Real numbers

Definition

Real numbers are the set of numbers that include all the rational and irrational numbers, which can be represented on the number line. They encompass integers, whole numbers, fractions, and decimals, providing a complete framework for expressing quantities and measuring continuous values. This broad set is fundamental in various mathematical contexts, particularly in understanding functions and their properties.

5 Must Know Facts For Your Next Test

1. Real numbers include positive and negative values, zero, and can be represented in decimal form, both terminating and non-terminating.
2. The set of real numbers is complete, meaning that there are no 'gaps' on the number line; every interval contains real numbers.
3. Real numbers can be used to model polynomial functions, which are expressed as sums of terms involving real number coefficients raised to non-negative integer powers.
4. The concept of limits in calculus relies heavily on the properties of real numbers, enabling precise definitions of continuity and convergence.
5. In the context of fields, real numbers form a field under standard addition and multiplication operations, satisfying all field properties.

Review Questions

• How do real numbers relate to polynomial functions and their properties?
  • Real numbers are essential in defining polynomial functions, as these functions consist of terms with real coefficients raised to non-negative integer powers. The behavior of these polynomial functions is closely linked to the properties of real numbers, such as continuity and the existence of roots. For example, the Fundamental Theorem of Algebra states that every non-constant polynomial function with real coefficients has at least one real root or complex roots that can also be represented using real numbers.
• Discuss the significance of completeness in relation to real numbers and how it impacts mathematical analysis.
  • Completeness is a crucial property of real numbers that ensures every bounded set has a least upper bound, which allows mathematicians to avoid gaps in their analysis. This completeness underpins many concepts in mathematical analysis, such as limits, continuity, and convergence. For instance, the ability to define limits relies on knowing that every sequence of real numbers that is bounded above converges to a limit within the set of real numbers.
• Evaluate how the structure of real numbers as a field impacts their use in algebraic operations and mathematical modeling.
  • As a field, real numbers support both addition and multiplication operations that adhere to certain properties such as associativity, commutativity, distributivity, and the existence of multiplicative inverses (for non-zero elements). This field structure allows for robust algebraic manipulations and simplifies many mathematical models. For example, when solving equations or working with functions in calculus, the ability to manipulate real number expressions freely helps facilitate deeper analysis in various scientific applications.

Related terms

Rational Numbers:

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

Irrational Numbers: Numbers that cannot be expressed as a simple fraction; their decimal representation goes on forever without repeating.

Completeness: A property of real numbers stating that every non-empty set of real numbers that is bounded above has a least upper bound (supremum).