0^0
It's very important to note here that 0^0=1 is a <i>shorthand</i> and not a <i>truth</i>.<p>Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 <i>is</i> 1" in the sense that they say "1+1 <i>is</i> 2" or "0.999... <i>is</i> 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why mathematics is the most powerful tool for determining truth that humanity has either discovered or invented (hat tip to anyone familiar with that debate).<p>One of the cold, austere, beauties of mathematics* is that if you do not accept a definition, you can reject it as false and reason with the result. To give the traditional example, if you accept as true that two parallel lines never intersect, then along with the other 4 of Euclid's Axioms you can prove all of Euclidian Geometry (what you learn in high school). If you do not accept it as true (an explicitly accept it as false), then you can prove all of Hyperbolic Geometry (one form of non-Euclidian Geometry).<p>In the case here, you're free to reject the convention that defines 0^0 as 1 and reason with the result; <i></i>you will not break any mathematics<i></i>. But you should know that mathematicians have never run into any issues with this convention--or can handle it trivially when they arise--so you're only adding a lot of work for yourself.<p>When you really think about it, it fills you with awe. It's awesome.<p>*“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” -Bertrand Russell