Numbers, operations, structures
Numbers are at the center of the mathematical edifice. After a long period where they were apprehended through intuition, the need arose to conceive them with the help of a rigorous axiomatic system; the one introduced by Peano for natural numbers is the finest example. Set theory subsequently led to the construction of rationals, reals, and complex numbers. Drawing inspiration from these methods and generalizing them, it enabled the definition of more general structures like the notion of group and to provide a rigorous foundation for geometry within the framework of vector space theory.
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Join the groups!
The concept of a group first emerged from efforts to solve equations in the 19th century and soon became indispensable, highlighting parallels between situations that at first seem quite different. Let's see why mathematicians are so group-minded.
Constructing numbers: a long history
In the beginning was number... If we go back to the very origins, these objects were represented by pebbles before being encoded by symbols. In fact, there are numbers to suit every taste! As everyone knows, when you love something, you don't count the cost...
The ternary Cantor set
Cantor constructed a fractal set before fractals had a name, showing that a subset of the real line can have the cardinality of the continuum, have measure zero, and have empty interior.