Why is 3 0 equal to 1

There are lots of different ways to think about it, but here's one: let's go back and think about what a power means. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all!

What does that mean? Well, let's go even farther back to the simplest case: addition. What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same.

So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number.

So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1. It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always well almost always good reason for them. Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google.

Okay, enough, onto your question:. Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'. For example, 2 3 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1.

Well that's gonna be one. Let's get even closer to zero: 0. Well, that also equals one. Let's get super close to zero: 0. Well once again, that also equals one. And it didn't even matter whether these were positive or negative. I could make these negative and I'd still get the same result.

Negative this thing divided by negative this thing still gets me to one. So based on this logic you might say, "Hey, well this seems like a pretty reasonable argument for zero divided by zero to be defined as being equal to one. Zero divided by 0. A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it, which equals one.

In general, the factorial of a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. It is pretty clear from these examples how to calculate the factorial of any whole number greater than or equal to one , but why is the value of zero factorial one despite the mathematical rule that anything multiplied by zero is equal to zero?

The definition of the factorial states that 0! This typically confuses people the first time that they see this equation, but we will see in the below examples why this makes sense when you look at the definition, permutations of, and formulas for the zero factorial. The first reason why zero factorial is equal to one is that this is what the definition says it should be, which is a mathematically correct explanation if a somewhat unsatisfying one.

Still, one must remember that the definition of a factorial is the product of all integers equal to or less in value to the original number—in other words, a factorial is the number of combinations possible with numbers less than or equal to that number. Because zero has no numbers less than it but is still in and of itself a number, there is but one possible combination of how that data set can be arranged: it cannot.

This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! For a better understanding of how this makes sense mathematically, it's important to note that factorials like these are used to determine possible orders of information in a sequence, also known as permutations, which can be useful in understanding that even though there are no values in an empty or zero set, there is still one way that set is arranged.

A permutation is a specific, unique order of elements in a set. We could also state this fact through the equation 3! In a similar way, there are 4! So an alternate way to think about the factorial is to let n be a natural number and say that n!

This corresponds to 2! This brings us to zero factorial. The set with zero elements is called the empty set. Even though there is nothing to put in an order, there is one way to do this. Thus we have 0!