Rational Numbers

A rational number can be written as pq where p and q are integers and q ≠ 0. This means rational numbers include natural numbers, whole numbers, integers, fractions, and decimals that either terminate or repeat.

Rational numbers and fractions are closely related because both can be written as ratios. All fractions whether proper, improper, positive, or negative are rational numbers as long as the numerator and denominator are integers and the denominator is not zero.

Examples of rational numbers:

Fractions: \(12\frac{1}{2}21​\) (half a pizza), \(34\frac{3}{4}43\) (three-quarters of a chocolate bar)
 

Decimals that end: 0.25 (a quarter of a dollar), 3.5 (three and a half pencils)
 

Decimals that repeat: 0.333… (one-third of a chocolate), 0.666… (two-thirds of a cup of juice)
 

Whole numbers/integers as fractions: 4 = 41, -3 = -31.

A rational number is said to be in standard form when the numerator and denominator have no common factors other than 1, and the denominator is positive.

For example, consider the rational number 12/36. Both 12 and 36 have a common factor of 12, so we can simplify by dividing both the numerator and the denominator by 12. This gives us 1/3.

Since 1 and 3 have no common factors other than one, and the denominator is positive, 1/3 is the standard form of the rational number 12/36.

Rational numbers are fractions or integers that can be written as \(p\over q\) (q ≠ 0). They follow important properties like closure, commutative, associative, distributive, and identity, which help in performing calculations easily and correctly.

• Termination or Repeating Decimals: When expressed as decimals, Rational numbers as decimals are either terminating (end after a finite number of digits) or repeating (have a repeating pattern of digits).

• Additive Identity: The additive identity of rational numbers is 0, meaning that adding 0 to any rational number does not change its value (a + 0 = a).

• Multiplicative Identity: The multiplicative identity for rational numbers is 1 because multiplying any rational number by 1 results in the same number (a  × 1 = a). 
  
   
• Additive Inverse: Every rational number a has an additive inverse -a, such that when they are added together, the result is 0 (a + (-a) = 0).

• Multiplicative Inverse: Every nonzero rational number a has a multiplicative inverse \(1\over a\), meaning their product is 1(a × \(1 \over a\)\(1\over a\)\(1\over a\) = 1). However, 0 has no multiplicative inverse. 

• Closure under Addition, Subtraction, and Multiplication: Rational numbers are closed under addition, subtraction, and multiplication, meaning the result is always a rational number. 

• Division Property: When dividing two rational numbers, the result is also a rational number, if the divisor is nonzero.

• Distributive property: Rational numbers adhere to the distributive property, which states that a(b + c) = ab + ac. 

• Ordering: Rational numbers can be ordered on a number line, where for any two rational numbers, one is greater than, less than, or equal to the other.

A rational number can also be written in decimal form. For example, 1.1 is a rational number because it can be expressed as a fraction.

1.1 = 1110.

Similarly, consider a non-terminating decimal, such as 0.333. Since this repeating decimal can be written as

0.333 = 1/3,

It is also a rational number.
So, any non-terminating decimal with a repeating pattern after the decimal point can be expressed as a fraction, which means it is a rational number.

Yes, 0 is a rational number because it can be expressed as a fraction of two integers, such as 0/1, 0/-2, and many others. In every case, 0/5, 0/-2, 0/1, the value remains 0, which confirms that it fits the definition of a rational number.

From this information, we can understand that there are infinitely many rational numbers. Because of this, we cannot write down the complete list of all rational numbers. Still, we can give a few examples, such as 3, 4.57, 3/4, 0, -7, and many more. This shows that natural numbers, whole numbers, integers, fractions, terminating decimals, and repeating decimals all fall under the category of rational numbers.

To add or subtract rational numbers, we follow the same rules used for adding and subtracting integers. For example,
 

Solve 1/2 - (-2/3)

Step 1: Start by applying the rule that subtracting a negative number is the same as adding its positive form.
 

So,
 

1/2 - (- 2/3) = 1/2 + 2/3 
 

Step 2: Now, add the fractions.
 

