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:
- Terminating Decimal: The division of integers ends and produces a finite decimal.
- 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:
- All rational numbers have either a terminating or a repeating decimal expansion.
- 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.