Reciprocal Definition

The word ‘reciprocal’ comes from a Latin word ‘reciprocus', meaning returning. But in mathematics, this word refers to the multiplicative inverse of a number. In mathematics, it refers to swapping the numerator and denominator of a fraction. The resulting fraction after interchanging the numerator and denominator is called the reciprocal. 

For example, if a is the numerator and b is the denominator in a fraction a/b, then the reciprocal of that fraction is b/a, where b is the numerator and a is the denominator.
 

Finding reciprocals is very easy. Understanding these rules and remembering them while finding reciprocals will make the process easier. Let’s quickly list out the reciprocal rules.

• Reverse the numbers or fraction: To find the reciprocal, just switch the numerator and denominator. For example, 
  
    
• The reciprocal of a whole number is a fraction: For example, the reciprocal of 5 is 1/5. 
  
   
• The reciprocal of 1 is 1: Because 1 ÷ 1, the reciprocal of 1 is still 1.
  
   
• The reciprocal of 0 does not exist: Division by zero (0) is impossible, so 0 has no reciprocal. 

• Multiplying a number by its reciprocal always equals 1: For example, 2 × 1/2 = 1.

To find the reciprocal of any given number, simply follow the steps given below.

• Zero: The reciprocal of zero cannot be determined, so it is undefined.
   

• Natural Number: If it’s a whole number, write it as a fraction.
  For example, 
  The number 5 can be written as 5/1
   

• Negative Numbers: If it is a negative number, for example, -5.
  First, write it as a fraction with 1 as the denominator. 
  -5 can be written as -5/1.
  Swap the numerator and denominator, 
  -5 is -1/5
  Add a negative sign to the result. The reciprocal of –5 is -1/5.

• Fraction: Reverse the fraction (swap the numerator and denominator).
  The reciprocal of 5/1 is 1/5.
   

• Mixed Fractions: First, convert the mixed fractions to an improper fraction. 
  For example, 8 1/2 =  17/2 
  Reverse the numerator and denominator
  Thus, the reciprocal of  8 1/2 is 2/17
   

• Decimals: For decimals, convert to a fraction first, then reverse it.
  For example, 
  0.5 = 1/2, so the reciprocal is 2/1 , which is 2. 
   

Let’s understand it better using the table given below. 

Reciprocal play an important role in our daily life without us even realizing it. From calculating speed to sharing things equally, and even understanding probability, reciprocals help us solve real-life problems easily. 

• Speed and Time: If you drive at 60 miles per hour, the reciprocal 1/60 represents the time in hours to travel 1 mile, i.e., 1 minute per mile.

• Sharing and Division: If you share a pizza among 4 people, each person gets 1/4 of the pizza. The reciprocal of 4 is 1/4.

• Probability and Odds: If the probability of winning a game is 1/5, the reciprocal is 5, meaning one win is expected for every 5 attempts on average.