Section 3: Real Number Arithmetic
Our discussion of real number arithmetic will build off the Additive Inverse Property introduced above. The additive inverse of any number is the number you add to something to get 0. If we place a number on a number line, the additive inverse is the number on the opposite side of 0 from the original number. For any number a, the additive inverse (or just "inverse") is labeled -a. Because of their relative positions on the number line, additive inverses are sometimes called"opposites" and the -sign can be thought of as the "opposite sign" .
Closely related to additive inverses is the concept of absolute value.
Definition 9 The absolute value of a number is the number's distance from 0 on the number line. The symbol for absolute value is | | . Thus we have the following: For any real number x, |x| = -x if x is non-negative and |x| = -x (the inverse of x) if x is negative.
Thus, the outcome of an absolute value will always be a positive number. For example, both 5 and -5 are the same distance from 0 so both have an absolute value of 5.
3.1 Addition of Real Numbers
Rules for Addition of Real Numbers
1. The sum of any two positive numbers is positive.
2. The sum of any two negative numbers is negative. Simply add the absolute values and take the additive inverse.
3. To add a positive and negative number with different absolute values, find the difference between the absolute values. Then take the sign of the number with the largest absolute value.
3.2 Subtraction of Real Numbers
With the above rules, most people can handle addition with positive and negative numbers. On the other hand, subtraction with a mixture of positive and negative numbers throws many people off. To get around this, we re-define subtraction in terms of addition.
Definition 10 Subtraction: The difference m - n (or "subtracting n from m") is the same as the sum of m and the additive inverse of n or m + (-n).
Once you've made the translation from subtraction notation to addition notation, then all the addition rules can be used to evaluate the expression. Admittedly, these rules iuclude a circular definition. Rule three for addition says to find a difference (i.e. subtract) and I've just told you that subtraction is really addition of opposites. Don't let this bother you. In the case covered by rule three above, you're finding a difference of two positive numbers because of the absolute values. That isn't difficult to do, especially with a calculator. The redefinition of subtraction is a tool to be used when the mixture of signs confuses you.
Example 3 5 - 4 = 5 + (-4) = 1
Example 4 5 - (-4) = 5+(-(-4)) = 5 + 4 = 9
Example 5 -5 - 4 = -5 + (-4) = -9
Example 6 -5 - (-4) = -5 + (-(-4)) = -5 + 4 = -1
The second example shows us another property of real numbers: the opposite of the opposite of any number is itself (or the additive inverse of the additive inverse). This is commonly called a double negative. To put this in terms of the subtraction definition, if n was negative, then -n is positive.
Multiplication and Division of Real Numbers
Rules for Multiplying and Dividing Real Numbers:
1 The product or quotient of any two positive numbers is positive.
2 The product or quotient of any two negative numbers is positive.
3 The product or quotient of any positive number and any negative number is negative. The order of the signs has no impact on the sign of the outcome.
Most people find it easier to keep track of signs when multiplying and dividing than when adding and subtracting. To state it simply: if the signs match the answer is positive, if the signs differ the answer is negative. For these rules, multiplication and division were combined. The reason is that division can be defined as multiplication by the reciprocal (just like subtraction was defined as addition of the opposite).
Definition 11 For any Real numbers, p and q, dividing p by q, or p ÷ q, is the same as multiplying p by the reciprocal of q. Symbolically, p ÷ q = p × 1/q = p/q.