Do you frequently find yourself at restaurants with friends fumbling 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 silicon 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 visualize 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.

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 apparently empty spaces between the integers on the number line are actually teeming with infinitely 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 vibrant terms like this are rare in math, I couldn’t help but introduce it to you. The word “vulgar” here is used as a synonym 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).

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 perfectly 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 followers and Facebook fans. So please join our growing community of social networking math fans, ask questions, and chat with other math enthusiasts. Check it out and see how you can win a free book!