Numbers: How Many are Enough? Mathematics was born when people first started counting. It began with numbers. Later it came to be about more -- shapes and logic, for example -- but not at first. Your own mathematical evolution probably began with plain old numbers, too. In the Beginning... First there were the numbers you learned from watching Sesame Street, or from counting your fingers and toes: 1, 2, 3, 4, 5, and so on. When you first started to quantify the things in the world around you, these numbers sufficed. Most of the things you dealt with -- people, gumdrops, cookies -- came in these amounts. Some math books call these the counting numbers. Others call them the natural numbers. (This sounds a little more impressive, don't you think?) At first, the natural numbers seemed like a self-sufficient society. They were complete unto themselves, and for awhile you never even imagined there were other types of numbers. You could add any two natural numbers, or you could multiply any two natural numbers, and the result was always a natural number. You learned your addition and multiplication tables and never had to worry about anything "unnatural." A set of numbers is said to be closed under an operation if performing that operation on members of the set always produces a result that is also a member of that set. The set of natural numbers is closed under addition and multiplication. But with subtraction, you found that the natural numbers weren't enough. Keeping Your Integer Integrity Even when you were little, you were able to see that "2 take away 1" would equal 1. But what's the result of "2 take away 2" or "2 take away 3"? To cover the results of subtracting any two natural numbers, including such cases as "2 - 2" and "2 - 3," you needed to expand your number system a bit and admit 0 and the negative whole numbers. This expanded set of whole numbers is called the integers. Warning! This is the stage at which some people begin to lose touch with math. What are these negative numbers anyway? Aren't they just figments of some mathematician's imagination? Do they really exist? Do they have any application to real-life experience? In fact, they do. For example, they are useful in distinguishing debts from assets and in representing very cold temperatures. Perhaps some of us prefer to remain blissfully ignorant about negative numbers because of their traditional associations with such depressing phenomena as debts or cold. Maybe we could cheer ourselves up and overcome this block by thinking about happier applications, like subpar golf scores, weight loss, or cash rebates. Negative numbers are really both logical and practical. Adding and Subtracting Positives and Negatives You've known how to add, subtract, multiply, and divide pairs of positive integers since elementary school. One thing you may still be a little tentative about is applying these basic operations to integers that are not all positive. This book assumes that you know how to add and subtract positives. Here are the rules for adding and subtracting positives and negatives. To Add a Pair of Negatives: First add the number parts, and then put a minus sign in front of the result. Negative plus negative will always be negative. Question: -23 + (-41) = ? Solution: Add 23 and 41 to get 64, and then put a minus sign in front: -23 + (-41) = -64 To Add a Positive and a Negative: First ignore the signs and find the positive difference between the number parts. Then attach the sign of the original number with the larger number part. The sum is positive if the positive number "outweighs" the negative, and the sum is negative if the negative number "outweighs" the positive. Question: 23 + (-34) = ? Solution: First ignore the minus sign and find the positive difference between 23 and 34 -- that's 11. Then attach the sign of the number with the larger number part -- in this case it's the minus sign from -34. So, 23 + (-34) = -11. To Subtract Signed Numbers: Turn subtraction into addition and proceed as above. Question: -17 - (-24) = ? Solution: Subtracting a negative is the same as adding a positive, so -17 - (-24) = -17 + 24. Now it's a matter of adding a negative and a positive. The positive difference between 17 and 24 is 7, and the positive 24 outweighs the negative 17, so the answer is positive: -17 - (-24) = -17 + 24 = 7 To Add or Subtract a String of Positives and Negatives: First turn everything into addition. Then combine the positives and negatives to reduce the string to the sum of a single positive number and a single negative number. Question: 3 + (-4) - (-5) - 6 + (-7) = ? Solution: First turn the subtractions into additions: 3 + (-4) - (-5) - 6 + (-7) = 3 + (-4) + 5 + (-6) + (-7) Then combine all the positives into one positive and all the negatives into one negative to reduce the string to the sum of a single positive and a single negative: 3 + (-4) + 5 + (-6) + (-7) = (3 + 5) + [(-4) + (-6) + (-7)] = 8 + (-17) = -9 Multiplying Positives and Negatives This book assumes you know how to multiply positive integers. It's only slightly more complicated to multiply a positive and a negative. The computational part of the task is the same as when both numbers are positive; the only complication is figuring out whether to attach a minus sign to the result. Here's the general rule. To Multiply a Positive and a Negative: When multiplying two signed numbers, if the signs are different -- one positive and one negative -- the product is negative. If the signs are the same -- both positive or both negative -- the product is positive. The rules for multiplying signed numbers are not arbitrary. They make sense. Suppose, for example, you're on a weight-loss plan that takes off 2 pounds each week. Five weeks from now you will weigh 10 pounds less than you do now, and you can represent this fact arithmetically this way: 5 Y (-2) = -10 Three weeks ago you weighed 6 pounds more than you do now, and you can represent this fact arithmetically this way: (-3) Y (-2) = +6 The rules for multiplying signed numbers can be generalized, as follows. To Multiply a String of Positives and Negatives: When multiplying any number of signed numbers, attach a minus sign to the result if there is an odd number of negatives. Question: (-2) Y (-3) Y (-5) = ? Solution: First multiply the number parts: 2 Y 3 Y 5 = 30. Then go back and note that there were three negatives. That's an odd number of negatives, so the product is negative (the answer is -30). Dividing Positives and Negatives The rule for dividing is pretty much the same as for multiplying. The computational part is the same as when the numbers are positive. You just have to figure out what sign to attach. To Divide a Positive and a Negative: When dividing signed numbers, the result is negative if the original numbers have different signs, and the result is positive if they have the same sign. Question: (-12) ? (-3) = ? Solutions: First divide the number parts: 12 ? 3 = 4. Then go back and note that the original numbers had the same sign (both negative), so the result is positive: (-12) ? (-3) = +4 Understanding Rational Numbers You hit another big bump in the "I understand math" road when you start dividing integers. You find that the integers are insufficient. You can add, multiply, or subtract any two integers and the result is always an integer. But what about division? Sometimes when you divide integers you get integer results. These are cases of "divisibility." 12 ? 3 = 4 (-10) ? (-5) = 2 But if you just pick a couple of integers at random, you are more likely to come up with a case where division produces a noninteger result: 12 ? 5 = (-10) ? (-6) = These numbers may be nonintegers, but they're still special in their own way. They are members of a bigger, yet still exclusive group. A number that can be expressed as the ratio of two integers is called a rational number. All integers are rational numbers because they can b
