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February 2009Zero Raised to the 0th Power (00)For an example of the vagaries and effect of opinion in mathematics let's look at the seemingly mundane topic of 00 (0^0 or 0 raised to the 0th power). Review any history on this topic and you will see how much of popular mathematics is "indeterminate", "controversy" and "consensus". Personally I prefer my mathematics to be based on, well... mathematics. Many fail to distinguish between the ordinal and quantity uses of zero in this expression. This is another case where a dodgy view of zero can be misleading. For A0 any non-zero number "A" raised to the power of zero is generally agreed to equal 1. This result of 1 is the same as the number of dimensions in the number A. Popularly, using the { 0 } view of zero, 00 is supposed to equal 1 (or it's undefined depending on your opinion). Taking the null view, when A=0 null cannot be raised to any power, so the result is null or zero. Zero is also the number of dimensions in our view of zero. When we view the quantity as { 0 } we assume there is a number to work with, when using the { } view we assume there is no number. As an ordinator we use it as the set of a single integer halfway between -1 and 1, i.e. { 0 }. It may also be an empty quantity { } we wish to raise to this 0th power. Each of these uses of zero has different rules. 0^0 is also discussed in Ordinals and Quantities. Controversies such as this show that our understanding of mathematics is not as complete as we would like, however it is the purpose of these pages to focus on the mathematics of physics. Can anyone think of a valid use of 00 in physics? Let me know.
What's hot?One of our objectives in these pages is to uncover other controversial areas where mathematics and physics collide. If there's another raging topic on zero or infinity you'd like to discuss please contact us.
Regards, AJ Corcoran
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