ࡱ> )` R&5bjbj`!#q8J V8vv$hIv6-@6c|~n8vj u~h|rXy0lrl^vv` 66BXnn by Jason Marshall Lets kick things off with a bit of a strange question: If you were asked to walk up two stairs and then immediately back down three, how would you describe the total number of stairs youd climbed relative to where you began? Do any numbers in your numerical arsenal provide a satisfactory solution to this conundrum? Have you climbed one stair? No, not really. In fact, you havent net climbed any stairs since youve actually ended up below the place where you started. Really what youve done is to descend one stair. In the same way that climbing and descending describe opposite actions, the natural numbers that we discussed in the last episode have a sort of opposite too. And these oppositescalled negative numbersare just the ticket we need to answer our stair-climbing riddle. Yes, it sounds a bit odd, but in this situation you could say youve climbed negative one stair. Just in case youre wondering what impact this information might have on your life outside staircaseswell, there are many answers, but one in particular might grab your attention: money. Want to understand the flow of cash in and out of your wallet? Well get to a real life application later in the episode. But first, in preparation, we need to get a bit negativewith numbers, that is. Answering the Last Articles Math Problem Okay, before we jump head-first into the sea of negativity, lets quickly review how we got here. How did we get herenumerically? Well, in the last episode we talked about the natural, or counting, numbers. These are the numbers that you, quite naturally, use to count and order thingsevery whole number from zero on up is a natural number that, when combined with a bit of arithmetic, can be used to solve lots of everyday problems. In fact, how did you do with the SAT-inspired library book problem from the end of the last episode? In case you dont remember it, the problem went something like: If two books are checked-out from the library every minute, and one is returned every five minutes, how many fewer books are in the library after 20 minutes? Did you calculate that there would be 36 fewer books in the library after 20 minutes? Hopefully you did, but if you didnt, heres how it works... Since 2 books are checked-out every minute, after 20 minutes, 40 books (thats 2 times 20) are checked-out; and since 1 book is checked-in every 5 minutes, after 20 minutes, 4 books (thats 20 divided by 5) are checked-in; so if 40 books are checked-out, and 4 books are checked-in, then after 20 minutes, the library has 36 (thats 40 minus 4) fewer books on its shelves. Congratulations if you got it right, and worry-not if you struggledthings will start coming together soon enough. Where Did Negative Numbers Come From? Okay, that brings us up to dateso what comes next? Well, thats exactly the question that some of the worlds earliest mathematicians began asking themselves a few thousand years ago in China. Just like the question about how you would label the number of stairs youd climbed after your quick up-and-down jaunt, our mathematical ancestors started asking themselves questions like: If I can subtract 2 from 3 (leaving 1), shouldnt I also be able to subtract 3 from 2? And if this can be done, what would the resulting number look like? Well, there arent any natural numbers that satisfy this condition, right? Right. You wont find a solution amongst them. So, the story goes, upon realizing this was a question begging for a solution, the pioneering mathematicians came up with onenegative numbers. What are Negative Numbers? In the same way that the natural numbers start at zero (Is zero really a natural number? See "Can a Math Problem Have More Than One Right Answer?") and increase forever in increments of one, the negative whole numbers start at -1 and get ever more negative as you continually subtract one: -1, -2, -3, -4, -5, and so on up to as large a negative whole number as you can think of. For example, if you subtract 3 from 2, you get -1. How about subtracting 10 from 3? Its -7, right? Yes. Start at 3 and count backward: 3 to 2, 2 to 1, 1 to 0, 0 to -1, and so on ten times in total until we finally go from -6 to the answer: -7. With this extension to the natural number system, our ancestral mathematicians were finally able to solve the problem 2 minus 3, just as they previously had been able to solve the problem 3 minus 2. A small aside: sometimes youll hear people call a negative number, such as negative seven, minus seven instead. This is fine and not incorrect, and most people will certainly know what youre talking about, but I think its better to call it negative seven so that it doesnt get confused with the idea of subtraction. And I just think it makes you sound a bit smarter too. Negative Numbers and Temperature I should mention that negative numbers play a big role in something youre already quite familiar withthe temperature scale. That will be particularly familiar to those who live in a place that gets extremely cold in the winter. But regardless, everyone should have some familiarity with the fact that sometimes, in some places, the temperature outside can be below zeromeaning that its described with a negative number. But what does that mean? Well, when the Swedish astronomer Anders Celsius defined the temperature scale now bearing his name back in the 1740s, he defined zero degrees to be the temperature at which water freezes. But its entirely possible for ice to be colder than zero degrees. Thats right, some ice is indeed colder than others. So whats the temperature of ice thats colder than the point at which it freezes? It must be negative. How Were Negative Numbers First Used? Continuing with the negativity, in 7th century India, negative numbers were first used to represent debtsa practice that made its way within a few hundred years to the Islamic world, and then to Europeand eventually, after several additional centuries, its an all-too-integral part of our modern financial world. For example, heres a financial problem (in more ways than one) you may have encountered as a student. Youre completely broke and you need to buy books. Being a resourceful individual, you decide to have a garage sale in an attempt to rectify the situation. You make $50 at your sale, but you remember that you owe three friends $20 each. Easy come, easy goyou pay two friends the full amount you owe them, and you pay your unlucky last friend the remaining $10 you havewhich is, of course, only half of what you actually owe. So, whats youre net financial worth at this point? Well, you have $0 in your pocket. But no, thats not your net worth because you still owe your buddy $10. Thats right, the situation is even worse than you thought. Since you owe a debt of $10, your net worth is actually negative $10. What are Integers? Okay, lets take a moment to reflect upon where were at. Whether youve realized it or not, youve now been formally introduced to all the members of the very important group of numbers known as the integers. The integers are the group of numbers consisting of the natural numbers: 0, 1, 2, 3, and so on, and their negative counterparts: -1, -2, -3, etc. Imagine a big line extending to your left and right with equally spaced tick marks and the number 0 positioned squarely on the mark directly in front of you. Thats the number line youre imaginatively looking at, and the numbers at each tick mark to your right and left represent the positive and negative integers, respectively. Moving tick by tick along the line to the right of zero is analogous to counting up the positive integers in increments of one, while moving along the line to the left of zero is akin to counting backwards towards ever larger negative integers. Wrap Up Alright, I think thats enough for now. Well talk a lot more about the number line in the next few episodes, and well use it to help make arithmetic with integers easier. In particular, well talk about how to add, subtract, multiply, and divide positive and negative integersall while keeping the signs straight! And heres the kickerwere going to learn how to do thisget ready for itwithout relying on a calculator. Trust me, itll actually make things easier. So check out the next article to learn how to start kicking your calculator dependency. But until then, here are a couple of problems dealing with integers for you to think about. First: Are there any integers that are neither positive or negative? If so, how many are there? And second: Put the following four integers in order from smallest to greatest101, -1, 32, and -2010. Give these problems a shot and check out the next article to see if you get the right answers. Alright, that s all for now. Please email your questions and comments to000000follow the Math Dude on Twitter at000000and become a fan on Facebook. 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