Rational and irrational numbers are two fundamental categories in the field of mathematics, particularly in number theory.
Rational Numbers: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, a rational number can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
Example: The number 3.14 is a rational number because it can be expressed as the fraction \( \frac{314}{100} \), which is the result of multiplying both the numerator and denominator by 100 to eliminate the decimal. This fraction simplifies to \( \frac{157}{50} \), confirming that 3.14 is indeed rational.
Irrational Numbers: An irrational number, on the other hand, cannot be expressed as a simple fraction. This means that it cannot be written as a quotient of two integers; its decimal representation is non-terminating and non-repeating.
Example: Pi (π) is an example of an irrational number. Its decimal expansion begins as 3.14159 and continues infinitely without repeating. Since you cannot express π as a fraction of two integers, it is classified as an irrational number.
Summary:
- Rational Number: Can be expressed as a fraction of two integers (e.g., 3.14 = \( \frac{314}{100} \)).
- Irrational Number: Cannot be expressed as a fraction and has a non-terminating, non-repeating decimal representation (e.g., π).
Understanding the distinction between these two types of numbers is essential in various branches of mathematics and helps in appreciating the diversity of numbers within the real number system.
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