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If I'd read this book as a teenager, maybe I would've passed Calc I on my first try as opposed to my third. With a C-.
I would second this by "Concrete Mathematics" by Graham, Knuth and Patashnik. This is actual university course book with very formal proofs and theory, but the subject matter is still largely accessible to serious high school students and demonstrates beautiful reasoning examples throughout. It is also very practical book, after covering techniques in this book, one can often times calculate exact sums of infinite series quicker than estimating their bounds. If your high school student decides to study math at university level, the techniques and skills taught in this book will prove invaluable in broad areas of study.
https://openstax.org/subjects/math
Some are even downloadable to a Kindle (for free) on Amazon.
Here's why:
- Your goal of the gift is something more than a plain textbook. The Princeton Companion is something your child will return to throughout their math career. It will be an anchor book that will remind them of your support for them when they were still a budding mathematician.
- Relatedly, the book is far too broad to be consumed as a textbook. Hence it will be more like a friend (or companion :) ) on their journey. Even a really amazing textbook (like Baby Rudin) in contrast is just a snapshot of where they are now.
https://artofproblemsolving.com/store/list/aops-curriculum
I've got a PhD in bioengineering but I'm currently going through Introduction to Counting and Probability and I'm really enjoying it.
Some others (not AOPS series):
Nelsen - Proofs Without Words
Polya - How to Solve it
Strogatz - Nonlinear Dynamics and Chaos
https://www.amazon.com/Mathematical-Circles-Dmitry-Fomin/dp/... It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the creation of groups of students, teachers, and mathematicians called "Mathematical Circles". The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport-without necessarily being competitive. T…