When you manipulate algebraic expressions and equations you will need to understand the underlying properties of real numbers; these include the Commutative Properties, Associative Properties, Distributive Property, and Reflexive Property. This appendix provides more indepth discussion of these properties than is provided in the body of this tutorial.
The Commutative Properties
These properties tell us that the order of numbers does not matter when performing addition or multiplication. The word commutative comes from the word commute, which means to move around, exchange, and change order.
First, assume that a and b are real numbers.
1. a + b = b + a Commutative property of addition
|For example: If you were to add 3 and 5, you would get 8. Likewise, if you add 5 and 3, you will also get 8.||
5 + 3 = 3 + 5
8 = 8
If you were to add the variable x and 5, you would express
this as x + 5. If you add 5 and x, you will get
5 + x. By the commutative property we know that the order of terms can be reversed.
|x + 5= 5 + x|
2. a b = b a Commutative property of multiplication
For example, multiplying 4 by 12 yields 48, as does
multiplying 12 by 4.
(12)(4) = (4)(12)
48 = 48
|With algebraic terms, for example, multiplying x by 12z can be written as x12z. We also know by the commutative property of multiplication that the order of terms can be reversed as shown on the right.||x 12z = 12z x|
NOTE: the commutative property does not hold for subtraction or division. For example:
|Compare:||53 = 2||and||35 = 2||2 ¹ 2|
|Compare:||1÷2 = 1/2||and||2÷1 = 2/1 = 2||1/2 ¹ 2|
The associative properties also hold only for addition and multiplication. The associative properties demonstrate that the order in which terms are grouped does not matter, as long as the order in which the terms appear is not changed. The word associative is derived from associate, which means to join together, connect, combine, or unite.
First, assume that a, b, and c are real numbers.
1. a + (b + c) = (a + b) + c Associative property of addition
|For example, if you add 7 to the sum of 2 and 10 you will obtain 19. You will get the same answer if you first add 7 and 2 together, and then add 10.||
7 + (2+10) = 7 + 12 = 19
(7 + 2) + 10 = 9 + 10 = 19
|The same applies when we add algebraic terms, as shown on the right.||
(2x + 3x) + 4x = 5x + 4x =
2x + (3x + 4x) = 2x + 7x = 9x
2. a (b c) = (a b) c Associative property of multiplication
|There is also an associative property of multiplication. It is similar in principle to the associative property of addition. For example, if we multiply 4 by 3, then multiply the result by 5, we get 60. We get the same result when we multiply 5 by 4, then multiply the result by 3.||
5 (4 3) = 512 = 60
(54) 3 = 20 3 = 60
|The same principles apply in multiplication of algebraic terms. For example, if we multiply 5 by the product of 4x by 3z, we get the same result if we multiply 5 by 4x, then multiply the result by 3z.||
5 (4x 3z) = 512xz = 60xz
(54x) 3z = 20x 3z = 60xz
NOTE: The associative properties do not apply to subtraction or division. Observe what happens when we try to apply this property to both a division and subtraction problem.
|Compare:||(4÷2) ÷ 3 = 2 ÷ 3 = 2/3||and||
4 ÷ (2 ÷ 3) = 4 ÷ 2/3 = 12/2 = 6
2/3 ¹ 6
|Compare:||(5 1) 3 = 4 3 = 1||and||
5 (1 3) = 5 ( 2) = 7
1 ¹ 7
As you can see in both these examples, the result for each is different. These examples demonstrate why the associative property cannot be applied to division and subtraction problems.
This property tells us that we may distribute, or apply by multiplication, a term outside the parentheses to each term within the parentheses. First, assume that a, b, and c are real numbers.
For example, if we multiply 3 by the sum of 5 and 2 we get the same result as if we multiply 3 by 5 then add this to the result of multiplying 3 by 2.
3(5 + 2) = 3 7 = 21
3 5 + 3 2 = 15 + 6 = 21
|If we multiply 3 by the sum of x and 2, we get the same result when we multiply 3 by x then add this to the result of multiplying 3 by 2.||
3(x + 2) = 3x + 6
3 x + 3 2 = 3x + 6
In essence, the process of distributing a term from outside a set of parentheses to terms inside the parentheses is the opposite of factoring out a common term.
This is demonstrated on the right with equations that contain only numerical terms.
3 7 + 3 13 is equal to 3 (7 + 13)
3 (7 + 13) = 3 (20) = 60
|If we are given an algebraic equation, this is applied as shown on the right. Note that in the second expression, the common factor of 4 is taken out.||4x + 4y is equal to 4(x + y)|
When performing the distributive property, we will be putting a common term back in, rather than taking it out. You may wonder why we would ever want to do this. The answer is that often it is easier to perform two smaller multiplications and then take the sum, than it is to perform one large multiplication.
For example, suppose you are asked to multiply 13 by 14. Instead of reaching for the calculator, you can use mental math to multiply 13 (10 + 4). Note that this is still 13 by 14.
|13 10 + 13 4 = 130 + 52 = 182.|
|When we have subtraction within the parentheses, the property still holds. For example, suppose you are buying six notebooks for $1.97 each. The total price is then 6(1.97). A simpler way to compute this may be to rewrite 1.97 as (2.00 .03). Now find the total price by using the distributive property.||
T = 6 (1.97) = 6 (2.00 .03) =
6 (2.00) 6 (.03) = 12.00 .18 = $11.82
When working with algebraic expressions, factoring may often be more useful in simplifying complex expressions than distributing the common factor to all terms. For example, if we are given the expression 3x + 3y)/3, this expression can be simplified using factoring.
|In this case, the 3 on the top and bottom of the fraction cancel each other out. (If you feel you should review how to simplify fractions, review.)||
This property shows us if two expressions are set equal to one another, it does not matter in which order they are presented. First, assume that a and b represent real numbers or variables.
For example, if we are given that 5 is equal to x, then it is also true that x is equal to 5.
|If x = 5, then 5 = x|
|This also holds for any algebraic equation.||if 2x +4z = y, then y = 2x +4z|
This property is commonly used when working with algebraic equations. For example, if we are adding two equations x + 2z = 23 and 16 = x + 5z, we write these as:
But it would be more advantageous to write these as:
The reflexive property shows us that these two representations are equivalent.