Rational and irrational numbers are two distinct types of real numbers. Here are five main differences between them:
Definition:
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction b is not equal to zero. In other words, rational numbers are those that can be written as fractions of integers.Irrational Numbers: Irrational numbers cannot be expressed as fractions of integers. They cannot be represented as b are integers. Instead, they have non-repeating and non-terminating decimal expansions.
Decimal Representation:
Rational Numbers: The decimal representation of a rational number is either terminating (ends after a finite number of decimal places) or repeating (the decimal part repeats indefinitely).
Irrational Numbers: The decimal representation of an irrational number goes on forever without repeating. For example, the square root of 2
Closure under Operations:
Rational Numbers: Rational numbers are closed under addition, subtraction, multiplication, and division (except when dividing by zero or when the denominator becomes zero).
Irrational Numbers: When irrational numbers are combined through basic arithmetic operations, the result may be either rational or irrational. For example, adding or subtracting two irrational numbers may result in either a rational or irrational number.
Density in the Real Number Line:
Rational Numbers: The set of rational numbers is dense in the real number line, meaning that between any two distinct rational numbers, there exists another rational number. However, there are gaps in the real number line where irrational numbers reside.
Irrational Numbers: The set of irrational numbers is also dense in the real number line. Between any two distinct irrational numbers, there exists another irrational number. The combination of rational and irrational numbers forms the entire real number line.
In summary, rational numbers can be expressed as fractions of integers with terminating or repeating decimals, while irrational numbers cannot be expressed as fractions and have non-terminating, non-repeating decimals. The distinction is based on the nature of their decimal representations and their properties under mathematical operations.