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by Jason Marshall
In the last article, we discovered that integers alone are not sufficient to fully1 describe the world around us—we need the fractions existing between the integers too. (In learning that, we also learned that fractions are not integers.) In fact, eventually we’re going to discover that even fractions won’t fully satisfy our needs! But we’ll leave that for a future article—there’s still much to learn about the fractional world first.
In particular, today we’re going to take a deeper look at fractions and learn a few quick tips to help you understand exactly what numerators and denominators tell you.
What Are Fractions?
Before we get too far into the details of what the various parts of a fraction mean, let’s briefly2 review their anatomy3. First, a fraction is made up of two integers—one on the top, and one on the bottom.
（见图一）
The top one is called the numerator, the bottom one is called the denominator, and these two numbers are separated by a line. The line can be horizontal or slanted—they both mean the same thing and simply serve to separate the numerator from the denominator.
How to Pronounce Fractions in English
If you’ve known about fractions for a while, it’s probably been some time since you’ve contemplated4 the names we use to describe them. But they aren’t exactly obvious, so it’s worth spending a minute or two thinking about them.
Here’s the quick and dirty tip to help you remember how to pronounce them all: The numerator is always spoken first, and you pronounce it exactly as you pronounce the number. For example, in 1/2 the numerator, 1, is just pronounced “one;” or in 45/77 the numerator, 45, is simply pronounced “forty-five.” Easy enough. But denominators are a bit trickier5. They use the following convention:
2 is pronounced “half”
3 is pronounced “third”
4 is pronounced “fourth” (or “quarter”)
5 is pronounced “fifth”
6 is pronounced “sixth”
7 is pronounced “seventh”
8 is pronounced “eighth”
9 is pronounced “ninth”
10 is pronounced “tenth,” and so on.
So, for the fraction written 1/2, the denominator, 2, is pronounced “half,” and the entire fraction is therefore “one-half.” A little less obvious: For the fraction 45/77, the denominator, 77, is pronounced “seventy-seventh,” so the entire fraction is “forty-five seventy-sevenths.” An easy way to remember this is that with the exceptions of “half” and “quarter,” the words used to describe the denominator of a fraction are the same used to put things in order—for example, the order in which runners finish a race: “third,” “fourth,” “fifth,” etc.
What Is a Denominator?
Now let’s take a closer look at the different parts of a fraction. First, the bottom part—the denominator. The word “denominator” is derived6 from the Latin word “nomen,” which means “name” (and also shows up in words like “nominate” and “nomenclature”). And that’s pretty much what the denominator of a fraction does: it “names,” or indicates, the type of fraction that is described by the numerator (the top part).
What Does the Denominator Tell You?
Here’s what I mean. The denominator of a fraction tells you how many parts a whole is broken into. It can be a whole pineapple, a whole song, or a whole anything. If the denominator of a fraction is, say, 4, then that indicates that the whole whatever is broken up into 4 equally-sized pieces.
（见图二）
Or, if the denominator is 12, that means the whole whatever is broken-up into 12 equally-sized pieces. But how exactly does that “name” the type of fraction? Well, that leads us to the meaning of the numerator…
What Is a Numerator and What Does It Tell You?
The word numerator comes from the Latin verb “enumerate,” which we still use in English to mean “to count.” So, the numerator of a fraction counts the number of equally-sized pieces identified by the denominator that are contained in the fraction. How then do we put this all together to understand the meaning of fractions? Here’s the quick and dirty tip: Going back to our examples from before, the fraction 1/2 means “one piece of a whole object divided into two equally sized parts.” The denominator indicates that two parts make a whole, and the numerator counts off the fact that the fraction 1/2 contains one of those parts. Similarly, the fraction 45/77 means “forty-five pieces of a whole object that is divided into seventy-seven equally sized parts.”
What Does It Mean if the Numerator is Bigger than the Denominator?
In all the examples so far, the numerator has always been smaller than the denominator. In other words, in 1/2 and 45/77, 1 and 45 are smaller than 2 and 77, respectively. But what would it mean if the numerator were bigger than the denominator? Something like 7/4?
Well, let’s try interpreting this the same way as before. The denominator, 4, indicates that a whole is divided into four equally sized parts, and the numerator, 7, indicates that we have seven of those parts. So, if four parts make a whole, and we have seven, then we must have a whole object plus three more of the equally sized parts. So 7/4 is equivalent to 1 3/4—also known as “one and three-quarters”—and we now know that a fraction whose numerator is greater than its denominator represents a number that is greater-than one. In case you’re wondering, that type of fraction is called “improper,” whereas fractions like 1/2 with numerators less-than denominators are called “proper.”
What Does It Mean if the Denominator is Less than One?
So far we’ve only talked about fractions with denominators that are greater-than one. At the end of the last article, I asked the “brain-teaser” question: “Why can’t the denominator of a fraction be zero?” To find out the answer to that question, and take a peak at how fractions with denominators less-than one work, check out last week’s Math Dude Video Extra! episode posted to YouTube and the videos section of the Math Dude’s Facebook page.
Wrap Up
That’s all the math fun we have time for today. But before I go, here’s a “brain-teaser” problem for you to think about until next time:
How can you tell if one fraction is bigger or smaller than another?
Look for my explanation in this week’s Math Dude Video Extra! episode on YouTube and Facebook.
Also, throughout the month of March, we’re giving away one free book each week to lucky Math Dude Twitter followers7 and Facebook fans. So please join our growing community of social networking math fans, ask questions, and chat with other math enthusiasts8. Check it out and see how you can win a free book!
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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fully
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| adv.完全地，全部地，彻底地；充分地 | |
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briefly
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| adv.简单地，简短地 | |
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anatomy
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| n.解剖学，解剖；功能，结构，组织 | |
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contemplated
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| adj. 预期的 动词contemplate的过去分词形式 | |
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trickier
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| adj.狡猾的( tricky的比较级 )；（形势、工作等）复杂的；机警的；微妙的 | |
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derived
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| vi.起源；由来；衍生；导出v.得到( derive的过去式和过去分词 )；（从…中）得到获得；源于；（从…中）提取 | |
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followers
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| 追随者( follower的名词复数 )； 用户； 契据的附面； 从动件 | |
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enthusiasts
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| n.热心人，热衷者( enthusiast的名词复数 ) | |
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