Real numbers

Definition

Real numbers are all the numbers that can be found on the number line, including both rational and irrational numbers. They encompass integers, fractions, and decimals, providing a comprehensive system for measurement and calculation. The real number system is crucial for various mathematical concepts, allowing operations such as addition, subtraction, multiplication, and division.

5 Must Know Facts For Your Next Test

1. Real numbers can be classified into two main categories: rational and irrational numbers, with rational numbers being expressible as fractions and irrational numbers not.
2. The set of real numbers is often represented by the symbol ℝ and includes positive and negative numbers, zero, and all decimal expansions.
3. Real numbers are complete in that they do not have any gaps; between any two real numbers, there exists another real number.
4. Every real number can be represented on a number line, which provides a visual representation of their order and spacing.
5. Real numbers play a key role in defining fields in mathematics, particularly because they allow operations such as addition and multiplication to be performed without leaving the set of real numbers.

Review Questions

• How do real numbers differ from rational and irrational numbers in terms of their properties?
  • Real numbers include both rational and irrational numbers, which differentiates them from each other. Rational numbers can be expressed as fractions with integer numerators and non-zero integer denominators. In contrast, irrational numbers cannot be expressed as such fractions; their decimal expansions are non-repeating and non-terminating. Thus, while all rational and irrational numbers fall under the umbrella of real numbers, the key distinction lies in their ability to be expressed as ratios of integers.
• Discuss how the concept of cardinality relates to the set of real numbers compared to the set of natural numbers.
  • Cardinality refers to the size of a set, and when comparing the cardinality of real numbers to natural numbers, we find that the set of real numbers has a greater cardinality. While natural numbers are countable (they can be put in one-to-one correspondence with the positive integers), real numbers are uncountable due to their density on the number line. Between any two natural numbers, there are infinitely many real numbers; hence, the set of real numbers is considered to have a higher cardinality than the set of natural numbers.
• Evaluate how real numbers contribute to forming fields in mathematics and their importance in mathematical operations.
  • Real numbers form a field because they allow for operations such as addition, subtraction, multiplication, and division (except by zero) to be performed while still resulting in real numbers. This property is essential for creating a structured mathematical environment where solutions to equations exist within the same set. The completeness of real numbers further ensures that limits exist for sequences and functions, making them fundamental for calculus and analysis. Understanding real numbers is crucial for mathematicians as they provide a foundation for more advanced concepts in various branches of mathematics.

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, having non-repeating, non-terminating decimal expansions.

Cardinality:

A concept in set theory that refers to the number of elements in a set, which helps distinguish between different sizes of infinity.