Algebraic Number

Definition

An algebraic number is a complex number that is a root of a non-zero polynomial equation with rational coefficients. This concept connects deeply with the study of numbers, including integers, rational numbers, and certain types of irrational numbers, and provides insight into their relationships through polynomial equations.

5 Must Know Facts For Your Next Test

1. Algebraic numbers include all integers and rational numbers, which are roots of linear polynomials, as well as many irrational numbers like $$\sqrt{2}$$, which is a root of $$x^2 - 2 = 0$$.
2. The set of algebraic numbers is countably infinite, while the set of all real numbers is uncountably infinite, showing that there are many more real numbers than algebraic ones.
3. Not all roots of polynomial equations are algebraic; for example, the number $$\pi$$ and $$e$$ are transcendental numbers because they are not roots of any polynomial with rational coefficients.
4. The field of algebraic numbers plays a key role in number theory, particularly in understanding the properties and relationships between different kinds of numbers.
5. Algebraic numbers can be used to construct field extensions, which are important for solving polynomial equations and studying their properties.

Review Questions

• How does the definition of an algebraic number relate to other types of numbers like rational and irrational numbers?
  • An algebraic number is defined as any complex number that is a root of a polynomial equation with rational coefficients. This definition encompasses rational numbers, which can be expressed as roots of linear equations, and certain irrational numbers such as $$\sqrt{2}$$ that are roots of quadratic equations. Thus, while all rational numbers are algebraic, not all algebraic numbers are rational, highlighting the broader category that includes specific irrational numbers.
• In what ways do polynomial equations help us understand algebraic numbers and their properties?
  • Polynomial equations serve as the foundation for identifying algebraic numbers since these numbers are defined as the roots of such equations with rational coefficients. By analyzing different types of polynomials, mathematicians can categorize algebraic numbers into various classes based on their degrees and properties. This analysis aids in understanding how these numbers interact with each other and their relationships within the wider realm of number theory.
• Evaluate the implications of having more real numbers than algebraic numbers on mathematical theories and practices.
  • The fact that there are uncountably more real numbers than algebraic numbers has significant implications in mathematics. It indicates that while algebraic methods can provide insights into many numerical phenomena, they cannot capture all aspects of real-number analysis due to the existence of transcendental and irrational numbers. This distinction pushes mathematicians to explore broader frameworks like calculus and topology to accommodate these non-algebraic entities, ultimately enriching mathematical theory and expanding our understanding of numerical systems.

Related terms

Rational Number: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

Irrational Number: An irrational number is a number that cannot be expressed as a simple fraction, meaning it cannot be the root of any polynomial equation with rational coefficients.

Polynomial Equation: A polynomial equation is a mathematical expression involving variables and coefficients, typically written in the form of an equation set to zero.