The Bernoulli numbers are so exotic that one might well wonder how they came to be discovered. We are fortunate that this has been explained quite clearly in the posthumously published [Bernoulli:1713]. The story begins with the question, what is the formula for the sum of k-powers Pn 1 n k? The case k = 0 is trivial and the case k = 1 is well known:
1 + 2 + · · · + n = n(n + 1)/ 2 .
At the time Bernoulli took up the problem, many more cases were known, but it seems to have been he who first attacked it systematically. The key to his method is Pascal’s triangle, which lays out the coefficients in the expansion
Bernoulli's numbers play an important and quite strange role in mathematics and in various places like analysis, number theory, and differential topology. They first appeared in Ars Conjectandi, a famous (and posthumous) treatise published in 1713, by Jakob Bernoulli (1654-1705) when he studied the sums of powers of consecutive integers
and in the expansion of many usual functions as tan(x), tanh(x), 1/sin(x), ¼
Perhaps one of the most important results is the Euler-Maclaurin summation formula, where Bernoulli's numbers are contained and which allows accelerating the computation of slow converging series. They also appear in numbers theory (Fermat's theorem) and in many other domains and have caused the creation of a huge literature.
According to Louis Saalschültz, the term Bernoulli's numbers was used for the first time by Abraham De Moivre (1667-1754) and also by Leonhard Euler (1707-1783) in 1755.