Properties
When you work with real numbers, you do so according to a set of rules that mathematicians often describe as properties. The properties of a number system pertain to how the numbers behave when you carry out operations using them. Figure 2.6 illustrates the basic properties of the real numbers. These properties apply to all the number systems you have dealt with so far. However, not all properties are applicable to all number systems. The next few sections review these properties in detail.
Figure 2.6. Six groups of properties allow you to work with the real and other number systems.
Commutative Property
When you work with two numbers, a and b, the order in which you add or multiply them is commutative. The commutative property pertains to addition and multiplication. Consider these equations:
a + b = c 3 + 5 = 8 a × b = c 3 × 4 = 12
b + a = c 5 + 3 = 8 b × a = c 4 × 3 = 12
If you start with 3 and then add 5 to it, you obtain the same result as when you start with 5 and add 3. If you multiply 3 by 4, you obtain the same result as when you multiply 4 by 3. The commutative property allows you to change the order in which you carry out an addition or multiplication operation without changing the outcome of the operation.
Associative Property
When you work with three or more numbers, a, b, and c, you can group them in different ways as you perform the addition or multiplication operations that join them together. How you associate them depends on the type of operations you perform with them. The associative property applies to addition and multiplication. Consider these equations:
a + (b + c) = (a + b) + c
3 + (5 + 7) = (3 + 5) + 7
3 + (12) = (8)+ 7
15 = 15
When you add 3 to the sum of 5 and 7, you obtain the same result as when you add the sum of 3 and 5 to 7. The difference is in the order of operations. The associative property allows you to alter the way that you group items in an expression as you reduce or solve the expression.
Distributive Property
Distribution allows you to reorganize the terms of an expression so that you can more easily work with them. This property applies to addition and multiplication. Consider the following equations:
b (a − c) = b (a) − b (c)
5(8 − 3) = 5(8) − 5(3)
5(5) = 40 − 15
25 = 25
In this instance, you can solve the problem in two different ways. What the distributive property enables is for you to distribute the multiplication activity so that you multiply 8 by 5 and then 3 by 5. Alternatively, you could just as easily subtract 3 from 8 and multiply the result by 5.
Now consider the following expression:
3(x + 2)
In this expression, note that you may not just add 2 to x because you do not know the value of x. However, you may distribute the 3. Here is how you would apply the distributive property:
3(x + 2)
= 3x + 3(2)
= 3x + 6
Exercise Set 2.2Here are a few equations that relate to commutative, associative, and distributive properties. For each equation, identify the property or properties that best explain the operations.
Use the associative property to write an expression equivalent to each of the following:
Use the commutative property to write an expression equivalent to each of the following:
Use the distributive property to factor each of the following. Check by multiplying.
|
Identity and Inverse Properties
When working with real numbers, there are properties that pertain specifically to the addition and multiplication of 0 and 1. These properties are called identity and inverse properties.
Additive Identity
When you add zero to a number, the outcome is the number itself. Zero in this case is called the additive identity, such that:
a + 0 = 0 + a = a
Multiplicative Identity
When you multiply a number by 1, you get the number itself. 1 in this case is called the multiplicative identity, such that:
a × 1 = 1 × a = a
Consider the following equations:
1 × a = a −1 × a = −a
If you multiply a by 1, you get a. If you multiply a by −1, you still get a, but the sign of a is now changed to match the sign of 1.
Additive Inverse
For a given number a, if you add a value to a that is equal to and opposite in value to a, then the result is zero. The number you add to a is called the additive inverse, or negative, of a such that:
a + (−a) = (−a) + a = 0
The additive inverse of a is −a. The additive inverse of −a is a.
Multiplicative Inverse
A number multiplied by its inverse is 1. The inverse of a number is the fraction (or ratio) by which you can multiply the number to create the value of 1. For a given number a, as long as a ≠ 0, there is a number 
, called the multiplicative inverse, or reciprocal of a, such that:
a · (1/a) = (1/a) · a = 1
Consider the following equations:
The multiplicative inverse of a is 
. The multiplicative inverse of 5 is 
. The multiplicative inverse of 4000000 is 
. The last example proves a little more involved than the first three. Consider that you can rewrite the equation as 
. Inverses prove useful as ways to reduce the complexity of problems.
Multiplicative Property of Zero
When you multiply a number by zero, the result is zero.
a × 0 = 0 × a = 0
Division Involving Zero
While division by zero is undefined, if you divide zero by any number, the result is zero. The rules for division involving zero are
Exercise Set 2.3Here are a few problems that involve working with additive and multiplicative identities, inverses, and the zero properties. Solve each problem.
Change the sign (find the opposite or additive inverse) of each number:
Find −(-x) when x is each of the following:
|
