Rational Numbers - Math Steps, Examples & Questions (2024)

Here you will learn about rational numbers, including the definition of a rational number, examples of rational numbers and how to identify rational numbers.

Students will first learn about rational numbers as part of the number system in 6th grade.

Every week, we teach lessons on rational numbers to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

What are rational numbers?

Rational numbers are numbers that can be expressed in the form \cfrac{a}{b} where a and b are integers (whole numbers) and b ≠ 0. They include natural numbers, whole numbers and integers.

Rational numbers come in four forms. Below are examples of each. Each example has been expressed as a fraction in the form \frac{a}{b} to show that it is rational.

Key points on rational numbers

• A rational number must have a non-zero denominator

For rational numbers expressed as fractions in the form \cfrac{a}{b}, \; b must be a non-zero integer because zero cannot be a divisor. (Try 5 \div 0 on your calculator and it will give you an error message)

The letter a however can be equal to 0 as you can divide 0 by any real number and get the solution 0. This means that 0 itself is a rational number.

• Numbers that are not rational are called irrational numbers

If a number cannot be represented as a fraction in the form \cfrac{a}{b} where a and b are integers, then the number is irrational.

There are a several famous irrational numbers including

\text { Pi }(\pi=3.141 \ldots) \text {, The Golden Ratio }(\varphi=1.618 \ldots) \text {, and Euler's Number }(e=2.718 \ldots)

• Not all fractions are rational numbers

All rational numbers can be expressed as a fraction, but not all fractions are rational numbers.

For example, \cfrac{5}{\sqrt{2}} is a fraction but it is not rational. The numerator is an integer but the denominator is not (the square root of 2 is irrational). Therefore this fraction does not meet the definition of a rational number.

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – The Number System (6.NS.C.6)
  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

How to identify rational numbers

In order to identify and then show that a number is rational:

1. Identify if the number is any of the following. If it is then it is a rational number.
   ● An integer
   ● A terminating decimal
   ● A repeating decimal
   ● A fraction in the form \bf{\cfrac{\textbf{a}}{\textbf{b}}} or a mixed number in the form \bf{C\cfrac{\textbf{a}}{\textbf{b}}} where \textbf{a} and \textbf{b} are
   integers
2. Show that the number is rational by writing it as a fraction in the form \bf{\cfrac{\textbf{a}}{\textbf{b}}} where
   \textbf{a} and \textbf{b} are integers.

Example 1: showing that a terminating decimal is rational

Show that 6.7 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

1. Identify if the number is any of the following. If it is then it is a rational number.
   ● An integer
   ● A terminating decimal
   ● A repeating decimal
   ● A fraction in the form \bf{\cfrac{\textbf{a}}{\textbf{b}}} or a mixed number in the form \bf{C\cfrac{\textbf{a}}{\textbf{b}}} where \textbf{a} and \textbf{b} are
   integers

6.7 is a terminating decimal and is therefore rational.

2Show that the number is rational by writing it as a fraction in the form \bf{\cfrac{\textbf{a}}{\textbf{b}}} where
\textbf{a} and \textbf{b} are integers.

6.7 = 6\cfrac{7}{10} = \cfrac{67}{10}

Example 2: showing that a terminating decimal is rational

Show that 0.045 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

Example 3: showing that an integer

Show that -17 is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

Example 4: showing that a mixed number is rational

Show that 3\cfrac{4}{5} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

Example 5: showing that a repeating decimal is rational

Show that 0 . \overline{6} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

Example 6:showing that a repeating decimal is rational

Show that 0 . \overline{4} is a rational number by expressing it as a fraction in the form \cfrac{a}{b} where a and b are integers.

Teaching tips for rational numbers

• Let students spend time exploring rational numbers using a calculator. This gives them an opportunity to notice patterns and also easily convert between fractions and decimals.

• Have students create a number line of their life with rational numbers – their birth starting at 0. They should plot things that happened before their birth with negative rational numbers and things that happened after their birth with positive rational numbers. Challenge them to represent each rational number on their number line in as many ways as possible.

• Just as developing whole number sense takes time and a variety of experiences in elementary school, so does understanding of rational numbers in middle school. Fit in quick reviews to rational numbers throughout the year or make connections as they appear in other standards to help deepen student understanding over time.

Our favorite mistakes

• Thinking zero is not a rational number
  Zero is a rational number because it can be written as \cfrac{0}{n} for any value of n other than 0.

• Thinking that all fractions are rational numbers
  All rational numbers can be written as fractions but not all fractions are rational numbers. If the fraction is in the form \cfrac{a}{b} and a and b are integers and b ≠ 0 then the number is rational. However if a or b are not integers then the fraction could represent an irrational number.
  E.g.
  \cfrac{\sqrt{2}}{3} is a fraction which is irrational.

• Non-terminating decimals and rational numbers
  Some non-terminating decimals are irrational and others are rational.
  E.g.
  \sqrt{3} = 1.73205… \; This is a non terminating irrational number.
  \cfrac{7}{9} = 0.777… \; This is a non terminating rational number.

Related types of numbers lessons

• Types of numbers
• Absolute value
• Prime numbers
• Natural numbers
• Prime and composite numbers
• Number sets
• Integers
• Even numbers
• Odd numbers
• Whole numbers

Can rational numbers be expressed in decimal form?

Yes, rational numbers can be shown as terminating or repeating decimals.

How many rational numbers are there?

There are an infinite number of rational numbers within our number system. In fact, even between any two given numbers there are an infinite number of rational numbers.

Are there other types of numbers besides rational?

Yes, there are irrational numbers, which students learn about later in middle school. In high school students also learn about real numbers, imaginary numbers and complex numbers.

The next lessons are

• Rounding numbers
• Factors and multiples
• Fractions

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