While studying algebra, one will come upon certain properties early on which will be used throughout the learning experience. These properties so valuable that they help to shape the learners understanding, and their ability to perform the necessary problems as needed. Some of these very early and basic properties shall be discussed here in order to establish material for study and gain a better understanding of both the properties, and algebra as a whole.
Properties of Real Numbers
This first section deals with the properties of real numbers. Of course, if you don’t know what a real number is, we’ll also review the definition.
Real numbers are all the numbers which correspond to a specific point on a number line. This means that real numbers can be positive or negative, and zero is a real number as well.
Commutative Properties: these properties exhibit the properties of order of numbers. Meaning, the order of addition and multiplication of two different numbers doesn’t matter since the resulting product or sum will remain the same. For example: 5 x 3 = 15, 3 x 5 = 15. The same is true for addition. 3 + 5 = 8, and 5 + 3 = 8. Order does not change the product.
A formula can also be used to show this:
X + Y = Y + X
A x B = B x A
Now, does the commutative property apply to subtraction, and why or why not?
Of course the answer is no, and the reason is because changing around a subtraction problem will result in a different product. For example: 7 – 4 = 3, while 4 – 7 = -3. 3 and -3 are on opposite ends of the number line and are not equal terms, it is important to remember this.
Associative Properties: Again the associative properties deal with addition and multiplication ( see a pattern here? This is one good thing to remember about these basic properties of algebra). When dealing with addition, the associative property states that the way in which three numbers are grouped in a problem does not change the resulting sum or product. This can be illustrated by the formula: (x + y) + z = x + (y + z). You’ll notice here the order of addition has change, however, the product does not. Let’s look at this with real numbers.
1 + (2 + 3) = 2 + (3 + 1)
6 = 6
The same is true for multiplication as per the associative property of multiplication. This is illustrated by the following formula: (a x b) x c = a x (b x c).
Now, with real numbers:
1 x (2 x 3) = 2 x (3 x 1)
6 = 6
Distributive Property: Multiplication over Addition
This property starts into solving equations. As you move into algebra expressions such as the following will become common: 5(3+3). What exactly do you do first? Well, those versed in the distributive property know that multiplication comes first, hence the title “multiplication over addition.” So in solving this problem we first multiply 5 x 3, and 5 x 3.
15 + 15
This can be illustrated by a simplified formula: x(z + y) = xz + xy
The same is true for expressions larger than two numbers as well.
x (z + y + w) = xz + xy + xw
and so on…