Rationals

Definition

Rationals are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. This category includes fractions, whole numbers, and terminating or repeating decimals, making them an essential part of number theory and mathematical operations. Understanding rationals helps in comparing sizes of numbers and working with mathematical concepts such as cardinality, where the sizes of different sets can be analyzed through the lens of rational numbers.

5 Must Know Facts For Your Next Test

1. Rational numbers can be represented on a number line, with each point corresponding to a specific rational value.
2. Every integer is also a rational number because it can be expressed as a fraction (e.g., 5 can be written as 5/1).
3. The set of rational numbers is denser than the integers, meaning that between any two rational numbers, there exists another rational number.
4. Rational numbers can be added, subtracted, multiplied, and divided (except by zero), following specific arithmetic rules.
5. The cardinality of the set of rational numbers is countably infinite, which means they can be put into one-to-one correspondence with the natural numbers.

Review Questions

• How do rationals fit into the larger context of different types of numbers, such as integers and irrationals?
  • Rationals are a subset of real numbers that include all integers and fractions. While integers are whole numbers without fractions or decimals, rationals encompass both whole numbers and fractions. In contrast, irrationals cannot be expressed as fractions and have decimal expansions that neither terminate nor repeat. This relationship highlights how rationals bridge the gap between whole numbers and more complex number types.
• Discuss how the concept of cardinality applies to rational numbers compared to other number sets.
  • Cardinality helps categorize sets by determining their size or number of elements. The set of rational numbers has a countably infinite cardinality, which means it can be matched one-to-one with natural numbers. In comparison, irrational numbers have an uncountably infinite cardinality, indicating that their size is larger than that of the rationals. This distinction emphasizes important aspects of mathematical theory regarding different infinities and how we understand size in various sets.
• Evaluate the importance of understanding rational numbers in relation to mathematical concepts such as operations and comparisons.
  • Understanding rational numbers is crucial for performing mathematical operations like addition, subtraction, multiplication, and division. Since rationals can represent fractions and whole values, this understanding allows for comparisons in magnitude across different contexts. Additionally, grasping how to manipulate rational expressions leads to better problem-solving skills and deeper insights into number theory. Ultimately, rationality forms a foundation upon which more advanced mathematical concepts build.

Related terms

Integers: Integers are whole numbers that can be positive, negative, or zero but do not include fractions or decimals.

Irrationals: Irrational numbers are numbers that cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal expansions.

Cardinality: Cardinality refers to the number of elements in a set, which can help distinguish between different types of infinity and the sizes of various sets of numbers.