Zero to the power of zero

Zero to the power of zero, denoted as 00, is a <a href="/facts/Mathematical_expression/MPrMlYbE">mathematical expression</a> with different interpretations depending on the context. In certain areas of mathematics, such as <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a> and <a href="/facts/Algebra/nMdqczjo">algebra</a>, 00 is conventionally defined as 1 because this assignment simplifies many <a href="/facts/Formula/BosTcKnv">formulas</a> and ensures consistency in operations involving <a href="/facts/Exponents/pac6QY2g">exponents</a>. For instance, in combinatorics, defining 00 = 1 aligns with the interpretation of choosing 0 elements from a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> and simplifies <a href="/facts/Polynomial_expansion/oIe5fBKr">polynomial</a> and <a href="/facts/Binomial_expansions/hd4SmkUd">binomial expansions</a>.
However, in other contexts, particularly in <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, 00 is often considered an <a href="/facts/Indeterminate_form/5Y3JWKjS">indeterminate form</a>. This is because the value of xy as both x and y approach zero can lead to different results based on the <a href="/facts/Limit_process/j8JBhj4e">limiting process</a>. The expression arises in limit problems and may result in a range of values or diverge to <a href="/facts/Infinity/cF506uD7">infinity</a>, making it difficult to assign a single consistent value in these cases.
The treatment of 00 also varies across different <a href="/facts/Computer_programming_language/cWTYbgWa">computer programming languages</a> and <a href="/facts/Software/RP79lVSW">software</a>. While many follow the convention of assigning 00 = 1 for practical reasons, others leave it <a href="/facts/Undefined_(mathematics)/Bc1Q3px4">undefined</a> or return errors depending on the context of use, reflecting the ambiguity of the expression in mathematical analysis.

Zero to the power of zero open-in-new

Zero to the power of zero