Algebraic Numbers

Definition

Algebraic numbers are numbers that are roots of non-zero polynomial equations with integer coefficients. This means they can be expressed as solutions to such polynomial equations, highlighting their importance in understanding the structure of numbers and the properties of polynomial functions. Algebraic numbers include all rational numbers and many irrational numbers, playing a crucial role in the completeness of number systems.

5 Must Know Facts For Your Next Test

1. Every rational number is an algebraic number because it can be the root of a polynomial equation with integer coefficients, specifically of degree one.
2. Algebraic numbers can be classified into two categories: algebraic integers, which are roots of monic polynomials (leading coefficient is 1), and non-integers.
3. The set of algebraic numbers is countable, while the set of real numbers is uncountable, showing that there are many more real numbers than algebraic ones.
4. Algebraic closure refers to the property of extending a field such that every non-constant polynomial equation has at least one root in that field; this is significant for understanding algebraic numbers.
5. The completeness property states that every non-empty set of real numbers that is bounded above has a least upper bound, connecting with algebraic numbers as they often serve as critical examples in discussions about limits and bounds.

Review Questions

• How do algebraic numbers relate to rational and irrational numbers within the context of polynomial equations?
  • Algebraic numbers include both rational and certain irrational numbers. Rational numbers can be roots of linear polynomials like $ax + b = 0$, while irrational algebraic numbers can be roots of higher degree polynomials. This connection illustrates how algebraic numbers serve as solutions to polynomial equations, showing their fundamental relationship to both rational and irrational numbers.
• Discuss the significance of algebraic closure in relation to algebraic numbers and polynomial functions.
  • Algebraic closure is significant because it ensures that every non-constant polynomial function has a root within an extended field of algebraic numbers. This concept highlights how the structure of algebraic numbers allows for the existence of solutions to polynomial equations, which is essential for mathematical consistency and completeness. It shows that while real numbers may exist outside this closure, understanding algebraic numbers aids in solving polynomials effectively.
• Evaluate the implications of the countability of algebraic numbers compared to the uncountability of real numbers in terms of their mathematical importance.
  • The countability of algebraic numbers implies that they are relatively sparse compared to the uncountable set of real numbers. This contrast highlights the unique role algebraic numbers play in mathematical analysis and number theory. The fact that there are infinitely many more real numbers than algebraic ones emphasizes the richness and complexity of the real number system, leading to discussions about completeness and limits that are essential in higher mathematics.

Related terms

Rational Numbers:

Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers; they have non-repeating, non-terminating decimal expansions.

Polynomial Functions: Functions that can be expressed in the form of a polynomial, which is a sum of terms consisting of a variable raised to a non-negative integer power multiplied by a coefficient.