Rational and irrational numbers are two categories of numbers distinguished by their properties, particularly regarding how they can be expressed.
Rational Numbers:
A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers (with \( b \neq 0 \)). Rational numbers have decimal expansions that either terminate (come to an end) or repeat periodically.
Example: 3.14
- The number 3.14 can be expressed as \( \frac{314}{100} \), which is a fraction of two integers. This shows that 3.14 is a rational number.
- Its decimal form is finite (it has a fixed number of decimal places), which is another characteristic of rational numbers.
Irrational Numbers:
An irrational number, on the other hand, cannot be expressed as the quotient of two integers. These numbers have non-terminating, non-repeating decimal expansions. This means that they continue indefinitely without forming a repeating pattern.
Example: Pi (π)
- Pi (approximately equal to 3.14159) is an example of an irrational number. It cannot be expressed as the fraction of two integers—no matter how closely you try to approximate it, you cannot find two integers \( a \) and \( b \) such that \( \pi = \frac{a}{b} \).
- The decimal representation of pi goes on forever without repeating: 3.14159265358979… and so on, making it clear that it is not rational.
Summary:
- Rational Numbers (like 3.14) can be expressed as fractions of integers and have decimal expansions that either terminate or repeat.
- Irrational Numbers (like π) cannot be expressed as fractions of integers, and their decimal expansions are non-terminating and non-repeating.
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