We all know how numbers look. And sometimes it is even hard to imagine that they can have a different form that what is familiar to us. But while math is a rather ancient science our number system and our numbers are not very old. This photo shows their development in time:

As you can see even numbers in the middle ages were different from modern numbers. And I am sure that you would not even think that there is any connection between our numbers and Hindu.

It is a surprising fact however that there is just one number system. You will probably say it is not correct. But it is. Yes, there were a lot of different number system in history. Even in the present we have more then one. Take the binary system for example.

But if we want a number system that is good enough for both algebra and geometry, that is a system that lets us measure distance we discovery that there is only one such system - they are all isomorphic to each other. It means that you can give different names to numbers, you can have it in base 17 (for example) and not in base 10 but it will still be the same thing. In mathematical language - we have only one mathematical structure that is a complete ordered field.

Grothendieck seminar at the ENS

21 hours ago

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