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Originally posted on What's new:

Mathematicians study a variety of different mathematical structures, but perhaps the structures that are most commonly associated with mathematics are the *number systems*, such as the integers $latex {{\bf Z}}&fg=000000$ or the real numbers $latex {{\bf R}}&fg=000000$. Indeed, the use of number systems is so closely identified with the practice of mathematics that one sometimes forgets that it is possible to do mathematics without explicit reference to any concept of number. For instance, the ancient Greeks were able to prove many theorems in Euclidean geometry, well before the development of Cartesian coordinates and analytic geometry in the seventeenth century, or the formal constructions or axiomatisations of the real number system that emerged in the nineteenth century (not to mention precursor concepts such as zero or negative numbers, whose very existence was highly controversial, if entertained at all, to the ancient Greeks). To do this, the Greeks used geometric operations…

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