ra
Reals-as-Oracles
| ra | Reals-as-Oracles | |
|---|---|---|
| 6 | 1 | |
| 53 | 1 | |
| - | - | |
| 7.1 | 6.9 | |
| about 1 month ago | 3 days ago | |
| TeX | TeX | |
| GNU General Public License v3.0 or later | - |
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ra
- Basic Analysis: Introduction to Real Analysis
- need a textbook recommendation for real analysis!
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Why don't they explain why Calculus works?
I remember posting a similar list for introductory analysis (like the kind commonly required for math majors) too, but I didn't bother to figure out where that was, so I just looked up Volume I of the book by Jiří Lebl and found another good explanation of it, including cases where the function to be integrated is not continuous.
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[Undergraduate] Please suggest me some good books on "Real Analysis"
I also found one where the only comment that would be useful to anyone other than the OP was mine; I recommended, as far as free resources go, the textbooks by Lebl and by Trench.
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Available LaTeX files for textbooks from which I can learn more about LaTeX?
Lebl's "Basic Analysis: Introduction to Real Analysis" is opensource and the sourcecode's on github here
Reals-as-Oracles
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Math Limitations
I think having a language that helps understand those limitations is a useful achievement. Much of mathematics does have that. A notable exception is the definition of real numbers. They are usually presented as a string of infinite decimals, or a converging sequence, or a set of numbers less than something. All of those notions obscure the basic limitation of knowing the real number and give a veneer of similarity to rational number. Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference.
This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers.
For details: https://github.com/jostylr/Reals-as-Oracles/
In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated).
It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.
What are some alternatives?
course-notes-core - This repo contains notes for (some) courses made during core years at IISERM. CAUTION: Contain some cool stuff too.
ml-pen-and-paper-exercises - Pen and paper exercises in machine learning
diffyqs - Notes on Diffy Qs, a textbook for differential equations
LaTeX-examples - Examples for the usage of LaTeX
maths_book - Planning for an entire maths LaTeX book
ent - Elementary Number Theory: Primes, Congruences, and Secrets
internal-methods - Notes on how to use the internal language of toposes in algebraic geometry
maths_formulae - A LaTeX File with Maths Formulae from High School Maths and Common Core Engineering
the_statistics_handbook - the statistics handbook open source repository