数学英语 03 How to Add Negative Integers

by Jason Marshall

Today’s article will walk you through the first steps of kicking your calculator dependency. This isn’t something we can accomplish in a single article, so we’ll be revisiting this topic periodically. Today, we’re kicking things off by using a number line, instead of a calculator, to help you keep your signs straight.
Review of Integers
But before we start working to break that calculator habit, let’s briefly¹ review what we talked about last time. The most important thing to take away was the idea that combining negative whole numbers with the natural numbers gives us the very important group of numbers known as integers. These integers can be arranged on a number line with big negative ones extending out indefinitely to the left, zero in the middle, and big positive ones extending out forever to the right. At the end of the article I asked if any integers exist that are neither positive nor negative. What do you think? If you take all the positive and negative integers off the number line, are there any integers left? How about that strange one right in the middle? Yep, that’s the one: zero.
How to Understand the Number Line
Okay, how about the other question? Did you figure out how to put the integers 101, -1, 32, and -2010 in order from smallest to largest? Once you understand the number line and how positive and negative numbers relate to each other, this shouldn’t be too hard. The number -2010 is the smallest since it’s the most negative, then comes -1, then 32, and finally 101 is the largest. But perhaps you’re thinking: How can -2010 be the smallest!? It’s a pretty big number...it has four digits²!
Well, here’s a quick and dirty tip to help you keep the relative size of numbers straight. Think about the number line again. Any number to the left of another on the number line must be the smaller of the two. Even though it might be a big number—in that it might have a lot of digits like negative one-trillion—it’s still smaller-than any number to the right of it. Even a seemingly puny³ one-digit number like zero.
How to Kick Your Calculator Dependency
Okay, with all that covered, let’s talk about our first calculator dependency kicking technique. This one is aimed at making addition of positive and negative integers easier. Let’s say you need to solve a problem like -46+16. Your first instinct might be to go grab your calculator and start punching numbers. Yes, that should give you the right answer, but you run the risk of not understanding the very important question of why it gave you the right answer. And, if you don’t understand “why,” how will you ever know if you’ve made some egregious⁵ error, resulting in sharing an embarrassingly ridiculous—and wrong—answer with everybody. Avoid this risk and learn how to do the problem in your head instead. That way you’ll know when something is fishy⁶ with a result.
Visualize⁷ the Number Line
Let’s start our journey by going back to the number line and thinking about some simple examples. First, imagine you’re standing⁴ at the zero marker of the number line with all the negative integers lined up to your left and the positive ones to your right. Really picture this—it’ll help if you have a vivid image in your head. Your number line could be on a beach, a football field, a pasture full of cows—or wherever else makes you happy. I know it seems kinda dorky. It is. But it’ll help.
How to Use the Number Line to Solve Problems
Alright, let’s kick things off with a super simple example: What’s 2+3? I know, I know. I told you it was going to be really simple, so pretend for a minute you don’t already know the answer. It turns out you can use your imaginary number line as a sort-of mental calculator to help solve problems like this.
Here’s how. Start by imagining you’re standing at the zero mark of your number line. Since the first number in the problem 2+3 is positive two, walk two steps in the positive direction (that’s to your right). You’re now standing at the position marked “2.” The second number in 2+3 is positive three, so you next need to walk three additional steps in the positive direction. Now, take a look at the number line and see where you’ve ended up. Of course, you’re at the position marked “5.” So—and I know this isn’t going to be a big surprise to you—you’ve calculated that 2+3=5.
How to Deal With Negative Numbers
Wow, who needs a calculator when you’ve got a number line, huh? Well, okay...I know that was a ridiculously easy problem, and there weren’t even any negative numbers in it. So here’s how to deal with adding positive and negative numbers. Instead of the problem 2+3, let’s say you need to solve the problem -2+3. It’s still pretty simple, but let’s think about how it works with the number line.
Once again, imagine starting at the zero mark on your number line. Since the first number in -2+3 is negative two, you first need to walk two steps in the negative direction (that’s to your left). So you’re now at the position marked “-2.” Now, since the second number in the problem -2+3 is positive 3, you next need to walk three steps in the positive direction. So, starting at “-2,” you take three steps in the positive direction and end up at the position marked “1.” Congratulations! You’ve calculated that -2+3=1.

Summary

And that’s the trick for adding positive and negative numbers. Admittedly, it’s not necessarily all that useful for the relatively⁸ easy examples we used to demonstrate the method. But it can be extremely helpful when tackling somewhat tougher problems like 3 + (-13) + 14 = ? or perhaps -9 + (-8) + (-3) + 7 = ?. These problem might seem a little intimidating⁹ at first, but try mentally walking through them and see if it helps you keep the signs straight. Just take it step by step, walking in the positive or negative direction according to the sign of the number. We’ll go over the answers next time, and we’ll also talk about extending this method to include not just adding negative numbers, but subtracting them too.
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1 briefly   

adv.简单地，简短地

参考例句：

• I want to touch briefly on another aspect of the problem.我想简单地谈一下这个问题的另一方面。
• He was kidnapped and briefly detained by a terrorist group.他被一个恐怖组织绑架并短暂拘禁。

2 digits   

n.数字( digit的名词复数 )；手指，足趾

参考例句：

• The number 1000 contains four digits. 1000是四位数。 来自《简明英汉词典》
• The number 410 contains three digits. 数字 410 中包括三个数目字。 来自《现代英汉综合大词典》

3 puny   

adj.微不足道的，弱小的

参考例句：

• The resources at the central banks' disposal are simply too puny.中央银行掌握的资金实在太少了。
• Antonio was a puny lad,and not strong enough to work.安东尼奥是个瘦小的小家伙，身体还不壮，还不能干活。

4 standing   

n.持续，地位；adj.永久的，不动的，直立的，不流动的

参考例句：

• After the earthquake only a few houses were left standing.地震过后只有几幢房屋还立着。
• They're standing out against any change in the law.他们坚决反对对法律做任何修改。

5 egregious   

adj.非常的，过分的

参考例句：

• When it comes to blatant lies,there are none more egregious than budget figures.谈到公众谎言，没有比预算数字更令人震惊的。
• What an egregious example was here!现摆着一个多么触目惊心的例子啊。

6 fishy   

adj. 值得怀疑的

参考例句：

• It all sounds very fishy to me.所有这些在我听起来都很可疑。
• There was definitely something fishy going on.肯定当时有可疑的事情在进行中。

7 visualize   

vt.使看得见，使具体化，想象，设想

参考例句：

• I remember meeting the man before but I can't visualize him.我记得以前见过那个人，但他的样子我想不起来了。
• She couldn't visualize flying through space.她无法想像在太空中飞行的景象。

8 relatively   

adv.比较...地，相对地

参考例句：

• The rabbit is a relatively recent introduction in Australia.兔子是相对较新引入澳大利亚的物种。
• The operation was relatively painless.手术相对来说不痛。

9 intimidating   

vt.恐吓，威胁( intimidate的现在分词)

参考例句：

• They were accused of intimidating people into voting for them. 他们被控胁迫选民投他们的票。
• This kind of questioning can be very intimidating to children. 这种问话的方式可能让孩子们非常害怕。

10 humble   

adj.谦卑的，恭顺的；地位低下的；v.降低，贬低

参考例句：

• In my humble opinion,he will win the election.依我拙见，他将在选举中获胜。
• Defeat and failure make people humble.挫折与失败会使人谦卑。