What is zero raised to the power zero (0⁰)?
There has been a lot of debates about 0⁰ - most people say that the value is undefined and has no significance, but is it true?
The general agreed-upon value is ‘1’ and there are definitely good mathematical arguments that support this claim, one or two are discussed in this article but what if there is actually a way to look upon this value through its physical significance.
First, let's see what Mathematics has in its pocket by plotting a graph of (a^a) where a will be gradually decreased from 1 to 0, or to be precise very close to 0.
The graph clearly states that as we approach closer to zero, a^a gets closer to 1. Well, numbers speak for itself and now that it's clear why most people say that 0⁰ is ‘1’ BUT WHY?? This graph and some other mathematical arguments are only restricted to the Limits, What about actual 0⁰.
For example, 4x3 has a significance (4 added 3 times), 3! has a significance (number of ways of arranging 3 three different things in a row), etc. What is the significance of 0⁰ or for that matter (a^a). To understand this, let's use the concept of Probability and dig deeper.
(0.5)⁵ signifies — Probability of tossing a coin 5 times and getting head each time(ie, 5 times).
(0.5)² signifies — Probability of tossing a coin 2 times and getting head each time(ie, 2 times).
(0.5)⁰ signifies — Probability of tossing a coin 0 times and getting head 0 times. Sounds Confusing and absurd right? But let's understand what this statement actually means, It says “what is the probability of not getting head, when a coin is not tossed” Absolutely the answer is 1, When a coin is not tossed any outcome would not come so the statement is always true, hence the probability would be 1. (It is similar to the concept of P(~A)).
Now It's clear that anything raised to the power 0 is 1, moving to the end of this mystery: A coin has only two outcomes — Head OR Tail, there is no third, let's assume an outcome “coin landing vertically on its edge” which would never happen, ie, P(coin landing vertically on its edge) = 0.
(0)⁰ signifies — Probability of tossing a coin 0 times and getting coin landed vertically on its edge 0 times. It says “What is the probability of not getting the coin land vertically on its edge, when the coin is not tossed?” Absolutely 1, when a coin is not at all tossed, probability of any outcome not happening would be 1. (similar to the last case)
Although there are too many arguments which might contradict the value. In this case, nothing is absolutely wrong and nothing is absolutely right — it's just the matter of opinions. Which side are you on, ‘0’, ‘1’ or ‘not defined’?
