0

To clarify, I am just a student in grade 12 so I only understand simple mathematical structures like sets, matrices, determinants and all the basic functions. My question is that, is there any one math book or something similar which defines everything from simple arithmetic to functions like logarithms or circular functions. Like a place which defines all mathematical structures exactly how they were meant to be defined?

For example there are many definitions of 'e', but what is that one definition which is the original one. Similarly, we can define trignometry using circles and triangles both, but which is the 'original' way to do it.
I hope you understand what I am trying to ask.

Chris Mzz.
  • 115
  • 8
Krrish Gupta
  • 70
  • Mathematics as a whole is far too vast for anyone to recommend you a general book for "everything". You'll probably come across an Analysis textbook, but that book maybe won't define structures you'd need for group theory. I know this comment isn't very helpful, but I hope you understand what I mean. – Chris Mzz. Jan 23 '24 at 16:04
  • 2
    "Original" sounds like you want the oldest definition, but that is often not the cleanest definition, nor even one that is even in use today. My guess is that that is not actually what you want, and it's more likely you want the most "fundamental" definition: explicit, clear, straight forward, with fewest assumptions. Can you clarify what you want? – JonathanZ Jan 23 '24 at 16:06
  • 2
    Maybe you might enjoy Amann and Escher's Analysis textbook. It has been many years and I no longer have my copy, but I recall them starting from essentially nothing. – Brian Shin Jan 23 '24 at 16:07
  • 1
    I think the approach of understanding things using the "original" definitions is not very fruitful at all. It happens quite often that the first papers where new things are introduced are very complicated to understand and retroperspectively not very well written as things were not very well understood in the beginning and years, maybe decades later these concepts are better understood and using these new notions very beautiful introductions are written. – Matthias Jan 23 '24 at 16:08
  • @JonathanZ by original I understand avoiding circular definitions and proofs – D S Jan 23 '24 at 16:36
  • I think you want any book on the topic of "real analysis." – Michael Jan 23 '24 at 16:45
  • As others have noted, Real Analysis is the first course where all the explicit details are gone into. But I did notice that you didn't mention limits, or calculus, in your question, and Real Analysis quickly gets to, and spends a lot of time on, the epsilon-delta type calculus proofs. If you want to include that then search for "real analysis text" and you'll find lots of posts with suggestions here. – JonathanZ Jan 23 '24 at 17:38
  • You may want to look at Fundamentals of Abstract Analysis by Andrew M. Gleason (1966; corrected reprint in 1991; review at MAA Reviews and internet archive copy), at least the first several chapters (which I think would be accessible to you; the last 2 or 3 chapters, not so much). See the amazon.com reviews and my comments here. – Dave L. Renfro Jan 23 '24 at 20:43

1 Answers1

3

In the hope I understood what your try to ask -- you don't want to start neither with OBM, Old Babylonian Math, nor RMP, Rhind Mathematical Papyrus -- you like to see the construction plan of the math logical building. Maybe Nicolas Bourbaki could be helpful.

m-stgt
  • 301
  • +1 when I saw the question, Bourbaki is the first name comes to my mind. – achille hui Jan 23 '24 at 16:47
  • The OP has said that he is in the 12th grade (i.e. in high-school) with a good but only elementary understanding of mathematics, so Bourbaki is definitely more than he can currently chew. At high-school level, one's mathematical knowledge is at the level of mid 19th century at best. Those who upvote this answer probably have not read the question carefully for context. – Alex M. Jan 23 '24 at 16:48
  • 1
    Yep, my comment was going to lead to mentioning the Bourbaki series, whose intentions are exactly what the OP asked for. But, as pointed out, without any motivation, discussion, or examples it's pretty hard to get a handle on things. I'd say that Bourbaki is exactly what the OP asked for, but they will discover it's not what they want. A good, slower-paced book on real analysis would probably make them happier. – JonathanZ Jan 23 '24 at 17:33
  • @Alex M. -- correct, I have no idea what "12th grade" actually is, but it never hurts to have a look past the horizon. Then, OP asked for the original way to find e or do trig (analytical or graphing?) -- then a book about history of math would be nice, e. g. how Euler did this and that, but I understood the OP as zest for know-why, not just know-how. Nevertheless, the question is closed now for to be too confusing ;) – m-stgt Jan 23 '24 at 18:22