1/2 + 2/3
 

Step 3: To add these fractions, convert them into like fractions by finding a common denominator.

 
The LCM of 2 and 3 is 6.

Convert the fractions.
 

1/2 = 3/6, 2/3 = 4/6
 

Now, add them.

3/6 + 4/6 = 7/6
 

This improper fraction can be written as a mixed number
 

1 1/6

The multiplication and division of rational numbers follow the same steps we use for fractions. To multiply two rational numbers, you multiply their numerators together and their denominators together, and then simplify the result. 

For example,

Multiply 3/5 × -2/7

Step 1: To multiply 3/5 by -2/7, start by multiplying the numerators.

Step 2: The numerators give 3 × (-2) = -6.

Step 3: Next, multiply the denominators: 5 × 7 = 35. So, the product is -6/35.

For division, we multiply the first fraction, which is the dividend, by the reciprocal of the second fraction, which is the divisor.

For example, divide 3/5 ÷ 2/7

Step 1: Replace the second fraction with its reciprocal. So, 3/5 ÷ 2/7

Step 2: Multiply the numerators. 3 × 7 = 21

Step 3: Multiply the denominators. 5 × 2 = 10.

So, the final answer is 21/10, which can also be written as the mixed numbers 2 1/10

Rational numbers are numbers that can be written in the form p/q where p and 𝑞 are integers and q=0. This group includes positive numbers, negative numbers, and zero, as long as they can be expressed as a fraction or as a decimal that either ends or repeats.
 

Example of Rational Number

Whole numbers: 4 (can be written as 4/1), 0 (0/1)
 

Fractions: ½ (half a pizza), 3/4 (three-quarters of a chocolate bar), –5/2 (negative five halves)
 

Terminating decimals: 0.25 (a quarter of a dollar), 3.5 (three and a half pencils)
 

Repeating decimals: 0.333… (one-third of a chocolate), 0.666… (two-thirds of a cup of juice).

Students often get confused between rational and irrational numbers, and they get stuck trying to differentiate them. Rational numbers can be expressed as pq, Where p and q are integers and q ≠ 0, while irrational numbers cannot. Let’s understand this better using a table.
 

Rational numbers have different types depending on their form and properties:
 

• Standard Form: The standard form of a rational number means that the numerator and the denominator have no common factor other than 1. 
  
  For example, \(18\over24\)is a rational number, but it can be simplified to \(3\over 4\), where the numerator and denominator share only 1 as a common factor. Therefore, the rational number \(3\over 4\)  is in its standard form. 

• Positive Rational Numbers: Positive rational numbers have a positive value, for example, \(4\over6\)or \({-6} \over {-9}\) (where negative signs cancel). If both are negative, dividing by –1 simplifies the fraction into a positive rational number. Examples of positive rational numbers include  \(4\over6\), \({-6} \over {-9}\), etc. For example,  \({{-6} \over {-9}} = {6\over 9}\), as the negative signs cancel out.

• Negative Rational Numbers: Negative rational numbers are those where either the numerator or the denominator is a negative integer. Examples of negative rational numbers include  \(4\over6\), \({6} \over {-9}\), etc.

• Terminating Rational Numbers: Rational numbers with a decimal representation that ends after a certain number of digits are known as terminating decimals. A rational number has a terminating decimal if its denominator, after simplification, is of the form \(2^m \times 5^n\), where m and n are non-negative integers.

• Non Terminating and Repeating Rational Numbers:  Repeating decimals are rational numbers whose decimal representation follows a repeating pattern. The decimal expansion of a non-terminating rational number does not end: instead, a single digit or a group of digits repeats at a fixed interval. For example, \(5\over 6\) = 0.8333 … , \(7\over 9\) = 0.777 … 

Since rational numbers are a subset of real numbers, they can be represented on a number line. The following steps explain how to place a rational number on the number line.

Steps to Represent a Rational Number on a Number Line

Step 1: Determine the sign
    
If the number is positive, it will be plotted to the right of zero. 

