Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. They can also be written in decimal form, either as a finite number of digits or a repeating pattern of digits.
Periodic decimal expansion refers to the repeating pattern of digits that occur in the decimal representation of a rational number. This occurs when the fraction has a denominator that is not a factor of 10.
A decimal is a rational number if it either terminates (has a finite number of digits) or has a repeating pattern of digits. For example, 0.25 is a rational number because it can be written as 1/4, while 0.333... is also a rational number because it has a repeating pattern of 3s.
No, irrational numbers cannot have a periodic decimal expansion. Irrational numbers are numbers that cannot be expressed as a fraction, and their decimal representations never repeat or terminate. Examples of irrational numbers include pi (3.14159...) and the square root of 2 (1.41421...).
Periodic decimal expansions are important because they help us to identify rational numbers and understand their properties. They also allow us to perform calculations with rational numbers, such as addition, subtraction, multiplication, and division, using the rules of fractions and decimals.