数学英语 17 What Are Fractions?

 

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by Jason Marshall

Do you frequently find yourself at restaurants with friends fumbling¹ to figure out how to split the bill and calculate the tip? Have you ever resorted to seeking advice from your “smart” phone? Don’t worry, we all have. Heck, a whole industry of iPhone Apps has emerged to help answer these very questions. But you shouldn’t really need them. As we’ll soon find out, the key to freeing yourself from your silicon² assistant and figuring these questions out by yourself is understanding how to calculate with percentages. But in order to do that, you first have to have a solid understanding of fractions. So with that in mind, today we’re kicking things off by answering the question: What are fractions?
Are There Numbers Between the Integers on the Number Line?
Let’s start by turning our thoughts back to integers and the number line. In a previous article, we established that integers are the group of numbers consisting of:
All the positive whole numbers: 1, 2, 3, and so on,
their negative counterparts: -1, -2, -3, etc., and
the neither positive nor negative number 0.
We also talked about how to visualize³ adding and subtractingpositive and negative integersby mentally walking step-by-step along the number line. But what would happen if you were joined on your imaginary stroll along the number line by two imaginary friends: one much taller than you, and another who’s a bit shorter? For every step you take—from zero to one, one to two, and so on—your taller friend takes a longer step and your shorter friend takes a smaller step—and they both continually fall between the integer marks.（见图一）

Let’s say you and your friends all walk ten steps in the positive direction, after which you stop squarely at the mark on the number line labeling the integer “10.” Your tall friend has traveled further than you and has stopped somewhere between “12” and “13,” and your shorter friend has stopped somewhere just shy of “8.” How far have your friends traveled?
Are Fractions Integers?
Well, sadly, there aren’t any integers that can answer this question because integers are whole numbers—like “12” and “13.” But surely there must be numbers between each of the integers—we know your friends have traveled some numerical distance. And, of course, there are numbers there. The apparently⁴ empty spaces between the integers on the number line are actually teeming⁵ with infinitely⁶ many fractional numbers—that is, numbers that have a fractional (or non whole number) part. You might know them better as fractions. Are fractions integers? No, they’re all the numbers between the integers.
What are Fractions?
The easiest type of fractions to understand are built by turning the integers on their heads. Every integer has what’s called a reciprocal which is obtained by dividing one by that integer. For example: the reciprocal of 1 is 1/1, the reciprocal of 2 is 1/2, the reciprocal of 3 is 1/3, and so on. You could conceivably create a list of all such fractions by walking positive integer steps along the number line and calling out the reciprocal of the integer at each position. Eventually, you’d start getting to big numbers: 1/99, 1/100, and then eventually even bigger: 1/999, 1/1000, and then even bigger, and bigger, forever.
You can think of all these fractions as pieces of a pie, or portions of a mile, kilometer, lifetime, or whatever. Adopting the pieces of pie analogy, the reciprocal of the integer 1 is 1/1, which is equivalent to 1—representing one whole pie. The reciprocal of the integer 2 is 1/2 which represents one piece of a pie that is evenly divided into two—in other words one-half a pie. Similarly, the reciprocal of the integer 3 represents 1/3 of a pie, and so on. The bigger the integer we start with, the smaller the reciprocal—and therefore the smaller the fraction. For example, a slice that’s 1/3 of a pie is much bigger than a slice that’s 1/12 of a pie. And a slice that’s 1/99 of that pie would be miniscule. No matter how small a fraction is, you can always find smaller fractions by taking the reciprocal of yet larger integers!
What are Common Fractions?
I must admit I’d never heard the term “vulgar fraction” until I started preparing to write this article; and since quirky and vibrant⁷ terms like this are rare in math, I couldn’t help but introduce it to you. The word “vulgar” here is used as a synonym⁸ for “common,” so the term “vulgar fraction” simply refers to common fractions. But what are these common fractions? Well, common fractions are all the numbers that have an integer in their numerator (the top number) and a non-zero integer in their denominator (the bottom number).
（见图二）

