For example, Trigonometry was written by Wolf-Prize winner Israel Gelfand, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks on some elementary undergraduate courses such as College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE as I would love my students to read textbooks written by the best mathematicians.

  • $\begingroup$ So you will teach them calculus using Courant and John? Seriously, Gelfand was unusual in this way: taking the time to do elementary textbooks. But before adopting Galfand for your students, take a look at it to see whether it is at the right level for them. $\endgroup$ – Gerald Edgar Oct 29 '19 at 10:06
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    $\begingroup$ textbooks written by the best mathematicians --- It seems to me that two often different attributes are being conflated here. Also, the textbook's audience is a crucial factor (e.g. Harvard's Math 55 vs. these courses). That said, despite what you might think from Gelfond's name, the analogy "Gelfond's book to trigonometry $::$ Courant/John's book to calculus" (or any of the other well known substitutions one might make for "Courant/John $:$ calculus") is far from being very accurate. $\endgroup$ – Dave L Renfro Oct 29 '19 at 11:22
  • $\begingroup$ @Dan Fox: Typo on my part. $\endgroup$ – Dave L Renfro Oct 29 '19 at 13:44
  • $\begingroup$ David Mumford appears among the list of "producers" of the Hughes-Hallett Mutlivariable Calculus book. $\endgroup$ – Michael Bächtold Oct 29 '19 at 14:39
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    $\begingroup$ Are you assuming that textbooks written by world-class mathematicians will be good textbooks? To quote an acquaintance of Serge Lang's, "We all knew he wrote terrible books, but we loved him anyway." $\endgroup$ – Daniel R. Collins Oct 30 '19 at 0:29

Books about middle school algebra and trigonometry written by someone like Gelfand are rare (I don't know other examples; for example V.I. Arnold's books nomially for children are aimed at very special children, their author's claims to the contrary), but there are many undergraduate textbooks written by superb mathematicians. I list below some that occur to me now. However, most of those I list are aimed at a level probably a bit above that which interests the OP.

Vladimir Arnold wrote an excellent textbook on ordinary differential equations.

Peter Lax wrote a (moderately advanced) book on linear algebra.

Serge Lang wrote textbooks on calculus, linear algebra, complex analysis, and some other undergraduate subjects. I don't particularly like his books, but that is a different matter.

Cliff Taubes wrote a book called "Modeling differential equations in biology".

While not exactly a textbook, the book of Persi Diaconis and Brian Skyrmes called "Ten great ideas about chance" is an excellent introduction to probabilistic thinking directed at a general undergraduate audience.

Marcel Berger wrote some textbooks directed at future math teachers that treat classical affine and projective geometry (they have been translated into English). They would be hard for a US educated audience.

Elias Stein has coauthored with Rami Shakarchi a series of books on real, complex, fourier, and functional analysis. The only one I've read is the book on Fourier analysis, and it is excellent and pedagogical, the book I always wished I had encountered as a student.

Yuri Manin wrote a linear algebra textbook (however, it is not afraid to discuss functors).

  • $\begingroup$ Thank you for this list! It is very helpful. $\endgroup$ – Zuriel Oct 29 '19 at 14:51

This may not qualify, as it is a stretch to classify this as a "textbook." It arose from a series of lecture Klein gave to high-school teachers.

Klein, Felix. Elementary Mathematics from an Advanced Standpoint: Geometry. Transl. from the Third German Ed. by ER Hedrick and CA Noble. Dover, 1939.


I love this (1939) opening "Gentlemen!" Preface sentence:

"Gentlemen! The course of lectures which I now begin will be an immediate continuation of, and supplement to, my course last Winter."

How times have changed!

I am less familiar with the content of the other volumes:

Klein, Felix. Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis. Vol. 1. Courier Corporation, 2004.


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