Lang, Serge (1983). Fundamentals of Diophantine geometry. New York: Springer-Verlag. doi: 10.1007/978-1-4757-1810-2. ISBN 0-387-90837-4. MR 0715605. Second edition of Diophantine Geometry (1962) [19] [20] Jorgenson, Jay; Lang, Serge (2009). Heat Eisenstein series on SL n(C). Memoirs of the American Mathematical Society. Vol.201. doi: 10.1090/memo/0946. ISBN 978-0-8218-4044-3. MR 2548067. Lang’s Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books."—NOTICES OF THE AMS

Algebraic Number Theory | SpringerLink Algebraic Number Theory | SpringerLink

Edit: I forgot to answer the part about trigonometry. Plane trigonometry can be divided roughly into two parts: (1) geometric applications to triangles, quadrilaterals, etc.; (2) analytic trigonometry, which involves various algebraic manipulations of trigonometric functions. For the more advanced aspects of (1), good knowledge of (2) is necessary. Fulton, William; Lang, Serge (1985). Riemann–Roch algebra. Grundlehren der mathematischen Wissenschaften. Vol.277. New York: Springer-Verlag. doi: 10.1007/978-1-4757-1858-4. ISBN 978-1-4419-3073-6. MR 0801033. Lang, Serge (1990). Cyclotomic fields I and II. Graduate Texts in Mathematics. Vol.121. With an appendix by Karl Rubin (Combined second edition of 1978/1980 originaled.). New York: Springer-Verlag. doi: 10.1007/978-1-4612-0987-4. ISBN 0-387-96671-4. MR 1029028. O'Connor, John J.; Robertson, Edmund F., "Serge Lang", MacTutor History of Mathematics Archive, University of St Andrews Lang, Serge (January 1995). "Mordell's review, Siegel's letter to Mordell, Diophantine Geomertry, and 20th century mathematics" (PDF). Gazette des mathématiciens (63): 17–36.The basic topics covered doesn't necessarily mean that the book will be easy; this was my first introduction to proofs, and to trying to create my own (which I often failed at doing). However, even if I knew most of the material already (like that a negative number multiplied by a negative number results in a positive number), I had never known the reasoning behind why this was true. The logical approach presented taught me a different, more effective way to learn mathematics. Lang even has an entire section (somewhere after chapter 3, I think) where he simply covers logic and some notation, to help the reader get a better grasp of the idea. Lang, Serge; Cherry, William (1990). Topics in Nevanlinna Theory. Lecture Notes in Mathematics. Vol.1433. With an appendix by Zhuan Ye. Berlin: Springer-Verlag. doi: 10.1007/BFb0093846. ISBN 3-540-52785-0. MR 1069755. Jorgenson, Jay; Krantz, Steven G., eds. (April 2007). "The Mathematical Contributions of Serge Lang" (PDF). Notices of the AMS. 54 (4): 476–497. From April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books: "Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." Jorgenson, Jay; Lang, Serge (1993). Basic Analysis of Regularized Series and Products. Lecture Notes in Mathematics. Vol.1564. Berlin: Springer-Verlag. doi: 10.1007/BFb0077194. ISBN 3-540-57488-3. MR 1284924.

Basic Mathematics by Serge Lang - Physics Forums Basic Mathematics by Serge Lang - Physics Forums

Serge Lang ( French: [lɑ̃ɡ]; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group. Mordell, L. J. (1964). "Review: Diophantine geometry. By Serge Lang" (PDF). Bull. Amer. Math. Soc. 70 (4): 491–498. doi: 10.1090/s0002-9904-1964-11164-2. Lang, Serge (1995). Introduction to modular forms. Grundlehren der mathematischen Wissenschaften. Vol.222. With appendixes by D. Zagier and Walter Feit (Corrected reprint of the 1976 originaled.). Berlin: Springer-Verlag. doi: 10.1007/978-3-642-51447-0. ISBN 3-540-07833-9. MR 1363488. [26] This book is truly one of a kind. This is a book on basic mathematics, written from the point-of-view of an advanced mathematician. Do not expect many plug-and-chug exercises that just drill the method into you. Do not expect silly notions such as "it is important to rationalize the denominator", it is not. What you should expect is a book that teaches what mathematics really is about. It teaches proofs and logic as a foundation of mathematics.

A lot of interesting things learnt in this book. Overall, it’s a good book though it is a bit misleading. It’s not basic at all but this book is a good introduction to proofs. I learnt a lot of about proofs and how theorems I take for granted work, including those on determinants, complex numbers, geometry, coordinate systems, etc.