The fractions we’ve dealt with so far like 1/3 and 1/4 certainly are common, but fractions like 2/3, 3/4, and 63/72 with numbers other than 1 in their numerator are common too. Also, all the fractions we’ve talked about so far have been smaller than one, but there’s no reason fractions can’t be larger than one too. So fractions like 4/3, 7/4, and an infinite number of others are all perfectly⁹ common too.
Wrap Up

That’s all the math we have time for today. But rest assured we’ll be talking a lot more about fractions and how to interpret and work with them in upcoming articles. In the meantime, here’s a problem for you to think about: Why can’t the denominator (that is, the bottom number) of a fraction be zero? Look for my explanation in the weekly “solutions” video posted each week to the videos section of the Math Dude’s Facebook page and to YouTube.
Also, throughout the month of March, we’re giving away one free book each week to lucky Math Dude Twitter followers¹⁰ and Facebook fans. So please join our growing community of social networking math fans, ask questions, and chat with other math enthusiasts¹¹. Check it out and see how you can win a free book!

 

1 fumbling   

n. 摸索,漏接 v. 摸索,摸弄,笨拙的处理

参考例句：

• If he actually managed to the ball instead of fumbling it with an off-balance shot. 如果他实际上设法拿好球而不是fumbling它。50-balance射击笨拙地和迅速地会开始他的岗位移动，经常这样结束。
• If he actually managed to secure the ball instead of fumbling it awkwardly an off-balance shot. 如果他实际上设法拿好球而不是fumbling它。50-50提议有时。他从off-balance射击笨拙地和迅速地会开始他的岗位移动，经常这样结束。

2 silicon   

n.硅(旧名矽)

参考例句：

• This company pioneered the use of silicon chip.这家公司开创了使用硅片的方法。
• A chip is a piece of silicon about the size of a postage stamp.芯片就是一枚邮票大小的硅片。

3 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。

4 apparently   

adv.显然地；表面上，似乎

参考例句：

• An apparently blind alley leads suddenly into an open space.山穷水尽，豁然开朗。
• He was apparently much surprised at the news.他对那个消息显然感到十分惊异。

5 teeming   

adj.丰富的v.充满( teem的现在分词 )；到处都是；（指水、雨等）暴降；倾注

参考例句：

• The rain was teeming down. 大雨倾盆而下。
• the teeming streets of the city 熙熙攘攘的城市街道

6 infinitely   

adv.无限地，无穷地

参考例句：

• There is an infinitely bright future ahead of us.我们有无限光明的前途。
• The universe is infinitely large.宇宙是无限大的。

7 vibrant   

adj.震颤的，响亮的，充满活力的，精力充沛的，(色彩)鲜明的

参考例句：

• He always uses vibrant colours in his paintings. 他在画中总是使用鲜明的色彩。
• She gave a vibrant performance in the leading role in the school play.她在学校表演中生气盎然地扮演了主角。

8 synonym   

n.同义词，换喻词

参考例句：

• Zhuge Liang is a synonym for wisdom in folklore.诸葛亮在民间传说中成了智慧的代名词。
• The term 'industrial democracy' is often used as a synonym for worker participation. “工业民主”这个词常被用作“工人参与”的同义词。

9 perfectly   

adv.完美地，无可非议地，彻底地

参考例句：

• The witnesses were each perfectly certain of what they said.证人们个个对自己所说的话十分肯定。
• Everything that we're doing is all perfectly above board.我们做的每件事情都是光明正大的。

10 followers   

追随者( follower的名词复数 )； 用户； 契据的附面； 从动件

参考例句：

• the followers of Mahatma Gandhi 圣雄甘地的拥护者
• The reformer soon gathered a band of followers round him. 改革者很快就获得一群追随者支持他。

11 enthusiasts   

n.热心人，热衷者( enthusiast的名词复数 )

参考例句：

• A group of enthusiasts have undertaken the reconstruction of a steam locomotive. 一群火车迷已担负起重造蒸汽机车的任务。 来自《简明英汉词典》
• Now a group of enthusiasts are going to have the plane restored. 一群热心人计划修复这架飞机。 来自新概念英语第二册