Understanding the Definition of Real Numbers and Polynomial Roots

Mathematics, a discipline that spans from basic arithmetic to advanced theorems, categorizes numbers into various sets. One of the most fundamental and extensive sets is that of real numbers. This article delves into the definition of real numbers and explores the conditions under which real numbers can be roots of polynomial equations with rational coefficients.

What Are Real Numbers?

The definition of a real number is as follows: real numbers consist of a complete ordered field that is vast and includes both rational and irrational numbers. Essentially, every real number can be represented as a decimal, possibly an infinitely long one. Rational numbers, which are fractions, either terminate (such as 1/2 0.5) or repeat (such as 1/3 0.3333). In contrast, irrational numbers are characterized by a decimal expansion that is infinitely long and non-repeating.

Algebraic and Transcendental Numbers

Within the set of real numbers, two specific subsets stand out: algebraic and transcendental numbers. Algebraic numbers are those that can be roots of polynomial equations with rational coefficients. For example, consider the polynomial equation (p(x) 0); any real number (x) that satisfies this equation is an algebraic number.

Transcendental numbers, on the other hand, are those values which are not the roots of any algebraic equation with rational coefficients. This means that transcendental numbers include values like (e) and (pi), which cannot be solutions to any polynomial equation with rational coefficients. An important distinction to note is that all transcendental numbers are irrational, whereas not all irrational numbers are transcendental. A classic example is (sqrt{2}), which, although irrational, is algebraic because it satisfies the equation (x^2 - 2 0).

Polynomial Roots and Rational Coefficients

The nature of the roots of polynomial equations with rational coefficients is closely tied to the concept of algebraic numbers. Specifically, any real number that is a root of a polynomial equation with rational coefficients is an algebraic number. This implies that such roots must satisfy a polynomial equation with rational numbers as its coefficients. Conversely, any number that is not a root of such a polynomial is considered transcendental.

To illustrate, consider a polynomial equation (a_nx^n a_{n-1}x^{n-1} cdots a_1x a_0 0), where (a_i) (for (i 0, 1, ldots, n)) are rational numbers. If there exists at least one real number (x) that solves this equation, that number is classified as an algebraic number. For example, the number (sqrt{2}) is a solution to the polynomial equation (x^2 - 2 0), and thus, it is considered an algebraic number.

Conclusion

In summary, real numbers encompass a broad range of mathematical entities, from the easily recognizable rational numbers to the more mysterious irrational numbers. The distinction between algebraic and transcendental numbers within the set of real numbers provides a deeper understanding of the structure and properties of these numbers. Recognizing the conditions under which real numbers are roots of polynomial equations with rational coefficients is crucial for grasping the intricate connections between algebra and number theory.