Nombres réels - partie 1 : nombres rationnels

Introduction

This tutorial explores the concept of rational numbers as part of the broader topic of real numbers. Understanding rational numbers is essential for students studying mathematics, as they form a foundational aspect of number theory. This guide will break down the key concepts presented in the video, including decimal representations and the distinction between rational and irrational numbers.

Step 1: Understanding Rational Numbers

Rational numbers are defined as numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.

• Examples of rational numbers include:
  • Fractions like 1/2, -3/4, and 5/1 (which is 5).
  • Whole numbers, since they can be expressed as a fraction (e.g., 3 = 3/1).
  • Terminating decimals (e.g., 0.75, which equals 3/4).
  • Repeating decimals (e.g., 0.333..., which equals 1/3).

Step 2: Decimal Representation of Rational Numbers

Rational numbers can be represented in decimal form, either as terminating or repeating decimals.

• Terminating Decimals: These decimals have a finite number of digits after the decimal point.

  • Example: 0.5 is a terminating decimal because it can be expressed as 1/2.

• Repeating Decimals: These decimals have one or more digits that repeat infinitely.

  • Example: 0.666... can be written as 2/3.

Step 3: Identifying Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction. They have non-repeating and non-terminating decimal expansions.

• Example of an irrational number:
  • The square root of 2 (√2) is an irrational number because it cannot be expressed as a fraction. Its decimal representation is approximately 1.41421356..., which continues indefinitely without repeating.

Step 4: Practical Applications of Rational Numbers

Understanding rational numbers is crucial in various real-world scenarios:

• Finance: Calculating interest rates, budgets, and investments often involves rational numbers.
• Measurements: In construction and engineering, precise measurements are often expressed as fractions.
• Statistics: Many statistical calculations rely on rational numbers for averages and probabilities.

Conclusion

In this tutorial, we covered the definition and properties of rational numbers, how to identify them through decimal representations, and distinguished them from irrational numbers. To further your understanding, practice converting fractions to decimal form and identifying whether given numbers are rational or irrational. For additional exercises and resources, consider visiting the Exo7Math website or other educational platforms.