Hello class and welcome to today's algebra lesson which is about the properties of real numbers. By the end of today's lesson you will be able to recognize and apply the properties that we go over. The vocab for today is our big list of properties. We're going to start with the reflexive property. This is saying that two things are exactly equal.
So for example, five is equal to five or it could be a math. problem. So it could say 3 plus 4 is equal to 3 plus 4. The next property is the symmetric property. This property allows you to change what is on the left side and the right side of the equal sign.
So for example, if I have 3 plus 4 is equal to 7, I can flip the left and the right sides to say that 7 is equal to 3 plus 4. So we simply switched what was on the left side of the equal sign with what was on the right side of the equal sign. Our next property is the transitive property. This allows more than one thing to be equal to the same thing.
So for example, I can say, starting off again, that 3 plus 4 is equal to 7. I can also say that 7 is equal to 2 plus 5. Now what I can do is take out the middle piece and I can conclude that 3 plus 4 is the same as 2 plus 5. So I eliminated that 7 from the middle and just said the two adding expressions are equal to each other. The transitive property works with any number of things so I also could have said 3 plus 4 is equal to 7, 7 is equal to 2 plus 5, and 2 plus 5 is equal to 1 plus 6 to conclude that 3 plus 4 equals 1 plus 6. So it can keep going as long as the middle pieces are always equal. The next property is called the commutative property. This is what allows you to switch the order of the numbers when you are adding or multiplying. So for example, the commutative property says that 3 plus 4 is the same thing as 4 plus 3. Also applies to multiplication.
3 times 4 is the same thing as 4 times 3. This property does not apply to subtracting or dividing. 3 minus 4 is negative 1. 4 minus 3 is positive 1. So, commutative property works for adding and multiplying. The next property, the associative property, allows me to change the groups when we're doing adding or subtracting.
So, for example, if I have 2 plus parentheses 3 plus 4, that is the same thing as putting the parentheses around the 2 and the 3 and then adding 4. We can double check this using PEMDAS. 3 plus 4 is 7, plus 2 is 9. If I solve the other one, I have 2 plus 3 is 5, plus 4 is 9. Again, this also works for multiplication. So if I have 2 times 3 times 4, that's going to give me 12 times 2, which is 24. Or I can change the parentheses to the first part and do 2 times 3, which is 6. then multiplying by 4, which still gives me 24. So, commutative and associative properties apply for addition and multiplication, but not for adding and subtracting. There are a few more specific properties for adding and multiplication.
The first one is the additive identity. That is saying that any number plus 0 is going to give you that number back. So zero, adding zero is the additive identity to keep our numbers the same.
The additive inverse is saying the number plus the opposite of the number or the negative of that number is going to give you zero. You can also see this as the number subtracting itself is equal to zero. So you can look at it either as...
adding the opposite or as doing the opposite operation. The identity of multiplication is similar to the identity of addition, but we're using a different value now. So the number times one is equal to itself.
So one is the multiplicative identity of numbers. When it comes to inverses for multiplication, that's where we have the fraction. So 4 times 1 fourth is equal to 1. Again, you can think of that as the reverse operation. 4 divided by 4. is going to give you one. Multiplication also has one more bonus property, and that is the zero property, which says that four times zero, or excuse me, any number times zero is going to give you a quantity of zero.
So this is a summary chart of all of the properties that we are going to be applying throughout the rest of this lesson. First, we wanna go through and find the missing value and then name the property used. So we have 2 plus 8 is equal to 8 plus something. First thing we need to do is figure out what that value is. In this case, it is going to be 2. Then you're going to want to refer back to your property list and see which one we are using.
We have the same numbers in a different order, so that's going to be our commutative property. That allows us to change the order when we are... adding or multiplying. For our next example here, it says if 3 plus 8 is equal to 11 and 4 plus x is equal to 11, then 3 plus 8 is equal to 4 plus x.
First thing I need to figure out, 4 plus something is 11. That something is going to be 7. Now I have two expressions that are equal to the same quantities and I eliminate that. quantity from the expression so that is going to be my transitive property. Go ahead and try these two on your own.
The first one we have a value of seven. It's the reflexive property because it is the same numbers in the same order on both sides of the equal sign. Next one we have the value is two and that is our associative property. Same numbers in same order, but we changed the grouping symbols. We changed those parentheses.
Now we're going to use the properties to help us evaluate an expression. So we're going to go through solving with PEMDAS and name any properties that we use. So according to PEMDAS, my first step here should be to solve the parentheses. 4 minus 3 is 1. And then I'm going to write out the rest of my expression. As I solved that step, if you refer back to your properties list, 4 minus 3 is not a specific property, so we're just going to call that simplifying.
So anytime you are not using a specific property, you can just call it simplifying. The next thing I'm going to do, according to PEMDAS, is 7 times 1. That's going to give me 7. In that step, I am multiplying by 1. 1 which is my multiplicative identity. My next step according to PEMDAS is going to be to do 5 times 1 fifth and when I do that I get an answer of 1. That is the multiplicative inverse, a number being multiplied by its inverse fraction. And then my final step here would be to do 7 plus 1. That's 8. there is no specific property for adding one.
So that again is just going to be simplifying. Go ahead and fill in the properties used to get from here to here for property one and here to here for property two. First one is our additive inverse, eight minus eight, a number minus its up. or excuse me, a number and its opposite being subtracted.
And then we have adding one, or excuse me, not adding one, multiplying by one with the one times three. And that's your multiplicative identity. If you have questions about this or anything else from the lesson, please feel free to reach out and let me know.