Basic Math and Pre-Algebra

PART 1. The World of Numbers

CHAPTER 5. Fractions

 

Your first encounter with the world of numbers was the simple act of counting. That world is larger than you could have imagined at that first encounter, and over the last few chapters, we’ve moved from the counting numbers, to the whole numbers, to the integers. When we worked with the integers, the positive and negative whole numbers, I suggested a number line as a way to help you think about them, and I promised to fill in the spaces between the integers soon. That time has come.

The spaces between the integers are filled with numbers that talk about parts of a whole. In this chapter, you’ll look at one of the two ways of representing those numbers that show parts. You’ll examine the different methods of arithmetic for fractions and learn to add, subtract, multiply, and divide fractions and mixed numbers.

 

The Rational Numbers

 

The counting numbers, or natural numbers, are the numbers you use to count. When you find that you need a symbol for nothing and add 0 to the counting numbers, you form the set called the whole numbers. When you introduce negative numbers, you have the integers. Now you’re turning your focus to numbers that represent parts of a whole, so what language do you use to describe all of this?

Fractions are the names our number system gives to symbols representing parts of a whole. A fraction is what we produce when we break a whole number into parts. Common fraction is the name we give to a way of writing a fraction as one number divided by another. The number 1/2 is an example of a common fraction, although usually it’s just called a fraction.

 

DEFINITION

A fraction is a symbol that represents part of a whole. Decimal fractions are written in the base ten system, with digits to the right of the decimal point. Common fractions are written as a quotient of two integers.

 

The set of numbers that includes all the integers plus all the fractions is called the rational numbers. The name rational comes from ratio, another name for a fraction or a comparison of two numbers by dividing. The rational numbers include any number you can write as a quotient of two integers. So 1/2 is a rational number, but so is 2, because you can write it as 1/2. Other examples of rational numbers are -4/13, 163/71, and 0.

 

DEFINITION

Rational numbers are the set of all numbers that can be written as the quotient of two integers.

 

Proper Fractions and Improper Fractions

When we talk about fractions, we are usually talking about rational numbers written as a quotient of two integers. When a fraction is written this way, the top number is called the numerator, and the bottom number is called the denominator.

 

DEFINITION

The denominator of a fraction is the number below the division bar. It tells how many parts the whole was broken into, or what kind of fraction you have. The numerator is the number above the bar, which tells you how many of that sort of part you actually have. In the fraction 1/2, 2 is the denominator and 1 is the numerator.

 

Because a whole number like 7 is a rational number, even though you wouldn’t usually think of it as a fraction, it can be written as the quotient of two integers. You could write 7/1, which you would think of as a fraction. You could actually write many different quotients that equal 7, like 42/6 or 700/100. Those look like the other quotients you think of as fractions, like 5/8, 17/3, and 4/51.

In those examples, there were two proper fractions, or fractions whose value is less than 1. Those were 5/8 and 4/51. You can spot a proper fraction because the numerator is less than the denominator. On the other hand, 17/3 and 7/1 (and all the fractions equal to 7/1) are improper fractions. They’re worth more than 1, because their numerators are larger than their denominators.

 

DEFINITION

A proper fraction is one whose value is less than one. An improper fraction is one whose value is more than one.

 

Fractions use their denominator to tell you how many parts the whole was broken into and their numerator to tell you how many of those parts you have. 5/8 says the whole was broken into 8 parts (called eighths), and you have 5 of them. That’s less than the whole, so it’s a proper fraction.

To interpret 17/3, you have to imagine that many wholes were each divided into 3 parts, called thirds, and you have a total of 17 of those parts. You could piece your parts back together and make 5 full wholes, with 2 of the pieces left over. You can write that as  which is called a mixed number Mixed numbers are a combination of a whole number and a fraction, a condensed way of saying 5 + 2/3.

 

DEFINITION

A mixed number is made up of a whole number and a fraction, written side by side. It tells you that you have a number of wholes plus part of another.

 

CHECK POINT

1. Change 18/5 to a mixed number.

2. Change 37/4 to a mixed number.

3. Change  to an improper fraction.

4. Change  to an improper fraction.

5. Change  to an improper fraction.