If the number is negative, it will be plotted to the left of zero.

For example,  -3 (negative number) and +5 (positive number) on a number line can be represented like this on the number line
 

Step 2: Identify the type of fraction

If the rational number is a proper fraction (numerator < denominator), it lies between 0 and 1 (for positive numbers) or 0 and -1 (for negative numbers).

If the rational number is an improper fraction (numerator ≥ denominator), convert it into a mixed fraction. The number will be located beyond its whole number part.

Step 3: Divide the number line

Identify the two consecutive whole numbers between which the fraction lies.

Divide the section into equal parts based on the denominator of the fraction.

Step 4: Locate the Desired Value

Count the required number of divisions as indicated by the numerator and mark the point. 

Example: Represent \(5\over 4\) on the number line.
 

Solution: The number \(5 \over 4\) is positive, so it will be placed on the right side of zero.

\(5 \over 4\) is an improper fraction. Converting it into a mixed fraction, we get \(1 {1\over 4}\). 
    
The number lies between 1 and 2

Divide the segment between 1 and 2 into 4 equal parts (since the denominator is 4).

Count 1 division beyond 1 (since \(5 \over 4\) = \(11 \over 4\)) to mark the point. 

Rational numbers can be identified using the following characteristics:

Includes integers, whole numbers, natural numbers, and fractions: All these numbers are considered rational because they can be written as a fraction.

Example: 4 (whole number) = 4/1, –3 (integer) = –3/1, ½ (fraction)

Decimals that terminate or repeat: If a number’s decimal form ends after a few digits or repeats in a pattern, it is a rational number.

Example: 5.6 (terminating decimal), 2.141414… (recurring decimal)

Decimals that never end or don’t repeat are irrational: Numbers with non-terminating, non-repeating decimals cannot be expressed as fractions.

Example: √5 = 2.236067977… is irrational

Can be expressed in the form p/q: A number is rational if it can be written as pq where p and q are integers and 

Example 1: Is 0.923076923076… a rational number?
Solution: The decimal 923076 repeats continuously, so it is a recurring decimal. Therefore, this is a rational number.

Example 2: Is √2 a rational number?
Solution: The decimal value of √2 = 1.414213562… is non-terminating and non-repeating. Therefore, it cannot be expressed as a fraction, making it an irrational number.

There are four common arithmetic operations of rational numbers. Let’s learn more about it below.

Addition of Rational Numbers: The addition of two rational numbers can be performed using a step-by-step method. Below, the sum \(5\over 8\) and \(2\over 5\) is explained as an example.
 

Step 1: Find the least common denominator (LCD)
 

The least common denominator of 8 and 5 is 40.
 

Step 2: We now convert the fractions to have the same denominator.

            
\({5\over 8} = {{5 \times 5} \over {8 \times 5}} = {25\over 40} \)
 

\({2\over 5} = {{2 \times 8} \over {5 \times 8}} = {16\over 40} \)

Step 3: Add the numerators
 

\({{25 \over 40} + {16 \over 40}} = {{25 + 16} \over 40} = {41\over40} \)

Step 4: Simplify the result if possible
 

\(41\over 40\) is an improper fraction and can be written as \(1 {1\over40}\)

Thus, the sum of \(5 \over 8\) and \(2\over 5\) is \(41\over 40\) or \(1 {1\over40}\)
 

Subtraction of Rational Numbers: The subtraction of two rational numbers can be performed using a step-by-step method. Below, the subtraction of \(7 \over 9\) and \(1\over 4\)is explained. 


Step 1: Find the Least Common Denominator (LCD)
 

The least common denominator of 9 and 4 is 36.
 

Step 2: We now convert the fractions to have the same denominator.

