Reciprocal: Definition, Meaning, Properties, and Solved Examples

The concept of a reciprocal is fundamental in mathematics and finds application in various fields, including algebra, geometry, and calculus. A reciprocal of a number is essentially its multiplicative inverse, meaning when a number is multiplied by its reciprocal (a x 1/a), the result is 1. Understanding reciprocals involves delving into their definition, exploring their meaning, and recognizing their properties. Important properties of reciprocals include the reciprocal of a fraction, the reciprocal of 1 being itself, and the reciprocal of zero being undefined. By mastering these properties, one can solve a variety of mathematical problems with greater ease and precision. This article provides a comprehensive overview of reciprocals, detailing their definition, meaning, and properties, along with solved examples to define their practical applications.

What is Reciprocal?

A reciprocal is the multiplicative inverse of a number. In simpler terms, if you multiply a number by its reciprocal, the result is always 1. For any non-zero number a, its reciprocal is 1/a​.

Properties of Reciprocals

1. Multiplicative Inverse:
   • The product of a number and its reciprocal is always 1. 

a×1/a = 1(where a≠0)

2. Reciprocal of a Fraction:
   • To find the reciprocal of a fraction, you simply swap the numerator and the denominator.

Reciprocal of a/b is b/a(where a,b≠0)

3. Reciprocal of 1:
   • The reciprocal of 1 is 1. 

a x a = a (1×1 = 1)

4. Reciprocal of a Negative Number:
   • The reciprocal of a negative number is also negative.

Reciprocal of −a is −1/a

5. Reciprocal of Zero:
   • The reciprocal of zero is undefined because division by zero is undefined.

1/0 is undefined

Also Read: What are Composite Numbers from 1 to 100?

Meaning of Reciprocal

The reciprocal of a number is its multiplicative inverse. This means that when you multiply a number by its reciprocal, the result is 1. The reciprocal of a number aaa is denoted as 1/a​.

Important Points

1. Multiplicative Inverse: The core idea of a reciprocal is that it is the number which, when multiplied by the original number, yields 1.
   a×1/a = 1
2. Applicability: The concept of a reciprocal applies to all non-zero numbers, whether they are integers, fractions, or decimals. The reciprocal of zero is undefined because division by zero is not possible.
3. Notation: The reciprocal of a number aaa is written as 1/a​.

Practical Applications

Understanding reciprocals is important in various mathematical operations, such as dividing fractions, solving equations, and working with ratios and proportions. For example, when dividing by a fraction, you multiply by its reciprocal:

⅔ ÷ ⅘ = ⅔ × 5/4 = 10/12 = 5/6

Properties of Reciprocal

The concept of reciprocals comes with several important properties that are fundamental in mathematics. These properties help in simplifying complex mathematical expressions and solving various types of problems.

1. Multiplicative Inverse:
   • The reciprocal of a number aaa is its multiplicative inverse. When a number is multiplied by its reciprocal, the result is always 1.

a×1/a = 1(where a≠0)

2. Reciprocal of a Fraction:
   • To find the reciprocal of a fraction, you simply swap the numerator and the denominator.

Reciprocal of a/b is b/a(where a,b≠0)

3. Reciprocal of 1:
   • The reciprocal of 1 is 1, because multiplying 1 by itself yields 1.

a x a = a (1×1 = 1)

4. Reciprocal of a Negative Number:
   • The reciprocal of a negative number is also negative. This means if aaa is a negative number, then its reciprocal is also negative.

Reciprocal of −a is −1/a

​

5. Reciprocal of Zero:
   • The reciprocal of zero is undefined because division by zero is not possible.

1/0 is undefined

6. Reciprocal of a Decimal:
   • The reciprocal of a decimal can be found by converting the decimal to a fraction and then swapping the numerator and the denominator. For instance, the reciprocal of 0.25 (which is 1/4​) is 4.

7. Product of Reciprocals:
   • The product of the reciprocals of two numbers is equal to the reciprocal of the product of the two numbers.

(1/a) × (1/b) = 1/(a×b)(where a,b≠0)

8. Reciprocal of a Reciprocal:
   • Taking the reciprocal of a reciprocal returns the original number.

Reciprocal of (1/a) is a

9. Reciprocal of Powers:
   • The reciprocal of a power of a number is the power of the reciprocal of the number.

(1/a)n = 1/an​

Also Read: Ascending Order

Reciprocal with Solved Examples

Here, we will explore the properties of Reciprocal and provide five solved examples.

Example 1: Whole Number

• Number: 5
• Reciprocal: 1/5

Example 2: Fraction

• Number: 3/4
• Reciprocal: 4/3

Example 3: Negative Number

• Number: -2
• Reciprocal: -1/2

Example 4: Decimal

• Number: 0.25 (which is equal to 1/4)
• Reciprocal: 4

Example 5: Mixed Number

• Number: 2 1/3 (which is equal to 7/3)
• Reciprocal: 3/7

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