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➕pre-algebra review

Real Number

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A real number is a number that can be represented on the number line. It includes all rational numbers (fractions and integers) as well as irrational numbers (such as pi and the square root of 2), and encompasses the complete set of numbers that we use in everyday life and mathematics.

5 Must Know Facts For Your Next Test

  1. Real numbers include both rational and irrational numbers, forming a complete, continuous set of numbers.
  2. Rational numbers can be expressed as fractions or terminating/repeating decimals, while irrational numbers have non-repeating, non-terminating decimal representations.
  3. Every real number has a unique position on the number line, which extends infinitely in both the positive and negative directions.
  4. Operations such as addition, subtraction, multiplication, and division can be performed on real numbers, following the rules of arithmetic.
  5. Real numbers are the foundation for many mathematical concepts and are essential for understanding and solving a wide range of problems in various fields.

Review Questions

  • Explain the key differences 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 3/4 or -7/2. They include all integers and fractions. Irrational numbers, on the other hand, are numbers that cannot be expressed as a simple fraction, and have non-repeating, non-terminating decimal representations. Examples of irrational numbers include pi (π), the square root of 2, and the golden ratio. While rational numbers can be represented as fractions or terminating/repeating decimals, irrational numbers have decimal expansions that go on forever without repeating.
  • Describe the relationship between real numbers and the number line, and explain how this visual representation helps in understanding the properties of real numbers.
    • The number line is a visual representation of the set of real numbers, which includes both rational and irrational numbers. Every real number has a unique position on the number line, which extends infinitely in both the positive and negative directions. This visual representation helps in understanding the properties of real numbers, such as their order (greater than, less than, or equal to), the distance between them, and the operations that can be performed on them. The number line also illustrates the continuity of the real number system, as there are no gaps or jumps between the numbers, allowing for a seamless representation of all real numbers.
  • Analyze the significance of real numbers in mathematics and everyday life, and explain how a deep understanding of this concept can contribute to problem-solving in various contexts.
    • Real numbers are the foundation for many mathematical concepts and are essential for understanding and solving a wide range of problems in various fields. In mathematics, real numbers are the basis for calculus, algebra, and other advanced mathematical disciplines. In everyday life, real numbers are used to represent quantities, measurements, and values in a variety of contexts, such as finance, engineering, physics, and more. A deep understanding of real numbers, their properties, and their relationships with other mathematical concepts can greatly enhance one's ability to solve complex problems, make accurate calculations, and model real-world phenomena effectively. By grasping the nuances of real numbers, students can develop stronger problem-solving skills, critical thinking abilities, and a more comprehensive mathematical foundation that can be applied across diverse domains.