\({7\over9} = {{7 \times 4} \over {9 \times 4}} = {28\over 36}\)
 

\({1\over4} = {{1 \times 9} \over {4 \times 9}} = {9\over 36}\)
 

Step 3: Subtract the numerators
 

\({28\over 36} - {9\over 36 } = {28 - 9 \over 36} = {19\over 36}\)
 

Step 4: Simplify the result if possible
 

\(19\over 36\) is already in its simplest form
 

Therefore, \({7\over9} - {1\over 4} = {19\over 36}\)
 

Multiplication of Rational Numbers: Multiplication of two rational numbers is done by simply multiplying their numerators and denominators. Below is a step-by-step method using \(-7 \over 2\) and \(3 \over 8\) as an example. 

 

Step 1: Write the rational numbers with a multiplication sign
 

\({-7 \over 2} \times {3 \over 8} \)
 

Step 2: Multiply the numerators and denominators individually 
 

\( {{(-7) \times 3} \over 2 \times 8} = {-21\over 16}\)
 

Step 3: Simplify the result if possible
 

\(-21 \over 16 \) is already in its simplest form
 

Thus, \({{-7\over 2} \times {3\over8}} = {-21 \over 16}\) 
 

Division of Rational Numbers: Division of two rational numbers is done by multiplying the first number by the reciprocal of the second number. Below is a step-by-step method using \({{5 \over 6} \div {2\over 9}}\) as an example.
 

Step 1: Write the rational numbers with the division sign
 

\({{5 \over 6} \div {2\over 9}}\)
 

Step 2: Change “÷” to “×” and take the reciprocal of the second rational number
 

\({{5 \over 6} \times {9\over 2}}\)
 

Step 3: Multiply the numerator and denominators individually
 

\({{5 \times 9} \over {6\times 2}} = {45\over 12}\)
 

Step 4: Simplify the result if possible
 

The greatest common factor (GCF) of 45 and 12 is 3.
 

\({{45 \div 3} \over {12 \div 3}} = {15\over 4}\)
 

Thus, \({5\over6} \div {2\over 9} = {15\over 4 }\) or \({3 {3\over4}}\) (as a mixed fraction).

Rational numbers might seem tricky at first, but with a few simple tricks, students can easily understand and use them. Parents and teachers can play a significant role in helping children feel confident with these numbers in both school and everyday life.
 

• A rational number is any number that can be written as p/q, where p and q are whole numbers, and q isn’t zero. This includes whole numbers, fractions, and decimals that either end or repeat. Parents and teachers can help students recognize these different forms in everyday examples.
   
• One easy trick is to simplify fractions. For instance, 4/8 becomes 1/2. Parents and teachers can guide children in practicing these simplifications step by step.
   
• Using a number line is another helpful tool. Students can see where numbers go, compare them, or even add them more easily. Parents and teachers can draw examples or use visual aids to make it clear.
   
• When dividing fractions, remember to flip the second fraction and multiply. For example: ½ ÷ ¼ = ½ × 4/1 = 2. Parents and teachers can walk kids through these steps to make them easier to remember.
   
• Finally, practice using real-life examples, such as measuring ingredients while cooking, counting money, or sharing items. Parents and teachers can create fun activities to show students how rational numbers are used every day.
   
• Parents should encourage children to practice multiplying and dividing simple fractions at home, using real-life examples such as recipes or sharing snacks.
   
• Teachers can use visual models, such as fraction bars or number lines, to help students understand how numerators and denominators change during multiplication and division.
   
• Children should remember the simple rules.

Rational numbers play a crucial role in various real-world situations. From managing money to measuring ingredients in cooking, they help us make accurate calculations and decisions.

These applications demonstrate the practicality of rational numbers. Let’s go through a few examples:

• Money and Finance: We commonly use rational numbers in tasks related to money and finance such as banking, shopping, calculating expenses, etc.

• Cooking: When cooking, we often measure ingredients using fractions. For example, doubling or halving a recipe can be done by multiplying or dividing rational numbers.

• Construction and Carpentry: Builders use fractions and decimals to measure wood, tiles, and bricks accurately. 

• Sports and Statistics: Players’ batting averages, shooting percentages, and game scores involve rational numbers.
  
   
• Temperature Measurement: Rational numbers are used to represent temperatures above or below zero, for example, +25 °C or -5 °C.