The next two properties that we'll look at are the identity and inverse properties. These two properties also apply to all real numbers, whether they're positive negative decimals, fractions, these properties will work. So first we have the identity properties. The identity property for addition tells us that the sum of any number and zero will equal that number. So basically, if we add zero to any number, we'll still get that number as the answer. For example, if we add five plus zero, it will still just equal five. Now, we can represent that as a rule using the variable A instead of just using the number five. So any number A plus zero will equal that number. The identity property for multiplication tells us that the product of any number and one will equal that number. So we can multiply any number times one and we'll still get that number as the answer. For example, three times one will equal three. And we can also represent that as a rule using the variable A. So any number, any real number a times one will equal that same number that we start with. Next we have the inverse properties. The inverse property for addition tells us that the sum of any number and its opposite equals zero. So what do we mean by the opposite of a number? The opposite of a number just means that we're changing the sign of the number. So if we have the number positive five, the opposite of that will be negative five. Or if we start with negative five, the opposite of that will be positive five. We basically just take that number and change the sign of it. So if we do add positive five plus its opposite, which is negative five, we'll get zero as our answer. We can represent that as a rule using the variable A as well. So positive A plus negative A will give us a sum of zero. Just a reminder - to find the opposite of a number, you just change its sign. Now, another term that we can use, instead of saying the opposite of a number, we can also call it the additive inverse. So if you ever see that phrase additive inverse, it really is just saying that it's the opposite of the number. Next, we have the inverse property for multiplication. And this one says that the product of any number and its reciprocal is one. So to understand what a reciprocal is, let's look at this example. We start off with the number four. We have the whole number four and then we're going to multiply it by its reciprocal, which is the fraction one over four, one-fourth. Now, if you look at the four and the fraction one-fourth, they do have some things in common. The one-fourth basically just took the number four and put it at the bottom as the denominator. And then we just put a one as the numerator. If we multiply four times its reciprocal one over four, we'll get one as our answer. If we write that as a rule, we can say that A times one over A will equal one. And that works for any real number. So here are the steps for finding the reciprocal of any number. First, you want to take that number and write it as the denominator under the fraction line, and then just put a one as the numerator. Now, if the number is already a fraction, you would just flip the fraction. So for example, if you start off with a fraction two over three, you flip it and it becomes three over two. Now, the reciprocal can also be called the multiplicative inverse. If you see that term, it's just a synonym for reciprocal. So just to review, the identity and inverse properties apply to all real numbers. The identity properties basically just tell us what we can do to a number so that the answer is still just equal to that number. So if we add zero to that number, the answer is just that number. And if we multiply the number by one, the answer will be that number. The inverse property for addition tells us what we can do to a number to get zero as the answer. So if we start off with a number and add its opposite, we'll get zero. And remember, to find the opposite, we just change the sign. If it's positive, it'll become negative. If it starts as negative, the opposite will be the positive of that number. And then the inverse property for multiplication tells us what we can multiply the number by so that we get one as the answer. And if we multiply by the reciprocal, as we just learned, we will get one as the answer. And the reciprocal is just the number becoming the denominator of the fraction with a one as the numerator. Or you can also think of it as just flipping the fraction. All right, so now you know how to apply the identity and inverse properties to real numbers.