Real Numbers

Definition

Real numbers are a comprehensive set of numbers that include all rational and irrational numbers. They represent the complete number line and are the foundation for most mathematical operations and concepts, including absolute value functions.

5 Must Know Facts For Your Next Test

1. Real numbers include both rational and irrational numbers, and they represent the complete number system.
2. The set of real numbers is denoted by the symbol $\mathbb{R}$, and it includes all numbers that can be plotted on the number line.
3. Absolute value functions, which are a key topic in 1.6, rely on the properties of real numbers to define the distance of a number from zero.
4. Operations such as addition, subtraction, multiplication, and division can be performed on real numbers, following the rules and properties of the real number system.
5. Understanding the properties of real numbers, such as ordering, density, and completeness, is crucial for working with absolute value functions and other mathematical concepts.

Review Questions

• Explain the difference between rational and irrational numbers, and provide examples of each.
  • Rational numbers are numbers that can be expressed as a ratio of two integers, such as fractions, whole numbers, and terminating or repeating decimals. Examples of rational numbers include 1/2, 3, and 0.375. Irrational numbers, on the other hand, are numbers that cannot be expressed as a ratio of two integers, such as $\pi$, $\sqrt{2}$, and non-repeating, non-terminating decimals. Irrational numbers have decimal representations that go on forever without repeating a pattern.
• Describe how the properties of real numbers are important for understanding absolute value functions.
  • Absolute value functions rely on the properties of real numbers to define the distance of a number from zero. The absolute value of a real number is the distance between that number and zero on the number line. Understanding the ordering and completeness of the real number system is crucial for working with absolute value functions, as it allows you to determine the magnitude of a number regardless of its sign. Additionally, the density of real numbers ensures that there are infinitely many real numbers between any two given real numbers, which is important for analyzing the behavior of absolute value functions.
• Analyze how the set of real numbers is related to the concept of a number line, and explain how this relationship is essential for working with absolute value functions.
  • The set of real numbers is directly related to the number line, as each point on the number line corresponds to a unique real number. This visual representation of the real number system is essential for understanding absolute value functions, which measure the distance of a number from zero on the number line. The number line allows you to clearly see the relationship between a real number and its absolute value, as the absolute value represents the distance of the number from the origin (0) on the number line. This understanding of the real number system and its geometric representation on the number line is crucial for analyzing the properties and behavior of absolute value functions.

Related terms

Rational Numbers: Rational numbers are numbers that can be expressed as a ratio of two integers, such as fractions, whole numbers, and terminating or repeating decimals.

Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a ratio of two integers, such as $\pi$, $\sqrt{2}$, and non-repeating, non-terminating decimals.

Number Line: The number line is a visual representation of the set of real numbers, where each point on the line corresponds to a unique real number.