In mathematics, a reciprocal is simply the inverse of a number or a value. If ‘n’ is a real number, then its reciprocal is expressed as 1/n. In other words, it is obtained by flipping the number. For instance, the reciprocal of 9 is written as 1/9. When a number is multiplied by its reciprocal, the result is always equal to 1. This property is also known as the multiplicative inverse.
Meaning of Reciprocal
The term ‘reciprocal’ is derived from the Latin word ‘reciprocus,’ which means ‘returning.’ This means that when we take the reciprocal of a reciprocal, we return to the original number. In this article, we will explore the definition of reciprocal, its properties, and how to find reciprocals of integers, fractions, decimals, and mixed numbers along with examples.
Table of Contents
In simple terms, the reciprocal of a number is 1 divided by that number. For any value \(\displaystyle a\), the reciprocal is written as \(\displaystyle \frac{1}{a}\). Multiplying a number by its reciprocal always results in \(\displaystyle 1\).
Example:
\(\displaystyle \text{Reciprocal of } 7 = \frac{1}{7}, \quad 7 \times \frac{1}{7} = 1\)
Other Definitions of Reciprocal
- Reciprocal is also called the multiplicative inverse.
- It can be thought of as flipping a number upside down.
- For fractions, it is obtained by interchanging numerator and denominator.
- Every non-zero number has a reciprocal.
- Reciprocal can also be expressed as \(\displaystyle x^{-1}\) for any number \(\displaystyle x\).
- Taking the reciprocal twice gives back the original number.
Reciprocal of Zero
Zero does not have a reciprocal because division by zero is undefined. Hence, every real number has a reciprocal except zero.
\(\displaystyle \text{For example, the reciprocal of } 5 \text{ is } \frac{1}{5}.\)
\(\displaystyle \text{For instance, the reciprocal of } -17 \text{ is } -\frac{1}{17}.\)
To find the reciprocal of a fraction, swap the numerator and denominator.
Example:
\(\displaystyle \text{Reciprocal of } \frac{2}{3} \text{ is } \frac{3}{2}\)
To find the reciprocal of a mixed fraction, first convert it into an improper fraction.
Example:
\(\displaystyle 4 \frac{1}{2} = \frac{9}{2}\)
\(\displaystyle \text{Its reciprocal is } \frac{2}{9}\)
The reciprocal of a decimal can be found by dividing 1 by the decimal or by converting it into a fraction.
Example:
\(\displaystyle 0.75 = \frac{3}{4}\)
\(\displaystyle \text{Its reciprocal is } \frac{4}{3}\)
When a number is multiplied by its reciprocal, the result is always 1 (unity).
Examples:
\(\displaystyle 2 \times \frac{1}{2} = 1\)
\(\displaystyle 10 \times \frac{1}{10} = 1\)
\(\displaystyle 100 \times \frac{1}{100} = 1\)
Reciprocals are very useful in division of fractions.
Example:
\(\displaystyle \left(\frac{2}{5}\right) \div \left(\frac{7}{5}\right) = \left(\frac{2}{5}\right) \times \left(\frac{5}{7}\right) = \frac{2}{7}\)
- \(\displaystyle \text{Reciprocal of } x \text{ is } \frac{1}{x}\)
- \(\displaystyle \text{Reciprocal of a fraction } \frac{x}{y} \text{ is } \frac{y}{x}\)
- \(\displaystyle \text{Reciprocal of } 2 \text{ is } \frac{1}{2}; \quad \text{reciprocal of } 9 \text{ is } \frac{1}{9}\)
- \(\displaystyle \text{Reciprocal of } \frac{3}{\frac{2}{3}} = \frac{9}{2}, \; \text{so reciprocal is } \frac{2}{9}\)
- \(\displaystyle \text{Reciprocal of } -\frac{5}{4} \text{ is } -\frac{4}{5}\)
- \(\displaystyle \text{Reciprocal of } 0.25 = \frac{1}{0.25} = 4\)
- \(\displaystyle \text{Reciprocal of } -45 \text{ is } -\frac{1}{45}\)
- Find the reciprocal of 29.
- \(\displaystyle \text{Find the reciprocal of } \frac{14}{15}\)
- \(\displaystyle \text{Find the reciprocal of } 1.25 \)
- \(\displaystyle \text{Find the reciprocal of } -80\)
- \(\displaystyle \text{Find the reciprocal of } ax^{2}\)
Conclusion
Reciprocals in math are a simple yet powerful concept that helps in solving equations, simplifying fractions, and understanding inverse relationships. By flipping a number or fraction upside down, you get its reciprocal, which when multiplied with the original number always equals 1. From fractions to decimals and even algebraic expressions, reciprocals play an important role in mathematics. Mastering this concept builds a strong foundation for higher-level math and everyday problem-solving.
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Ans: A reciprocal is the multiplicative inverse of a number. It is equal to 1 divided by that number.
Ans: \(\displaystyle \text{Interchange numerator and denominator. Example: Reciprocal of } \frac{3}{4} \text{ is } \frac{4}{3}.\)
Ans: \(\displaystyle \text{Convert to an improper fraction and then flip it. Example: } 2 \frac{3}{4} \;\to\; \frac{11}{4}, \;\text{reciprocal is } \frac{4}{11}.\)
Ans: No, zero does not have a reciprocal because division by zero is undefined.
Ans: \(\displaystyle \text{The reciprocal of infinity is } 0, \text{ since } \frac{1}{\infty} = 0.\)
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