Why Is A Number Raised To The Power Zero = One?

If you are the typical person, you have the general understanding that any number raised to the power zero equals one. When we try out a few examples on a calculator, our knowledge is reaffirmed. And because of this, most of us seldom question: Why? Very few of us actually understand why a number raised to the power zero equals one. In this article, I try to explain why exponentiation (a number raised to the power of another number) works this way by solving easy, incremental puzzles that anyone can understand. Furthermore, there is one exception where this cannot be applied. So, it is mathematically wrong to say that any number raised to the power zero equals one. We’ll also touch upon this point shortly towards the end.

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What is Exponentiation?

Let us begin with understanding exponentiation. It is easier to take a look at examples first and then move on to the theoretical reasoning behind them. Consider the example: 2^3 = 8. Here, the base (2) occurs in a multiplicative series the exponent (3) Number of times: 2*2*2. The answer is then the result of multiplication. The key point to note here is the process of exponentiation is first resolved into a multiplicative series. Then, carrying out the multiplication operation gives us the final result. This level of understanding is smoothly scalable for higher integers. For instance, for 2^100, we would have a multiplicative series with a hundred 2’s. So far, so good.

But what about 2^0. Does it lead to a multiplicative series with zero number of 2’s? If that were the case, the result should be zero, right? But how come we arrive at 1 as the answer? Before we answer that question, let’s try and move the exponents up and down – this makes it significantly easier to understand what’s going on here.

Increasing the Exponent

So far, we have 2^3 = 8. What if I wanted to use this to compute 2^4? In other words, what should I do to 2^3 in order to get 2^4? Since we know that the exponent represents the number of times the base occurs in the multiplicative series, we could say that 2^4 has one extra 2 compared to 2^3. More precisely, we need to multiply 2^3 with a 2 to get to 2^4. Here are a few other examples worked out to make the point clear:

The key point to note here is that multiplying one more two (or base, in general) to a series adds 1 to the exponent. The exponentiation converts multiplication into addition at the exponent level.

Decreasing the Exponent

Now, let us start with where we left off the last time: 2^6. What do I need to do to 2^6 in order to make it 2^5? I need to do the opposite of what I did to increase the exponent, which is: divide 2^6 by 2. Here are the corresponding examples worked out:

The key point to note here is that in order to reduce the exponent by 1, we divide by the two (or base, in general). The exponentiation converts division into subtraction at the exponent level.

Getting To 2^0

Now that we’ve solved the puzzle up to the point of decreasing the exponent, let’s just keep going. What happens if I wish to reduce 2^3 to 2^2, and so on? Let’s find out.

We have now understood that 2^0 actually just divides 2 by itself. And this holds true for all non-zero numbers. The reason why any non-zero number raised to the power zero equals one is that the number is divided by itself. When we understand exponentiation from this perspective, it makes logical sense, and we need not just take things for granted. I’ll prove the same point one more time using the division rule of exponentiation.

The Division Rule of Exponentiation

The division rule of exponentiation says that whenever we have a base raised to an exponent divided by the same base raised to another exponent, we just need to subtract the exponent of the denominator base from the exponent of the numerator base. This is actually just an extension of increasing and decreasing exponents we have seen so far. Here are a few examples to clarify how this works:

Now, let us try and prove 2^0 backward. Let’s start with 1 this time. We just proved previously that 2^0 is just 2^1 divided by 2. What if we apply the division rule of exponentiation here? Let’s find out:

This pretty much explains why 2^0 leads to 1 from the other side. You could try out a few other examples, and you’ll realise that this works for any non-zero number.

The Exception

The only exception where this does not work is with zero as the base. Zero is a strange number for a variety of reasons. If you try 0^0 in your calculator, you are likely to be greeted by some form of error message. The reasons for this are beyond the scope of this article (I will cover it in a future article though). For now, it suffices for us to be aware of the fact that 0^0 is an exception to the general understanding we have that a number raised to the power zero equals one. That is x^0 = 1, except for x=0.

Update: I just realised that there is one important point that might have been ambiguous in my original article, and wanted to clarify that. For negative numbers raised to the power zero, we have a situation where the computational rule known as order of operations applies. This means that when we use a digital computer to calculate a negative number raised to the power zero, we may get (for example): -1⁰ = -1. This is because, the rule of the order of operations treats -1⁰ as -(1⁰), and resolves the exponentiation first before applying the negation. However, mathematically speaking, a negative number raised to the power zero is indeed equal to 1. To achieve this result correctly on a digital computer, one has to use parentheses to include the negative sign inside the exponentiation like this: (-1)⁰. This will result in the correct computation. Thanks to John C Hicks for pointing out my original mistake. I hope this update clarifies any potential ambiguity that might have originally been there.

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Further reading that might interest you:

What Exactly Is Zero Raised To the Power Zero?

What Really Happens When You Divide By Zero?

Does Division By Zero Really Lead To Infinity?