Properties of Operations
The human brain, which is capable of incredibly complex thought processes, is a very simple organ when it comes to computation. Like a computer, the brain can deal with only two numbers at a time. The speed with which some people—and some computers—can do computations can be very impressive, but fundamentally, the brain is fairly simple. Do the following problem by speaking aloud as you work through the process of getting the answer.
Multiply: 2 × 3 × 4
Did you say, “2 × 3 is 6 and 6 × 4 is 24”? Or maybe you said, “4 × 3 is 12 and 12 × 2 is 24.” Then there is the rare person who will say, “2 × 4 is 8 and 8 × 3 is 24.” But in each case, do you see how the brain handles just two numbers at a time? Did you also notice that different approaches can lead to the same answer?
Addition and multiplication are much more flexible than subtraction and division in allowing for attaining answers using different approaches. For example, 2 + 3 gives the same answer as 3 + 2. In the same way, 2 × 3 = 3 × 2. The commutative property, which means that the operation can be performed in either order, exists for addition and for multiplication, but not for subtraction or for division. (Note that 2 − 3 is not the same as 3 − 2 and that 2 ÷ 3 does not equal 3 ÷ 2.) When operating with more than two numbers, we must decide which two to begin with. In mathematics, we often use grouping symbols to show the reader how a problem is done, and the most common grouping symbols are parentheses. Did you say 2 × 3 first or 3 × 4 first when you did the multiplication problem earlier? As we have seen, with multiplication we have some flexibility when determining the answer. Using grouping symbols, we see that (2 × 3) × 4 = 2 × (3 × 4), with the problem in parentheses computed first. The associative property for addition and for multiplication states that the manner in which we group the initial numbers to be computed does not change the outcome. Verify for yourself that the associative property does not work for subtraction or division by computing the values of (12 − 5) − 6 and 12 − (5 − 6) and then the values of (40 ÷ 4) ÷ 5 and 40 ÷ (4 ÷ 5).
The identity element for addition and subtraction is 0. Add 0 to any number or subtract 0 from any number, and the result is identical to the number with which you started. The identity element for multiplication and division is 1. The identity elements will prove crucial when you begin to solve equations. Getting back to the identities is also very important. Which number, when added to 6, gives the identity additive 0? Which number, when added to −7, gives the identity additive 0? You know that the answers are −6 and 7, respectively. Thus we say that the additive inverse of 6 is −6 and the additive inverse of 7 is −7.
You may have called additive inverse numbers opposites in the past, but this may lead to confusion because the word opposite can have a variety of interpretations. From now on, you should use the term additive inverse so that you will become accustomed to it.
What number, when multiplied by 6, gives the multiplicative identity 1? What number, when multiplied by −7, gives the multiplicative identity 1? The answers are 1/6 and −1/7, respectively. We say that multiplicative inverse of 6 is 1/6 and the multiplicative inverse of −7 is −1/7. In the past you may have called these numbers reciprocals. Because the word reciprocal has only this one meaning in mathematics, you may continue to use it, but be sure you understand that it has the same meaning as multiplicative inverse.
Distributive Operation of Multiplication over Division
The last of the properties to be examined in this section is the distributive operation of multiplication over division. How is the distributive property used? Compute 6 (7 + 3). According to the order of operations, do the work in the parentheses first: 7 + 3 = 10. You next do the problem 6 × 10 = 60. Note, however, that (6 × 7) + (6 × 3) = 42 + 18 = 60 is equivalent to the original problem.
Although this is frequently called the distributive property, be sure to remember that it is multiplication that is being distributed over addition.
An important example of the distributive property is the problem −6(-8 + 8). When we add first, the problem becomes −6(0) = 0. When we apply the distributive property, the problem becomes (-6)(-8) + (-6)(8). (-6)(-8) + (-48) must equal 0. Thus (-6)(-8) must equal 48, which reinforces the rule that the product of two negative numbers is a positive number.