My good friend David Cushing popped on Facebook messenger to ask me a question:

I did tweet it, and I got a lot of good responses. Before I tell you about those, I’ll quickly list the books we mentioned above, that *of course* a keen 13-year-old already has.

Alex Bellos‘s two books, *Alex’s Adventures in Numberland* and *Alex Through the Looking-Glass* are both superb, as is Matt Parker‘s book, *Things to Make and do in the Fourth Dimension*.

I also recommended *50 Mathematical Ideas You Really Need to Know* by Tony Crilly. It looks like a bog-standard pop maths compilation, but it’s packed full of really interesting stuff. Blame the publishers for the generic title.

Now, here’s what everyone else recommended.

Sue van Hattum suggested *Math Girls* by Hiroshi Yuki, *The Number Devil* by Hans Magnus Enzensberger, and *The Man Who Counted* by Malba Tahan.

*Math Girls* is a novel “combining mathematical rigour with light romance”, and a few people chimed in to agree that it would be a good present. *The Number Devil* is an illustrated children’s book. *The Man Who Counted* recounts the adventures of the character Beremiz Samir, who uses his mathematical know-how to settle disputes and win fame and fortune. There’s a nice article about Malba Tahan in the BBC Magazine).

Evelyn Lamb said that she read Douglas R. Hofstadter’s *Gödel, Escher, Bach: An Eternal Golden Braid* at around the same age as David’s cousin. While it’s certainly got lots of inspiring maths in it, to do with logic and the mathematical way of thinking, I have to say it’s a hard slog.

Javier Moreno suggested that *What Is The Name of This Book?* by Raymond Smullyan might be a more tractable option, in the same area.

Julia Collins said she enjoyed Paul Hoffman’s *The Man Who Loved Only Numbers*, a biography of Paul Erdős, the world’s most prolific mathematician. Apparently it has lots of puzzles in it.

Blake Stacey reminded me about Donald Knuth’s book *Surreal Numbers*, which it turns out we both read in our teens. It’s a book about an area of maths you’re extremely unlikely to need or even encounter in grown-up life, but presented through a most charming, whimsical story. Lots of the books in this post have been recommended by people who credit them with inspiring them to do maths in later life. *Surreal Numbers* is the book that did that for me.

Blake also recommended *The Cartoon Guide to Algebra* and *The Cartoon Guide to Calculus*, both by Larry Gonick, and had another excellent shout: a great way to get into mathematical thinking is to start programming computers.

Tom Edgar said he’s always liked *The Book of Numbers* by John Conway and Richard Guy. I don’t think I’ve ever read it to cover, but it’s my go-to reference for the names of really big numbers of the nonillion, decillion, … sort.

Marc Chamberland told me about his book, *Single Digits: In Praise of Small Numbers*, which I hadn’t heard of. It’s a collection of very small, independent sections on all sorts of topics, gathered together in chapters linked to single digits.

Colin Wright recommended Michael Spivak’s *Calculus* textbook. I haven’t read it, but I’ve heard it mentioned before and in a world full of awful doorstop calculus textbooks, one that’s aimed at giving a good first encounter with “real” maths is to be applauded. And if Colin says it’s good, it must be.

Dave Radcliffe suggests “any book by Martin Gardner”. I’ll pick *Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi*, Martin Gardner’s first book of mathematical puzzles and games. It’s got more than enough stuff in it to captivate a young mathematician for a good while, and it’s the first in a long series of similarly fascinating books. Dave said he also likes *Ingenuity in Mathematics* by Ross Honsberger. It’s a collection of 19 essays about different aspects of mathematical thinking.

Hanneli Tavante offered *The Manga Guide to Calculus* by Hiroyuki Kojima and Shin Togami. “Comics inside!” declares the cover – that would’ve appealed to a 14-year-old me.

Sue van Hattum popped in again to add a recommendation for *The Cat in Numberland* by Ivan Ekeland, a picture book about a cat. It looks a bit young for our target teenager. Sue also sent me a link to her list of mathematical books on her blog.

