數(shù)學(xué)英語(yǔ) 20 What are Negative Numbers and Integers?
by Jason Marshall
Let’s 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 you’d 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 haven’t net climbed any stairs since you’ve actually ended up below the place where you started. Really what you’ve 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 opposites—called negative numbers—are just the ticket we need to answer our stair-climbing riddle. Yes, it sounds a bit odd, but in this situation you could say you’ve climbed negative one stair. Just in case you’re wondering what impact this information might have on your life outside staircases…well, 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? We’ll get to a real life application later in the episode. But first, in preparation, we need to get a bit negative—with numbers, that is.
Answering the Last Article’s Math Problem
Okay, before we jump head-first into the sea of negativity, let’s quickly review how we got here. How did we get here…numerically? 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 things—every 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 don’t 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 didn’t, here’s how it works... Since 2 books are checked-out every minute, after 20 minutes, 40 books (that’s 2 times 20) are checked-out; and since 1 book is checked-in every 5 minutes, after 20 minutes, 4 books (that’s 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 (that’s 40 minus 4) fewer books on its shelves.
Congratulations if you got it right, and worry-not if you struggled—things will start coming together soon enough.
Where Did Negative Numbers Come From?
Okay, that brings us up to date—so what comes next? Well, that’s exactly the question that some of the world’s 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 you’d 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), shouldn’t I also be able to subtract 3 from 2? And if this can be done, what would the resulting number look like? Well, there aren’t any natural numbers that satisfy this condition, right? Right. You won’t 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 one—negative 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? It’s -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 you’ll 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 you’re talking about, but I think it’s better to call it “negative seven” so that it doesn’t 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 you’re already quite familiar with—the 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 zero”—meaning that it’s 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 it’s entirely possible for ice to be colder than zero degrees. That’s right, some ice is indeed colder than others. So what’s the temperature of ice that’s 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 debts—a practice that made its way within a few hundred years to the Islamic world, and then to Europe…and eventually, after several additional centuries, it’s an all-too-integral part of our modern financial world. For example, here’s a financial problem (in more ways than one) you may have encountered as a student. You’re 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 go—you pay two friends the full amount you owe them, and you pay your unlucky last friend the remaining $10 you have—which is, of course, only half of what you actually owe. So, what’s you’re net financial worth at this point? Well, you have $0 in your pocket. But no, that’s not your net worth because you still owe your buddy $10. That’s 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, let’s take a moment to reflect upon where we’re at. Whether you’ve realized it or not, you’ve 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. That’s the number line you’re 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 that’s enough for now. We’ll talk a lot more about the number line in the next few episodes, and we’ll use it to help make arithmetic with integers easier. In particular, we’ll talk about how to add, subtract, multiply, and divide positive and negative integers—all while keeping the signs straight! And here’s the kicker—we’re going to learn how to do this…get ready for it…without relying on a calculator. Trust me, it’ll 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 greatest—101, -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 to。。。。。。follow the Math Dude on Twitter at。。。。。。and become a fan on Facebook. You can also follow me, your humble host, on Twitter at。。。。。。
Let’s 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 you’d 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 haven’t net climbed any stairs since you’ve actually ended up below the place where you started. Really what you’ve 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 opposites—called negative numbers—are just the ticket we need to answer our stair-climbing riddle. Yes, it sounds a bit odd, but in this situation you could say you’ve climbed negative one stair. Just in case you’re wondering what impact this information might have on your life outside staircases…well, 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? We’ll get to a real life application later in the episode. But first, in preparation, we need to get a bit negative—with numbers, that is.
Answering the Last Article’s Math Problem
Okay, before we jump head-first into the sea of negativity, let’s quickly review how we got here. How did we get here…numerically? 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 things—every 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 don’t 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 didn’t, here’s how it works... Since 2 books are checked-out every minute, after 20 minutes, 40 books (that’s 2 times 20) are checked-out; and since 1 book is checked-in every 5 minutes, after 20 minutes, 4 books (that’s 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 (that’s 40 minus 4) fewer books on its shelves.
Congratulations if you got it right, and worry-not if you struggled—things will start coming together soon enough.
Where Did Negative Numbers Come From?
Okay, that brings us up to date—so what comes next? Well, that’s exactly the question that some of the world’s 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 you’d 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), shouldn’t I also be able to subtract 3 from 2? And if this can be done, what would the resulting number look like? Well, there aren’t any natural numbers that satisfy this condition, right? Right. You won’t 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 one—negative 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? It’s -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 you’ll 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 you’re talking about, but I think it’s better to call it “negative seven” so that it doesn’t 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 you’re already quite familiar with—the 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 zero”—meaning that it’s 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 it’s entirely possible for ice to be colder than zero degrees. That’s right, some ice is indeed colder than others. So what’s the temperature of ice that’s 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 debts—a practice that made its way within a few hundred years to the Islamic world, and then to Europe…and eventually, after several additional centuries, it’s an all-too-integral part of our modern financial world. For example, here’s a financial problem (in more ways than one) you may have encountered as a student. You’re 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 go—you pay two friends the full amount you owe them, and you pay your unlucky last friend the remaining $10 you have—which is, of course, only half of what you actually owe. So, what’s you’re net financial worth at this point? Well, you have $0 in your pocket. But no, that’s not your net worth because you still owe your buddy $10. That’s 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, let’s take a moment to reflect upon where we’re at. Whether you’ve realized it or not, you’ve 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. That’s the number line you’re 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 that’s enough for now. We’ll talk a lot more about the number line in the next few episodes, and we’ll use it to help make arithmetic with integers easier. In particular, we’ll talk about how to add, subtract, multiply, and divide positive and negative integers—all while keeping the signs straight! And here’s the kicker—we’re going to learn how to do this…get ready for it…without relying on a calculator. Trust me, it’ll 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 greatest—101, -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 to。。。。。。follow the Math Dude on Twitter at。。。。。。and become a fan on Facebook. You can also follow me, your humble host, on Twitter at。。。。。。
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