A huge conflict: What is zero raised to the power of zero?
What is zero raised to the power of zero? This is a question that asked more than 35 billion and 378 million times. And 98% of people hasn’t answered correctly.
First, what does 2⁵ mean? It means 2 times 2 times 2 times 2 times 2. In other words, multiply 2 by itself 5 times. Now, we can say 0⁰ means “multiply zero by itself 0 times”. Hmmm, that’s awkward.
Let’s go to different directions and find the other powers.
Once when we see an exponential equation such as 0⁹ = 0 , we will say “zero to the ninth power is zero”.
It looks like 0⁰ = 0. But 0 to the -5th power is 1 over 0 which is undefined and same with 0 to the -100th power. The negative exponents indicate 0⁰ should be undefined.
Let’s attack this from a different angle. Other numbers raised to 0 equal 1.
This pattern indicates that 0⁰ should also be 1. So, it looks like there isn’t certainly a particular accurate solution? Which is exact? Nonetheless depending on the situation, you work in one answer may be better than the others. The best explanation should be reliable, reduce needless complexity, and be beneficial.
Most theoreticians choose that in many cases, 1 is the finest definition for 0⁰. Let’s look at two reasons for this. a raise to b can be viewed as the number of sets of b elements that can be chosen from a set of a elements.
For example, 2¹ can be observed as the amount of sets of one element that can be chosen from the set of two elements.
And 0⁰ is the amount of sets of zero elements that can be selected from a set of zero elements. Which must be 1! So, 1 is the only definition reliable with this understanding of exponentiation.
In this perspective, any other definition would unnecessarily confound things. For another case where 0⁰= 1 is a beneficial definition, let’s look at the binomial statement.
As x = 0, this simplifies to 1 = 0⁰ • 1. In this item, the only explanation for 0⁰ that constructs the binomial statement correct is 1. Again 0⁰= 1 is the only definition that avoids needless complexity. Yet, depending on the sort of mathematics we are doing, 1 may not permanently be the finest definition.
For example let’s look at some limits. The limit of a function at point a is the value of the function approaches as its input approaches a. We’re involved in limits of the form 0⁰ when x = 0. A simple one is the limit of x⁰ as x approaches 0. Since x⁰ = 1 at all other points, its limit at 0 is 1 as well. This seems to verify that 0⁰ = 1.
Nonetheless there are other limits of the form 0⁰ with different values! The limit of 0 raise to x from the right is 0… And from the left it’s undefined. And other limits of the form 0⁰can be any value like this one which is e.
These conflicts are good reasons to call 0⁰ an “indeterminate form” or “undefined” when you’re dealing with limits. These are the only definitions that are consistent with the way we define limits.
So what is 0⁰? It depends! Often, 1 is the best answer. However, when dealing with limits, “undefined” or “indeterminate form” os more sensible. Depending on the type of math we are doing, even definitions and conventions can change!