Real numbers

Definition

Real numbers are a set of values that include all the rational and irrational numbers, essentially representing all the points on an infinite number line. This set allows for a complete representation of quantities and measurements, connecting arithmetic with geometric concepts. Real numbers play a crucial role in understanding size comparisons between different sets, their uncountability, and various mathematical properties such as continuity and ordering.

5 Must Know Facts For Your Next Test

1. Real numbers include both rational numbers (like 1/2 and 3) and irrational numbers (like √2 and π).
2. The set of real numbers is uncountable, meaning it cannot be put into a one-to-one correspondence with the set of natural numbers.
3. Real numbers can be used to define concepts like limits, continuity, and derivatives in calculus.
4. Cantor's diagonal argument demonstrates that there are more real numbers between any two given real numbers than there are natural numbers.
5. The properties of real numbers support the well-ordering theorem, which states that every non-empty set of real numbers has a least element.

Review Questions

• How does the concept of real numbers relate to Cantor's theorem and the comparison of set sizes?
  • Cantor's theorem shows that the set of real numbers is uncountable, which means it cannot be matched one-to-one with the natural numbers. This indicates that there are different 'sizes' or cardinalities of infinity. Real numbers fill every point on an infinite line without gaps, highlighting their complexity compared to countable sets. This understanding helps illustrate how infinite sets can vary in size.
• Discuss the significance of Cantor's diagonalization argument in demonstrating the uncountability of real numbers.
  • Cantor's diagonalization argument illustrates that no list can contain all real numbers by constructing a new real number not found in any given list. By altering the nth digit of the nth number in a sequence, Cantor proves that for any enumeration of real numbers, there will always be at least one real number missing. This establishes the uncountability of the reals compared to the countable infinity of natural numbers, showing a deeper understanding of set theory.
• Evaluate how the properties of real numbers connect to Zorn's lemma and the well-ordering theorem.
  • Zorn's lemma states that if every chain in a partially ordered set has an upper bound, then the whole set has at least one maximal element. The properties of real numbers support this principle through their completeness and order structure. Since every non-empty subset of real numbers has a least element (well-ordering), this further strengthens the application of Zorn's lemma within analysis and other mathematical contexts, providing powerful tools for proofs in various areas.

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 a simple fraction, meaning they have non-repeating and non-terminating decimal expansions.

Cardinality:

A measure of the 'size' of a set, representing how many elements are in that set, which helps to compare sizes of different sets.