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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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I.3 Some Fundamental Mathematical Definitions

The concepts discussed in this article occur throughout so much of modern mathematics that it would be inappropriate to discuss them in part III—they are too basic. Many later articles will assume at least some acquaintance with these concepts, so if you have not met them, then reading this article will help you to understand significantly more of the book.

1 The Main Number Systems

Almost always, the first mathematical concept that a child is exposed to is the idea of numbers, and numbers retain a central place in mathematics at all levels. However, it is not as easy as one might think to say what the word “number” means: the more mathematics one learns, the more uses of this word one comes to ...

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