The Only Negative Rational Number Equal to Its Reciprocal

When dealing with rational numbers, a common question arises: what is the only negative rational number which is equal to its reciprocal? Let us explore this concept in detail, proving why the answer is indeed -1.

Understanding the Reciprocal Concept

A reciprocal of a number (x) is defined as (frac{1}{x}). If a rational number (N) is equal to its reciprocal, we have:

(N frac{1}{N})

By multiplying both sides by (N), we get:

(frac{1}{N} cdot N frac{1}{N} cdot N Rightarrow 1 N^2)

This equation, (N^2 1), has two solutions: (N 1) and (N -1). However, since we are interested in the negative solution, we conclude that the only negative rational number which is equal to its reciprocal is (-1).

General Rational Number Case

For a more general case, let's consider (N frac{a}{b}), where (a) and (b) are integers, and (b eq 0). If (N) is equal to its reciprocal, then:

(frac{a}{b} frac{b}{a})

Multiplying both sides by (b times a), we get:

(frac{a}{b} cdot b cdot a frac{b}{a} cdot b cdot a Rightarrow a^2 b^2)

This simplifies to:

(a^2 b^2)

Therefore, (a b) or (a -b). Since we are looking for the negative solution, we have:

(a -b)

Substituting back, we get:

(frac{a}{b} frac{-b}{b} -1)

This confirms that the only negative rational number which is equal to its reciprocal is indeed -1.

Conclusion

In summary, the only negative rational number which is equal to its reciprocal is (-1). This result is derived from the fundamental properties of rational numbers and their reciprocals. Whether we use the specific form (N frac{1}{N}) or the general form (N frac{a}{b}), the solution consistently leads to (-1).

Additional Insights

The concept of reciprocals is not limited to rational numbers. It extends to real numbers, where the reciprocal of a non-zero real number (x) is (frac{1}{x}). However, the only rational number that is both negative and equal to its reciprocal is -1.

Keywords

- Negative rational number - Reciprocal - Rational number