What are properties of addition and multiplication? Properties are statements that are true for all numbers. First up, commutative properties. We’ll start with the commutative property of addition. The commutative property of addition states that the order in which two numbers are added does not change their sum. A variable representation of that is 𝑎 plus 𝑏 is the same thing as saying 𝑏 plus 𝑎. Or four plus three is the same thing as three plus four. The order doesn’t matter here.
The commutative property can also be applied to multiplication. Look at the changes that happened on the screen. Commutative property of multiplication states that the order in which two numbers are multiplied does not change their product. In this case, 𝑎 times 𝑏 equals 𝑏 times 𝑎. An example of that is five times four equals four times five.
One way to remember the commutative property is to think about the word commute. The word commute and the word commutative relates to exchange, substitution, and interchange. Here with the commutative property, we’re specifically talking about when we change the order in which we add or multiply. And in this case, it does not change the value depending on the order.
Next up, the associative property. And again we’ll start with addition. The associative property of addition states the way three numbers are grouped when they are added does not change their sum. Here’s what that looks like with variables: 𝑎 plus 𝑏 plus 𝑐 equals 𝑎 plus 𝑏 plus 𝑐. What that means is if we add 𝑏 and 𝑐 together first and then add 𝑎, that sum is the same as if we added 𝑎 and 𝑏 together first and then added 𝑐. In this case, one plus two plus three equals one plus two plus three. One plus five equals three plus three.
This property can also be applied to multiplication. Notice the changes here. The associative property of multiplication says the way three numbers are grouped when they are multiplied does not change their product. If we multiply 𝑏 and 𝑐 first and then take that value and multiply it by 𝑎, we will have the same product as if we’ve multiplied 𝑎 and 𝑏 and then that value by 𝑐. We can remember the associative property with the word associate. Association deals with groupings. The way that three numbers are grouped when they’re added or multiplied does not change their sum or their product.
And third on our list, identity properties. Starting with the identity property of addition, this property states that the sum of an addend and zero is the addend: 𝑎 plus zero equals 𝑎. Seven plus zero is seven. As we switch over to the identity property of multiplication, let’s look carefully at all the changes. The identity property of multiplication says the product of a factor and one is that factor. Our example, 𝑎 times one equals 𝑎. Notice that the identity property of multiplication is multiplying the factor by one.
Let’s look at these properties side by side. When we add zero to any value, we’re gonna get the same value back. And when we multiply by one, the same thing happens. You can remember the identity property by thinking about looking in the mirror. 𝑎 looks the same after you add zero. 𝑎 also looks the same after you multiply it by one.
Our last property is a little bit different. The distributive property shows us how we combine addition and multiplication. The distributive property says this: To multiply a sum by a number, multiply each addend of the sum by the number outside the parentheses. That’s a lot of words. Let’s see what that looks like.
Let’s start by looking at an example that uses numbers. The property tells us to multiply a sum by a number. Here’s our sum and here’s the number. We need to multiply each addend of the sum. Four and six are the addends of the sum. And we multiply that by the number outside the parentheses, in this case, the three. If you solve both sides of the equation, you end up with thirty equals thirty. The distributive property helps us take this three and distribute it across the four and the six, the two addends of the sum.
An algebra representation of the distributive property or a representation with variables would look like this: 𝑎 times 𝑏 plus 𝑐 equals 𝑎 times 𝑏 plus 𝑎 times 𝑐. What we’re doing here is we’re taking the 𝑎 and we’re distributing it across the addend 𝑏 and the addend 𝑐. This is also true in reverse. This is also true in reverse. In this example, we wanna take the 𝑎 out, put it outside the parentheses, and add the 𝑏 and 𝑐 first. Here’s that example with numbers: five times two plus five times three is the same thing as saying five times five or five times two plus three. Here’s a chart to help you summarize all the different properties. These properties will be the foundation of solving all kinds of equations.