0 why is it 1

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! Another reason for the definition of 0! This does not explain why zero factorial is one, but it does show why setting 0!

A combination is a grouping of elements of a set without regard for order. No matter how we arrange these elements, we end up with the same combination. There are other reasons why the definition of 0! The overall idea in mathematics is that when new ideas and definitions are constructed, they remain consistent with other mathematics, and this is exactly what we see in the definition of zero factorial is equal to one.

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Practice: Signs of expressions challenge. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples.

So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them. So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8.

So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1. So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times. So this is times negative 2 times negative 2 times negative 2.

So clearly these are the same number. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8. And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power. So let's think about that a little bit. What is positive 2 to the-- based on this definition-- to the 0 power going to be equal to?

Well, we just said. This says how many times are going to multiply 1 times this number? So this literally says, I'm going to take a 1, and I'm going to multiply by 2 zero times. Well, if I want to multiply it by 2 zero times, that means I'm just left with the 1.

So 2 to the zero power is going to be equal to 1. And, actually, any non-zero number to the 0 power is 1 by that same rationale. And I'll make another video that will also give a little bit more intuition on there.

That might seem very counterintuitive, but it's based on one way of thinking about it is thinking of an exponent as this. And this will also make sense if we start thinking of what 2 to the first power is. So let's go to this definition we just gave of the exponent.