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Reference.org
Ordinal arithmetic
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<p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, ordinal arithmetic describes the three usual operations on <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a>: <a href="/facts/Addition/apFWJBFB">addition</a>, <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>, and <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a>. Each can be defined in essentially two different ways: either by constructing an explicit <a href="/facts/Well-order/GrWEBWlW">well-ordered set</a> that represents the result of the operation or by using <a href="/facts/Transfinite_recursion/XlolgTDl">transfinite recursion</a>. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. </p>