas of first order predicate logic - if you haven't heard of first order predicate logic before, though, just ignore this remark). Hyperreals are nice because they, at least in my view, very clearly align with the intuitive picture many math students have, and a lot of things become much cleaner - continuity, for example, is very easy to define using hyperreals. You may define a function to be continous at a point _x_ if and only if any infinitesimal displacement _dx_ on the real number line produces only an infinitesimal change in the value of the function. This becomes particularly vivid if you introduce the operator _st(x),_ which basically "rounds" any hyperreal number to the real number "closest" to it. Then, continuity at a point _x_ simply means that _st(f(x + dx)) = f(x)_ for all infinitesimal _dx._ Compare this with how tricky it is to define continuity using epsilon and delta - and although it may not be as directly apparent, it's also pretty straightforward to generalize this definition to an arbitrary number of dimensions.
That was too long :))
Time to get real with Dr Peyam! 😃
Oh boy, i remember this from Analysis 1 by Tao
Loving analysis
Me tooo☺️☺️☺️☺️
I was wondering if you would do a video on finding green's functions to determine solutions to ode/pde problems with boundary conditions. I haven't been able to understand other videos and I feel like it would be clearer coming from you.
Maybe
Dr Peyam thank you for the consideration! I honestly look forward to anything you do on ode/pde problems.
Hey Dr.Peyam I'm wondering if you can make a video about summable families in the playlist series? There's like No video on it in TH-cam which is sad :( Thank you!
Sadly no
Doesn't it feel a bit handwavey to just say that minus minus is plus? 7:50
Nah, it just follows because x is an additive inverse of -x, since x + (-x) = 0
Its just like saying if you transpose a matrix or vector twice it is as if you did nothing
This is true by the uniqueness if additive inverses.
That was the previous theorem he proved.
1) The order of the dictionary is an order over C, but it doesn't satisfy the axioms O4 and O5.
2) An order has to be reflective (a
it seems like you are muted in all your videos recently
?
No, that is not the case, I hear all clearly.
Can you do the p-adic numbers?
I wish I knew more about them, haha 😅
Y= x ^ 2 is not compact
x ^ 2 + y ^ 2 = 1 is compact
is there a method to understand when
F (x, y) = 0 is compact?
perhaps a group is associated and if this group is compact then the design contained in the formula is also compact?
Thenks
In general, a set S \subset R^2 (for that matter, R^n) is compact if and only if it is closed and bounded (Heine-Borel Theorem). If F is continuous, (as in F(x,y)=x^2+y^2), then the set S={(x,y) l F(x,y)=0} is already closed, S is compact if and only if S is bounded, that is, all the solutions to F(x,y) is in some large rectangle [-M,M]^2 centered at the origin. Hope that helps.
wow guys we can now show that 0
Clickbait - title is "what is a real number?", reality is "what is an ordered field?"
Well, not really, I answered the question at the end 😉