Roots and Powers of Algebraic Expressions

What are Rational Numbers? - Definition & Examples

In this lesson, we will learn about rational numbers and their characteristics. We'll discover what they are, what they aren't and how to distinguish them from other types of numbers.

Definition

Mathematicians have taken all the numbers in the world and sorted them into categories, based on their characteristics. Generally, the categories, or sets, go from most to least complicated: complex numbers, imaginary numbers, real numbers, rational numbers, integers, whole numbers and natural numbers. Most numbers belong to more than one category.

Here, we'll talk specifically about the category, or set, of rational numbers. The set of rational numbers:
• Consist of positive numbers, negative numbers and zero
• Can be written as a fraction

The name rational is based on the word 'ratio.' A ratio is a comparison of two or more numbers and is often written as a fraction. A number is considered a rational number if it can be written as one integer divided by another integer. Sometimes this is referred to as a simple fraction.

The number 1/2 is a rational number because it is written as the integer 1 divided by the integer 2. The number 5 is a rational number because we can write it as 5/1. We can also write it as 15/3 or 50/10 because 15 divided by 3 or 50 divided by 10 both equal 5. The mixed number 1 ½ is also a rational number because we can write it as 3/2.

Any number that can be rewritten as a simple fraction is a rational number. This means that natural numbers, whole numbers and integers, like 5, are all part of the set of rational numbers as well because they can be written as fractions, as are mixed numbers like 1 ½.

Rational numbers can be positive, negative or zero. When we write a negative rational number, we put the negative sign either out in front of the fraction or with the numerator. That's the standard mathematical notation. For example, we would write -5/7 as opposed to 5/-7.

Examples of Rational Numbers

We mentioned earlier that natural numbers, whole numbers and integers are also rational numbers because they can be written as fractions. The simplest way to do this is to put the number over 1. For example: We can write 7 as 7/1 ; we can write -3 as -3/1 ; and we can write 0 as 0/1. Therefore, all of these numbers are rational numbers. .

Terminating decimals are rational numbers. A terminating decimal is a decimal that ends. All terminating decimals are rational numbers because they can be converted to fractions. We can write the decimal 1.2 as 12/10 or as 6/5. We can write 3.25 in a number of ways as a fraction, but one way is 325/100. .

Repeating decimals are rational numbers. Repeating decimals are decimals that do not end, but instead eventually repeat digits. It is possible to rewrite all repeating decimals as fractions. A great example of this is .33333. . . We can write that as the fraction 1/3. Try it for yourself - divide 1 by 3! You'll quickly see how the 3 repeats. .

Examples of Irrational Numbers

Just as numbers that can be written as one integer divided by another integer are rational numbers, there are also numbers that are irrational numbers. Irrational numbers are numbers that cannot be written as one integer divided by another integer. Irrational numbers cannot be written as fractions, cannot be written as terminating decimals, cannot be written as repeating decimals. .

There are two types of numbers that are irrational numbers. Roots and radicals are the first type. If a number does not have a perfect root, its root is an irrational number. The square root of 5 is a good example. Written as a decimal, the value of the square root of 5 begins 2.23606. . . Square root of 5 = 2.23606. . . .

However, the digits after the decimal will go on and on forever without a repeating pattern. We cannot find a way to write it as a fraction, so it is not a rational number. In contrast, the square root of 25 is a rational number because that has an exact value of 5, which can be written as a fraction. The square root of 25 = 5 = 5/1.

The other type of irrational numbers are special numbers like pi and e. These special numbers, like many radicals, go on and on and do not repeat or end. Pi, the ratio of a circle's circumference to its diameter, is the value 3.14159265. . ., which continues without repeating or ending. The value of e is 2.71828. . . and also goes on forever without repeating or ending. .

Lesson summary

The set of rational numbers includes all positive numbers, negative numbers and zero that can be written as a ratio (fraction) of one number over another. Whole numbers, integers, fractions, terminating decimals and repeating decimals are all rational numbers. Roots and radicals, and special numbers, like pi and e, can be written only as decimals that go on forever without a repeating pattern of digits and so are not rational numbers. .