Analysis I
by Terence Tao · first published 2006
82% recommend18% neutral
Why HN recommends it
Builds analysis from the ground up, starting with natural numbers and constructing the reals, with explanations of why proofs are done as they are; recommended for readers wanting a rigorous, patient foundation.
Summarised by Claude from recommending comments
This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean
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…to Datastructures and Algorithms [3] Introduction to Digital Design [4] Parallel Programming (using Java) [5] Optional but highly recommended, you'll probably find it completely out of scope: Real Analysis I [6] Real Analysis II [7] Introduction to Linear Algebra [8] Introduction to Physics [9] [1]: http://www.amazon.com/Touch-Class-Learning-Program-Contracts... [2]: http://ocw.mit.edu/courses/electrical-engineering-and-comput...…
on “DIY Computer Science Degree” ↗…references, I'll have a look. I can tell you what I don't want: books that throw theorems one after the other without any context, like Baby Rudin [1] for instance. I tried once to read Terence Tao's Analysis I [2], it's really good but my main problem was that I wasn't able to know if my proofs are correct when I do an exercise. So maybe the solution is to get a teacher. [1]: https://en.wikipedia.org/wiki/Principles_of_Mathematical_Ana...…
on “Mathematics: Its Content, Methods and Meaning” ↗I've tried reading Rudin a couple times, but it's a bit of a slog. There are easier analysis texts. Abbott's Understanding Analysis is good, though a bit basic. I'm currently reading Terence Tao's Analysis I, which is very good if you're in the right frame of mind for it. The first 150 pages are spent building the real numbers from…
on “Ask HN: Math books like SICP?” ↗…well-prepared for them with prerequisite courses. Terence Tao wrote two books to cover (with better exposition) what Rudin did in one. I think it would be pretty hard to read consistently three pages of Tao's Analysis I each day, before he even gets to limits. I think you're underestimating the intellectual effort here. In my opinion, even if you're reading a math book targeted to your level, committing to reading and…
on “Mathematics for Computer Science: Readings” ↗…definition of Limits https://youtu.be/K4eAyn-oK4M?t=331 If you really want to understand these things, start here: https://www.youtube.com/playlist?list=PL5A714C94D40392AB and watch them as you work through Analysis I by Terence Tao, skipping to chapter 2 where he begins from scratch introducing the naturals, integers, etc. Wildberger even shows you how to do algebraic calculus which will help to learn the Tao book…
on “Calculus Explained with GIFs and Pics (2014)” ↗Try Terence Tao's book Analysis I, start at chapter 2 if you forgot trig. Defines naturals, integers, rationals and reals from scratch teaching you how a proof works at the same time. When you get to the calculus chapters watch Gilbert…
on “Ask HN: What math course should I take?” ↗…this problem why the identity is true, and then getting all those 'How to prove it' style books filled with established techniques to see if your identity can be proved by them. Try Terence Tao's book Analysis I https://learnaifromscratch.github.io/math.html you pick up the gist of this kind of math doing basic proofs of natural numbers. You see and do enough of these that you will get the first proof here, where…
on “How do people learn advanced math” ↗…for a layman level, I feel it's crucial to at least make the readers understand why and how these concepts come to be, so that they can grasp at least a little bit of the 'logic' of math. Similarly, Analysis I by Terence Tao has a 'Why do analysis' that has made me actually enjoy analysis.
on “Ask HN: The book that did it for you in math and/or CS?” ↗I'd recommend starting from Terence Tao's Analysis I book. It really starts you from "zero" (literally), and constructing natural numbers, all the way to real numbers and integrals, and beyond.
on “Ask HN: How to learn math from zero for adults?” ↗Try Terence Tao's Analysis I and II.
on “Ask HN: If I work with applied mathematics should I still study analysis?” ↗…Physics major and lately I've been on a kick of filling in the mathematics that was used in my Physics classes but that I feel like was never really gone into in much depth. My current reading list is: - "Analysis I" (and II) by Terence Tao (I finished the first volume and am now on the second, but I consider them really one book) - "Understanding Analysis" by Stephen Abbott - "Topology Through Inquiry" by Starbird…
on “Ask HN: Technical Summer Reading List” ↗I'm about 100 pages into "Dune Messiah" (the 2 movies out are both for the first book, the 3rd movie is going to be this book), 50 pages into "Analysis I" by Terence Tao, 100 pages into "Thinking in bets"
on “Ask HN: What are you reading?” ↗You have to first prove that p is not undecidable. This is not being done in many proof by contra-positive based proofs in mathematics. First prove that p is not undecidable is not a step in Analysis I by Terence Tao for instance.
on “Proof by Contrapositive Is Wrong” ↗