What are Rational Numbers? - Definition & Examples

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  • 0:05 Definition
  • 2:07 Examples of Rational Numbers
  • 3:19 Examples of Irrational Numbers
  • 4:58 Lesson Summary
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Lesson Transcript
Instructor
Tara Quinn
Expert Contributor
Jerry Allison

Jerry holds a Doctor of Business Administration and a Master’s in Mathematics. He has taught business, math, and accounting for over 25 years.

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.

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Additional Activities

What Are Rational Numbers

Examples of Rational Numbers

The following are rational numbers because they are fractions made out of one integer divided by another integer:

1/3, -8/15, 6/31, 8 (or 8/1)

The following are also rational numbers because a decimal that stops (terminates) can be written as a rational number: 0.3, -0.25, 0.8976

The following are rational because every repeating decimal can be written as a fraction.

0.1111..., -0.254254254.., 0.837583758375...

The following are not rational numbers because the decimals go on forever without repeating any pattern:

7 + π, -6e, √(18)

Questions

Examine the following numbers. Use a calculator or a spreadsheet to help visualize the result.

1. Do you think π2 is a rational number or not? Why?

2. Do you think e + e is a rational number or not? Why?

3. Is √(7)2 a rational number or not? Why?

4. Is zero a rational number or not? Why?

5. Can two irrational numbers be combined using addition or subtraction to get a rational number?

6. Can two irrational numbers be combined using multiplication or division to get a rational number?

7. Can two rational numbers be combined using addition or subtraction to get an irrational number?

8. Can two rational numbers be combined using multiplication or division to get an irrational number?

Answers:

1. It is not a rational number. Every multiple of π is irrational.

2. It is not a rational number, since e added to itself is irrational.

3. This is a rational number. The square of a square root is the number inside the square root. So this would be 7, a rational number.

4. Zero is a rational number. It can be written as 0/1.

5. Two irrational numbers cannot be combined with addition or subtraction to get a rational number unless the irrationals cancel each other out as in π + -π.

6. This is a trick question. Two radicals, in some cases, can be multiplied to get a rational number. For example, √(8) * √(2) = √(16) = 4.

7. Two rational numbers cannot be combined using addition or subtraction to get an irrational number.

8. Two rational numbers cannot be combined using multiplication or division to get an irrational number.

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