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Chapter 1: Problem 57

Determine the reciprocal of the following numbers. $$ 213 $$

Short Answer

Expert verified
The reciprocal of 213 is \( \frac{1}{213} \).

Step by step solution

01

Understand the Concept of Reciprocal

The reciprocal of a given number is 1 divided by that number. If you have a number \( x \), its reciprocal is \( \frac{1}{x} \). It is essential to understand that reciprocals are multiplicative inverses, meaning if you multiply a number by its reciprocal, you get 1.
02

Set Up the Expression

Since we need to find the reciprocal of 213, we set up the expression as \( \frac{1}{213} \). This expression represents the reciprocal of 213.
03

Simplify, If Necessary

Check if the expression can be simplified. In this case, \( \frac{1}{213} \) is already in its simplest form because 213 cannot be simplified further when it is in the denominator of a fraction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplicative Inverse
The multiplicative inverse is a fascinating concept in mathematics. It is a number which, when multiplied by the original number, results in a product of 1. For any real number, its multiplicative inverse is simply its reciprocal. To find the multiplicative inverse of a number is straightforward:
  • If the number is an integer or whole number, like 7, its multiplicative inverse is \( \frac{1}{7} \).
  • If the number is already a fraction, like \( \frac{2}{3} \), then its multiplicative inverse flips the numerator and the denominator, resulting in \( \frac{3}{2} \).
This concept is particularly helpful in solving equations, as multiplying a term by its inverse simplifies the equation to hold a product of one.
Simplifying Fractions
Simplifying fractions is an essential skill in math. It makes fractions more manageable by reducing them to their simplest form. The fraction is simplified when the numerator and denominator have no common divisors other than 1. To simplify a fraction:
  • Determine the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and denominator by this GCD.
For example, the fraction \( \frac{15}{35} \) can be simplified. First, find the GCD of 15 and 35, which is 5. Then, divide both by 5, resulting in the simplified fraction \( \frac{3}{7} \). Simplifying fractions can often uncover the essence of the problem, making calculations much simpler.
Understanding Reciprocals
Reciprocals are fundamental to various mathematical operations and understanding this concept can make many topics easier. A reciprocal is merely a flipped version of the original number in a fractional form. For instance:
  • The reciprocal of a whole number, say 4, is \( \frac{1}{4} \).
  • The reciprocal of a fraction, \( \frac{5}{6} \), is \( \frac{6}{5} \).
When you multiply a number by its reciprocal, the result will always be 1. This special relationship is crucial for division in algebra and handling fractions in problems. Understanding and being able to find reciprocals quickly makes solving equations that involve fractions a breeze.

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