Editing Surreal number (section)

===Basic induction===
The induction steps for the surreal exponential are based on the series expansion for the real exponential,
<math display=block>\exp x = \sum_{n\ge 0} \frac{x^n}{n!}</math>
more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For {{mvar|x}} positive these are denoted {{math|[''x'']{{sub|''n''}}}} and include all [[partial sum]]s; for {{mvar|x}} negative but finite, {{math|[''x'']{{sub|2''n''+1}}}} denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For {{mvar|x}} negative infinite the odd-numbered partial sums are strictly decreasing and the {{math|[''x'']{{sub|2''n''+1}}}} notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.

The relations that hold for real {{math|{{nowrap|1=''x'' < ''y''}}}} are then{{ubl|{{nb5}}{{math|{{nowrap|1=exp ''x'' · [''y'' – ''x'']{{sub|''n''}} < exp ''y''}}}}}}and{{ubl|{{nb5}}{{nowrap|1={{math|exp ''y'' · [''x'' – ''y'']{{sub|2''n'' + 1}} < exp ''x''}},}}}}and this can be extended to the surreals with the definition

<math display=block>\exp z = \{0, \exp z_L \cdot [z-z_L]_n, \exp z_R\cdot[z-z_R]_{2n+1} \mid \exp z_R/[z_R-z]_n, \exp z_L/[z_L-z]_{2n+1} \}.</math>

This is well-defined for all surreal arguments (the value exists and does not depend on the choice of {{mvar|z{{sub|L}}}} and {{mvar|z{{sub|R}}}}).