What Is the Reciprocal of a Number?

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In mathematics, there are several classifications of numbers such as fractional, prime, even and odd. Reciprocal numbers are a classification in which the the number is the opposite of the primary number given. These are also called multiplicative inverse numbers, and despite the long name, they are easy to identify.

The Product of 1

A reciprocal number is a number that, when multiplied against the primary number, will result in the product 1. This reciprocal is often considered a reverse of the number. For instance the reciprocal of 3 is 1/3. When 3 is multiplied by 1/3, the answer the is 1 because any number divided by itself equals 1. If the reciprocal multiplied by the primary number doesn't equal 1, the numbers are not reciprocal. The only number that cannot have a reciprocal is 0. This is because any number multiplied by 0 is 0; you cannot get a 1.


Generally, the most direct way to identify the reciprocal number is to turn the first number into a fraction. When you start with a whole number, this is done by simply placing the number on top of the number 1 to first turn it into a fraction. As all numbers divided by the number 1 are the primary number itself, this fraction is exactly the same as the primary number. For example, 8 = 8/1. You them flip the fraction: 8/1 flipped over is 1/8. By multiplying these two fractions you now have the product 1. In the example, 8/1 multiplied by 1/8 yields 8/8, which simplifies to 1.

Mixed Numbers

The reciprocal of the mixed number is also the opposite or reverse of the fraction, but in mixed numbers, another step is needed to obtain the goal product of 1. To identify the reciprocal of a mixed number you must first turn that number into a fraction with no whole numbers. For instance the number 3 1/8 would be converted to 25/8 to then find the reciprocal of 8/25. Multiplying 25/8 by 8/25 yields 200/200, simplified to 1.

Uses in Math

Reciprocal numbers are often used to get rid of a fraction in an equation that contains an unknown variable, making it easier to solve. It is also used to divide a fraction by another fraction. For example is you wanted to divide 1/2 by 1/3, you would flip the 1/3 and multiply the two numbers for an answer of 3/2, or 1 1/2. They are also used in more exotic computations. For example, reciprocal numbers are used in a number of manipulations of Fibonacci's sequence and golden ratio.

Practical Uses of Reciprocals

Reciprocal numbers allow a machine to multiply to get an answer, instead of dividing, because dividing is a slower process. Reciprocal numbers are used extensively in computer science. Reciprocal numbers facilitate conversions from one dimension to another. This is useful in construction, for example, where a paving product might be sold in quantities of cubic meters, but your measurements are in cubic feet or cubic yards.


About the Author

Mary Green began writing professionally in 2003. She has written for websites such as The At Homer and PPC Mom. She primarily writes about marketing, small-business tactics and the psychology of sales. Green attended Mohawk Valley Community College and studied psychology.

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