Real numbers

Definition

Real numbers are the set of values that include all the rational and irrational numbers, representing quantities on a continuous number line. They encompass integers, fractions, and decimals, and can be either positive or negative, as well as zero. Real numbers are crucial in various mathematical contexts, especially when discussing limits, continuity, and completeness.

5 Must Know Facts For Your Next Test

1. Real numbers include both rational numbers, which can be written as a ratio of two integers, and irrational numbers, which cannot be expressed in such a way.
2. The real number line is a visual representation where every point corresponds to a unique real number, illustrating their density and continuity.
3. The concept of limits in calculus relies heavily on real numbers to define behavior as values approach a specific point.
4. Every real number can be approximated by rational numbers, showcasing the interplay between these two subsets of numbers.
5. Real numbers are essential for defining concepts such as intervals and continuity in functions, making them foundational in analysis.

Review Questions

• How do real numbers differ from rational and irrational numbers?
  • Real numbers encompass both rational and irrational numbers, meaning they include all integers, fractions, and decimals. Rational numbers can be expressed as fractions of integers, while irrational numbers cannot be represented as simple fractions and have non-terminating decimal expansions. Understanding these distinctions is important when analyzing properties of different types of numbers within the broader real number system.
• Explain how the completeness axiom applies to real numbers and why it is significant in mathematics.
  • The completeness axiom asserts that every non-empty set of real numbers that is bounded above has a least upper bound or supremum. This property ensures that there are no 'gaps' in the real number system, making it complete and allowing for rigorous analysis in calculus and other fields. The significance lies in its role in guaranteeing that limits exist for sequences of real numbers, which is fundamental for calculus.
• Analyze the importance of real numbers in the context of limits and continuity in calculus.
  • Real numbers play a critical role in calculus, particularly regarding limits and continuity. The behavior of functions as they approach certain points relies on the properties of real numbers to define limits accurately. Continuous functions must have no gaps in their range over intervals of real numbers, ensuring that small changes in input result in small changes in output. This connection underlines why understanding real numbers is essential for studying calculus and mathematical analysis.

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 simple fraction; they have non-repeating, non-terminating decimal expansions.

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