# What's the most amazing result in mathematics?

Euler’s identity, the Banach-Tarski paradox and the sum of all numbers up to infinity are all pretty amazing - find out why...

**Asked by: Simon Lewis, Bedford**

A popular choice is Euler’s identity, which shows that raising the endless number ‘e’ (roughly 2.718) to the power of pi, multiplied by the impossible square root of -1, and then adding the result to 1 produces… zero. How such a crazy mix of numbers leads to such a simple result defies common sense.

No less baffling is the Banach-Tarski paradox, which shows that a solid ball can be cut into five special shapes and reassembled to make two exact, perfectly solid replicas of the original ball. Admittedly, the shapes have to be pretty special – specifically, infinitely jagged, which isn’t possible in the real world.

Arguably, the craziest of all results does have real-world implications. It’s the sum of all the integers, 1+2+3+4 and so on, all the way to infinity. On the face of it, this must add up to infinity. The correct answer, however, isn’t even a positive whole number: it’s minus 1/12. This result emerges from something called analytic continuation of the Riemann zeta function. Physicists have successfully tested its implications in theories about the sub-atomic world.

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