Algebraic numbers

Definition

Algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients. This means that any number that can be expressed as a root of such equations, including both rational and certain irrational numbers, is considered algebraic. These numbers play a crucial role in number theory and help to bridge the gap between rational and irrational numbers.

5 Must Know Facts For Your Next Test

1. Every rational number is an algebraic number because it can be expressed as a solution to a simple polynomial equation, like x - a = 0.
2. The set of algebraic numbers is countable, which means there are infinitely many but they can be put into a one-to-one correspondence with the natural numbers.
3. Some famous examples of algebraic numbers include $$rac{1}{2}$$, $$rac{ ext{√2}}{1}$$, and the roots of equations like x² - 2 = 0.
4. While all rational numbers are algebraic, not all algebraic numbers are rational; for instance, $$ ext{√2}$$ is an irrational algebraic number.
5. Transcendental numbers like $$ ext{π}$$ and e are important in mathematics as they help define what algebraic numbers are not.

Review Questions

• How do algebraic numbers relate to both rational and irrational numbers?
  • Algebraic numbers encompass both rational and some irrational numbers. Rational numbers are a subset of algebraic numbers because they can be expressed as solutions to polynomial equations with integer coefficients. On the other hand, some irrational numbers, such as $$ ext{√2}$$, are also algebraic because they solve equations like x² - 2 = 0. Thus, while all rational numbers fit within the category of algebraic numbers, not all algebraic numbers are rational.
• What distinguishes algebraic numbers from transcendental numbers?
  • Algebraic numbers differ from transcendental numbers primarily in their relationship to polynomial equations. Algebraic numbers can be defined as roots of polynomial equations with integer coefficients, while transcendental numbers cannot be expressed as solutions to any such equation. For example, $$ ext{π}$$ and e are transcendental because there are no polynomial equations with integer coefficients that have them as solutions, highlighting a key distinction between these two categories of numbers.
• Evaluate the implications of the existence of algebraic versus transcendental numbers in the context of number theory.
  • The existence of both algebraic and transcendental numbers significantly impacts number theory by providing insights into the nature of different types of numbers. Algebraic numbers, being countable and capable of being described through polynomial relationships, allow mathematicians to explore various properties and relationships among them. In contrast, the existence of uncountably many transcendental numbers points to a vast complexity in the number system that goes beyond algebra. This duality encourages deeper investigations into how these types interact, leading to further developments in mathematical concepts such as field theory and Galois theory.

Related terms

Rational Numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

Irrational Numbers: Numbers that cannot be expressed as the quotient of two integers, meaning they cannot be written in the form a/b where a and b are integers.

Transcendental Numbers: Numbers that are not algebraic, meaning they cannot be solutions to any polynomial equation with integer coefficients.