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➕pre-algebra review

key term - Algebraic Numbers

Citation:

Definition

Algebraic numbers are the roots of polynomial equations with integer coefficients. They include both rational numbers and irrational numbers that can be expressed as solutions to algebraic equations.

5 Must Know Facts For Your Next Test

  1. Algebraic numbers include both rational and irrational numbers that are solutions to polynomial equations with integer coefficients.
  2. The set of algebraic numbers is a subset of the real numbers and is countable, while the set of real numbers is uncountable.
  3. Every rational number is an algebraic number, as it is the root of a linear polynomial equation with integer coefficients.
  4. Irrational numbers like π and √2 are also algebraic numbers, as they are the roots of polynomial equations with integer coefficients.
  5. The set of algebraic numbers is closed under the basic arithmetic operations of addition, subtraction, multiplication, and division.

Review Questions

  • Explain the relationship between rational numbers, irrational numbers, and algebraic numbers.
    • Rational numbers and irrational numbers are both subsets of the set of algebraic numbers. All rational numbers are algebraic numbers, as they are the roots of linear polynomial equations with integer coefficients. Irrational numbers, such as π and √2, are also algebraic numbers, as they are the roots of polynomial equations with integer coefficients. However, not all algebraic numbers are rational; the set of algebraic numbers includes both rational and irrational numbers that are solutions to polynomial equations with integer coefficients.
  • Describe the properties of the set of algebraic numbers.
    • The set of algebraic numbers has several important properties. First, it is a subset of the real numbers and is countable, meaning the elements can be put in a one-to-one correspondence with the natural numbers. Second, the set of algebraic numbers is closed under the basic arithmetic operations of addition, subtraction, multiplication, and division. This means that the result of any of these operations on two algebraic numbers will also be an algebraic number. Finally, the set of algebraic numbers is dense in the real number line, meaning that between any two real numbers, there exists an algebraic number.
  • Analyze the significance of algebraic numbers in the context of rational and irrational numbers.
    • Algebraic numbers play a crucial role in understanding the relationship between rational and irrational numbers. By definition, all rational numbers are algebraic numbers, as they are the roots of linear polynomial equations with integer coefficients. However, not all algebraic numbers are rational; the set of algebraic numbers also includes irrational numbers that are the roots of higher-degree polynomial equations with integer coefficients. This distinction highlights the fact that the set of algebraic numbers is a broader and more inclusive set than the set of rational numbers, and it provides a unifying framework for understanding both rational and irrational numbers as solutions to polynomial equations.