Real Numbers and their Decimal Expansions

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

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.

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 π 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.