Bernoulli number

Bernoulli numbers B±n<table><tbody><tr><th>n</th><th>fraction</th><th>decimal</th></tr><tr><td>0</td><td>1</td><td>+1.000000000</td></tr><tr><td>1</td><td>±⁠1/2⁠</td><td>±0.500000000</td></tr><tr><td>2</td><td>⁠1/6⁠</td><td>+0.166666666</td></tr><tr><td>3</td><td>0</td><td>+0.000000000</td></tr><tr><td>4</td><td>−⁠1/30⁠</td><td>−0.033333333</td></tr><tr><td>5</td><td>0</td><td>+0.000000000</td></tr><tr><td>6</td><td>⁠1/42⁠</td><td>+0.023809523</td></tr><tr><td>7</td><td>0</td><td>+0.000000000</td></tr><tr><td>8</td><td>−⁠1/30⁠</td><td>−0.033333333</td></tr><tr><td>9</td><td>0</td><td>+0.000000000</td></tr><tr><td>10</td><td>⁠5/66⁠</td><td>+0.075757575</td></tr><tr><td>11</td><td>0</td><td>+0.000000000</td></tr><tr><td>12</td><td>−⁠691/2730⁠</td><td>−0.253113553</td></tr><tr><td>13</td><td>0</td><td>+0.000000000</td></tr><tr><td>14</td><td>⁠7/6⁠</td><td>+1.166666666</td></tr><tr><td>15</td><td>0</td><td>+0.000000000</td></tr><tr><td>16</td><td>−⁠3617/510⁠</td><td>−7.092156862</td></tr><tr><td>17</td><td>0</td><td>+0.000000000</td></tr><tr><td>18</td><td>⁠43867/798⁠</td><td>+54.97117794</td></tr><tr><td>19</td><td>0</td><td>+0.000000000</td></tr><tr><td>20</td><td>−⁠174611/330⁠</td><td>−529.1242424</td></tr></tbody></table>
In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Bernoulli numbers Bn are a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> which occur frequently in <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a>. The Bernoulli numbers appear in (and can be defined by) the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> expansions of the <a href="/facts/Tangent_function/czej7ur0">tangent</a> and <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic tangent</a> functions, in <a href="/facts/Faulhaber%2527s_formula/bQJhbEnp">Faulhaber's formula</a> for the sum of m-th powers of the first n positive integers, in the <a href="/facts/Euler%25E2%2580%2593Maclaurin_formula/mN6uNCzN">Euler–Maclaurin formula</a>, and in expressions for certain values of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>.
The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by 
 
 
 
 
 B
 
 n
 
 
 −

{\displaystyle B_{n}^{-{}}}
  
 and 
  
    
      
        
          B
          
            n
          
          
            +

{\displaystyle B_{n}^{+{}}}
 
; they differ only for n = 1, where 
 
 
 
 
 B
 
 1
 
 
 −

=
        −
        1
        
          /
        
        2
      
    
    {\displaystyle B_{1}^{-{}}=-1/2}
  
 and 
  
    
      
        
          B
          
            1
          
          
            +

=
 +
 1
 
 /
 
 2
 
 
 {\displaystyle B_{1}^{+{}}=+1/2}
 
. For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the <a href="/facts/Bernoulli_polynomials/o3JBxUTN">Bernoulli polynomials</a> 
 
 
 
 
 B
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle B_{n}(x)}
 
, with 
 
 
 
 
 B
 
 n
 
 
 −

=
 
 B
 
 n
 
 
 (
 0
 )
 
 
 {\displaystyle B_{n}^{-{}}=B_{n}(0)}
 
 and 
 
 
 
 
 B
 
 n
 
 
 +
 
 
 =
 
 B
 
 n
 
 
 (
 1
 )
 
 
 {\displaystyle B_{n}^{+}=B_{n}(1)}
 
.
The Bernoulli numbers were discovered around the same time by the Swiss mathematician <a href="/facts/Jacob_Bernoulli/rUnwozsy">Jacob Bernoulli</a>, after whom they are named, and independently by Japanese mathematician <a href="/facts/Seki_Takakazu/pLD8pLIM">Seki Takakazu</a>. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his <a href="/facts/Ars_Conjectandi/gl1dTIsg">Ars Conjectandi</a> of 1713. <a href="/facts/Ada_Lovelace/s3c2GGRu">Ada Lovelace</a>'s <a href="/facts/Note_G/lklRwdhG">note G</a> on the <a href="/facts/Analytical_Engine/Ts9xduLI">Analytical Engine</a> from 1842 describes an <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> for generating Bernoulli numbers with <a href="/facts/Charles_Babbage/T3jklQcd">Babbage</a>'s machine; it is disputed <a href="/facts/Ada_Lovelace/s3c2GGRu">whether Lovelace or Babbage developed the algorithm</a>. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex <a href="/facts/Computer_program/e38jH9N9">computer program</a>.

Bernoulli number open-in-new

Bernoulli number