The set of real numbers

Definition

The set of real numbers is a collection that includes all the rational and irrational numbers, which can be represented on the number line. This set encompasses integers, fractions, and non-repeating decimals, providing a comprehensive framework for measuring continuous quantities and addressing many mathematical problems.

5 Must Know Facts For Your Next Test

1. The set of real numbers is denoted by the symbol ℝ and includes both positive and negative numbers, zero, and all types of decimals.
2. Real numbers can be represented as points on an infinite number line, where each point corresponds to exactly one real number.
3. The set of real numbers is uncountably infinite, meaning there is no one-to-one correspondence between the real numbers and the natural numbers.
4. Every real number can be classified as either rational or irrational based on whether it can be expressed as a fraction.
5. Cardinal numbers help compare different sizes of sets; for instance, the cardinality of the set of real numbers is greater than that of the set of natural numbers.

Review Questions

• How do rational and irrational numbers relate to the set of real numbers?
  • Rational and irrational numbers are both subsets of the set of real numbers. Rational numbers can be expressed as fractions where both numerator and denominator are integers, while irrational numbers cannot be expressed this way. Together, these subsets form the entirety of real numbers, making them essential for understanding continuous values and measurements in mathematics.
• What implications does the uncountability of the set of real numbers have in comparison to cardinal numbers?
  • The uncountability of the set of real numbers implies that its cardinality is greater than that of the natural numbers. While natural numbers can be listed in a sequence (countable), real numbers fill all gaps on the number line without any such sequence. This distinction illustrates different types of infinity and shows how infinite sets can vary significantly in size.
• Discuss how the concept of cardinality helps in understanding the differences between various sets, especially in relation to the set of real numbers.
  • Cardinality refers to the measure of the 'number of elements' in a set. In comparing sets like natural numbers and real numbers, cardinality reveals that while natural numbers are countable (finite or infinite but listable), the set of real numbers is uncountably infinite. This concept provides insight into different infinities; for example, Cantor's diagonal argument demonstrates that no matter how you attempt to list all real numbers, some will always remain unlisted, highlighting their unique cardinality compared to other sets.

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 cannot be written as a ratio of two integers.

Cardinal Number: A number that expresses quantity or size, which is used to compare the sizes of sets, including infinite sets.