Real Number System

Definition

The real number system is a complete ordered field that includes all the rational numbers (fractions) and irrational numbers (numbers that cannot be expressed as fractions, like $$\sqrt{2}$$ or $$\pi$$). This system allows for the arithmetic operations of addition, subtraction, multiplication, and division (except by zero), and supports concepts like limits and continuity, which are fundamental in analysis.

5 Must Know Facts For Your Next Test

1. The real number system is divided into rational and irrational numbers, with rational numbers being those that can be written as a fraction and irrational numbers being those that cannot.
2. The completeness property of the real number system ensures that any bounded set of real numbers has a supremum, which is essential for defining limits and ensuring convergence in analysis.
3. The real numbers are uncountably infinite, meaning they cannot be put into a one-to-one correspondence with the natural numbers, unlike the set of rational numbers which is countably infinite.
4. In the context of mathematical analysis, continuity is defined using the real number system, which requires the completeness property to make sure limits exist.
5. Every real number can be represented on the number line, which visually demonstrates their order and density; between any two distinct real numbers, there exists another real number.

Review Questions

• How does the completeness property of the real number system impact mathematical concepts such as limits and continuity?
  • The completeness property ensures that every non-empty set of real numbers that is bounded above has a least upper bound. This characteristic is crucial in defining limits, as it guarantees that sequences of real numbers converge to a specific limit within the real number system. Additionally, continuity relies on this property because it requires the existence of limits for functions at points within their domain, ensuring that small changes in input result in small changes in output.
• Discuss the differences between rational and irrational numbers and their significance within the real number system.
  • Rational numbers are those that can be expressed as fractions of two integers, while irrational numbers cannot be expressed this way. The significance lies in their contribution to the completeness of the real number system; together they form a continuum. This means that between any two rational numbers, there exist infinitely many irrational numbers, filling in gaps and allowing for a complete representation on the number line. This density supports many analytical concepts such as limits and continuity.
• Evaluate how the uncountability of the real number system contrasts with the countability of rational numbers and its implications for mathematical analysis.
  • The uncountability of the real number system means that there are more real numbers than natural numbers; this implies a richness in the structure of real analysis. While rational numbers are countably infinite, indicating that they can be listed or paired with natural numbers, irrational numbers fill in gaps on the number line without being countable. This difference highlights why certain properties such as limits and continuity can only be effectively addressed using real numbers, enabling deeper insights into calculus and functional analysis.

Related terms

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

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 a fraction of two integers; they have non-repeating, non-terminating decimal expansions.