Rational Numbers

A rational number can be written as p/q, where p and q are integers and q  0. This means that rational numbers consist of natural numbers, whole numbers, integers, fractions of integers, and decimals (terminating or repeating). 

Rational numbers and fractions are interrelated concepts, as both can be represented as ratios. Rational numbers include all fractions (proper, improper, positive, or negative) with integer numerators and non-zero denominators. In a fraction, the numerator and denominator are integers, with the denominator not zero. However, in a rational number, the numerator and denominator can be any integers, as long as the denominator is not zero. A rational number is any number expressible as p/q, where p and q are integers and q ≠ 0, including natural numbers, whole numbers, integers, fractions, and terminating or repeating decimals.
 

• 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/a, meaning their product is 1(a  1/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.
   

Students often get confused between rational and irrational numbers, and they get stuck trying to differentiate them. Rational numbers can be expressed as p/q, 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, 18/24 is a rational number, but it can be simplified to 3/4, where the numerator and denominator share only 1 as a common factor. Therefore, the rational number 3/4 is in its standard form. 

• Positive Rational Numbers: Positive rational numbers have a positive value, e.g., 4/6 or -6/-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 4/6, -6/-9, etc. For example, -6/-9 = 6/9, 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 -4/6, 6/-7, 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 of its denominator, after simplification, is of the form 2^m × 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, 5/6 = 0.8333 … , 7/9 = 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 a 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 5/4 on the number line.

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

    5/4 is an improper fraction. Converting it into a mixed fraction, we get 114. 
    
    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 5/4 = 11/4) to mark the point. 

Rational numbers can be recognized using the following characteristics:

Type of numbers: All integers, whole numbers, natural numbers, and fractions with integer numerators and denominators are rational numbers.

• Decimal Representation: A number is rational if its decimal form is either terminating or repeating (e.g., 5.6 or 2.141414…).

• Irrational Numbers: If the decimal form is non-terminating and non-repeating, the number is irrational. For example, 5 = 2.236067977… is an irrational number.

• Fraction Form: Any number that can be written as p/q is considered rational if p and q are integers and q  0.

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 5/8 and 2/5 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.
            
        5/8 = 5 × 5/8 × 5 = 2540

        2/5 = 2 × 8/5 × 8  = 1640

    Step 3: Add the numerators

        25/40 + 16/40 = (25 + 16)/40 = 41/40

    Step 4: Simplify the result if possible

        4140 is an improper fraction and can be written as a mixed number.

        1140

    Thus, the sum of 5/8 and 2/5 is 41/40 or 11/40.

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

        79 = 7 × 4/9 × 4 = 28/36

        14 = 1 × 9/4 × 9 = 9/36

    Step 3: Subtract the numerators

        28/36 – 9/36 = (28 -9)/36 = 19/36

    Step 4: Simplify the result if possible

        1936 is already in its simplest form

    Therefore, 7/9 – 1/4 = 19/36

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 -7/2 and 3/8 as an example. 

    Step 1: Write the rational numbers with a multiplication sign

        -7/2 × 3/8 

    Step 2: Multiply the numerators and denominators individually 

        (-7) × 3/2 × 8 = -21/16

    Step 3: Simplify the result if possible

        -2116 is already in its simplest form

    Thus, -7/2 × 3/8 = – 21/16.

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 5/6  2/9 as an example.

    Step 1: Write the rational numbers with the division sign

        5/6 × 2/9

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

5/6   9/2

Step 3: Multiply the numerator and denominators individually

    5 × 9/6 × 2 = 45/12

Step 4: Simplify the result if possible

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

    45  3/12  3 = 15/4

Thus, 5/6  2/9 = 15/4 or 3 3/4 (as a mixed fraction).

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.