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Dear Christopher, Rachel, Sylvain, Yaelle, and all the others!
I am writing this to you because I came to schools like yours, in France as well as in Canada, to talk mathematics with students who could have been your own friends, and who thus contributed to a joint enterprise. I wanted to show them beautiful mathematics, at the level of your class, but conceived the way a mathematician does it. In most school books, the topics are usually treated in a way which I find incoherent. They pile up one little thing on another, without rhyme or reason. They accumulate technical details endlessly, without showing the great lines of thought in which technique can be inserted, so that it becomes both appealing and meaningful. They don't show the great mathematical lines, similar to musical lines in a great piece of music. And it's a great pity, because to do mathematics is a lively and beautiful activity.
This book is made up of several lectures, or rather dialogues, which have been transcribed as faithfully as possible from the tapes, to preserve their lively style. It gave me great pleasure to have this kind of exchange with all the students, in different classes. The subjects concern geometric and algebraic topics, understandable at the ninth and tenth grade level. I even gave one of the talks to an eighth grade class ! If students at those levels could understand and enjoy the mathematics involved, so can you. Each dialogue is self-contained, so you don't have to read this book continuously from beginning to end. Each topic forms a single unit, which you can enjoy independently of the others. If, while reading anyone of them, you find the going too rough, don't let that put you off. Keep on reading, skip those passages that don't sink in right away, and you will probably find something later in the lecture which is easier and more accessible to you. If you are still interested, come back to those passages which gave you trouble. You will be surprised how often. after sleeping on it, something which appeared hard suddenly becomes easy. Just browse through the book, pick and choose, and mostly get your mind to function.
Who is Serge Lang?
Dear Christopher, Rachel, Sylvain, Yaelle, and all the others.
What is Pi?
Volumes in higher dimension.
The volume of the ball.
The length of the circle.
The area of the sphere.
Pythagorean triples.
Infinities.
Postscript