1.3 Real Numbers and their Decimal Expansions

Description

Quick Overview

This section explores the decimal expansions of real numbers, distinguishing between rational and irrational numbers.

Standard

In this section, we learn about the decimal representations of rational and irrational numbers. We observe that rational numbers have terminating or repeating decimal expansions, while irrational numbers exhibit non-terminating, non-repeating decimals. This understanding helps identify different types of numbers on the number line.

Detailed

Real Numbers and their Decimal Expansions

In this section, we delve into the decimal expansions that represent real numbers (both rational and irrational), focusing on how these expansions can help distinguish between the two types of numbers.

Rational Numbers

Rational numbers, by definition, can be expressed as the fraction
\[ \frac{p}{q} \]
where both p and q are integers and q is not equal to zero. When we look at their decimal expansions, they can either be:
- Terminating: These decimals have a finite number of digits, such as 0.875 or 2.56, which conclude after a certain point.
- Non-terminating recurring: These decimals continue indefinitely but eventually start to repeat, for example, 0.333... or 0.142857142857... .

Key Observations about Rational Numbers:

  1. Terminating Decimal: The division of integers ends and produces a finite decimal.
  2. Recurring Decimal: The remainder during division leads to the same cycle of digits, creating a repeating decimal.

Irrational Numbers

In contrast, irrational numbers cannot be expressed as fractions of integers. Their decimal expansions are characterized as:
- Non-terminating non-recurring: These decimals do not repeat and never settle into a repeating pattern, e.g., the decimal expansions of Ο€ and \( \sqrt{2} \).

Important Properties:

  1. All rational numbers have either a terminating or a repeating decimal expansion.
  2. Any decimal that is non-terminating and non-repeating corresponds to an irrational number.

By distinguishing between these two types of decimal expansions, we can accurately identify the nature of real numbers represented on the number line.

Key Concepts

  • Rational and Irrational Numbers: Distinct categories of real numbers determined by their ability to form fractions.

  • Decimal Expansions: The representation of numbers in decimal format can help identify their type.

  • Properties of Rational Numbers: They have either terminating or non-terminating repeating decimal expansions.

  • Properties of Irrational Numbers: They exhibit non-terminating non-repeating decimal expansions.

Memory Aids

🎡 Rhymes Time

  • Rational numbers end or repeat, their patterns can’t be beat!

πŸ“– Fascinating Stories

  • Imagine walking along a path covered with decimal signs. Some paths end abruptly, while others are winding and loop back on themselves; the endings are rational, while the wandering paths signify the irrationals.

🧠 Other Memory Gems

  • T-RaR (Terminating-Rational or Repeating-Rational, Non-terminating Non-repeating - Irrational).

🎯 Super Acronyms

R.I.P (Rational = Includes Patterns) and I.N.R (Irrational = No Repeating).

Examples

  • 0.5 is a terminating decimal and thus a rational number.

  • The decimal expansion of Ο€ is non-terminating and non-repeating, making it an irrational number.

Glossary of Terms

  • Term: Rational Numbers

    Definition:

    Numbers that can be expressed as the fraction p/q where p and q are integers, and q β‰  0.

  • Term: Irrational Numbers

    Definition:

    Numbers that cannot be expressed as a fraction of integers and have non-terminating, non-repeating decimal expansions.

  • Term: Terminating Decimal

    Definition:

    A decimal number that ends after a finite number of digits.

  • Term: Nonterminating Recurring Decimal

    Definition:

    A decimal number that continues indefinitely, observing a repeating pattern.

  • Term: Nonterminating Nonrecurring Decimal

    Definition:

    A decimal number that goes on forever without repeating.