Vincent Pantaloni suggested Cédric Villani’s autobiographical essay *Birth of a Theorem.* I’ve heard it’s good!

Steve wrote in very late on to suggest *The Emperor’s New Mind* by Roger Penrose, and *Cakes, Custard and Category Theory* by Eugenia Cheng. That reminds me that I was given Jim Henle’s *The Proof and the Pudding* recently. It’s a very good book!

Finally, Pieter Belmans asked, “Isn’t that the age when Pierre Deligne was given Bourbaki’s *Théorie des ensembles*?” Let’s not get carried away with ourselves!

So we got loads of book recommendations. David’s original question was about books that actually teach real maths, which not all of the books above do. Of the ones I know well enough to recommend, I think Hofstadter’s *Gödel, Escher, Bach*, Knuth’s *Surreal Numbers*, Conway and Guy’s *The Book of Numbers* and Gardner’s *Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi* are our best bets. I’m going to add quite a few of the others to my wish list, though!

Wow, GEB at age 14?! Kids are so smart these days.

Lockhart’s Measurement is worth adding to your list.

Off the beaten track, Mathsemantics is probably my favorite book that no one else has ever heard about. However, I guess a keen 14 yo will think it too easy and miss the point.

That’s a fabulous book list for a teen, so I hesitate to even mention another… but… I will ;-)

Stanley Farlow’s 2014 “Paradoxes in Mathematics” (paperback from Dover), is a quick overview of ~30 classic paradoxes/problems that are a great, fun introduction to mathematical thinking for a young person.

One more recent book aimed at bright teenagers interested in seeing not just calculus/university level mathematical analysis for the first time is Tom Körner’s beautifully written

Calculus for the AmbitiousCompared to Spivak’s textbook it’s much more affordable, easier to carry (i.e. less useful as a doorstop), less intimidating to the beginner and rather more light-hearted. For comparison’s with other texts see:

https://www.dpmms.cam.ac.uk/~twk/Which.pdf

Concepts of Modern Mathematicsby Ian Stewart.FYI: Mike Lawler branched the discussion with his own list here: More great books

Colin Wright and I have been exchanging suggestions on the topic of books for 14 year olds. Spivak is at the “very ambitious” end of the scale but bear in mind there are 11 year olds who do A-level Further Maths and get an A. Not very many, to be sure, and I suppose if your 14 year old had already done FM at 11 you wouldn’t need someone else to suggest books. But anyway.

I think Spivak would be ok for 14 year olds who e.g. wanted a book with actual stuff to do in it rather than being told stuff existed but not how any of it worked. I got Spivak as a birthday present myself, for what it’s worth. The price is of course a problem. Maybe you can find it second hand. You would need to be pretty sure Spivak was a good idea in the case of your particular 14 year old, though.

“Sets and Groups” by J A Green is something a friend recommended to me for this purpose many years ago. It’s a “real maths but not calculus for a change” contender.

The Mystery of the prime numbers by Matthew Watkins @SoC_trilogy. I am currently reading the first book. It seems a good introduction to the primes and what makes them interesting. I want to see where, and how far, he takes it in the other two books. Discovered in local library. The blurb on the back of the book includes high praise from Sir Roger Penrose and Brian Josephson.

After looking at the other list David Wells’ “You are a mathematician” came to mind.

Also, look at “The Pleasures of Counting” by Tom Korner, and the extensive list of book suggestions (with comments as to why) at the back of that. Clearly, since the book is from 1997 or so there are a lot of things not mentioned there but there are plenty of belters.

Late to the thread, but at the age of David’s cousin I found “100 Great Problems with Elementary Solutions” both refreshing and interesting

http://store.doverpublications.com/0486613488.html

I can’t say I followed more than a small amount, but it made an impression on me, and it might complement some of the “pop maths” suggestions since it does actually prove things, rather than just mentioning famous puzzles or people.

(On the “pop maths” side, I always liked Ian Stewart’s books, but they may have dated somewhat.)