Defining a Rational Number
A rational number is any number that can be expressed as a fraction (ratio) p/q, where 'p' is an integer (whole number, positive, negative, or zero) and 'q' is a non-zero integer. The term 'rational' comes from 'ratio', highlighting its fractional nature.
Key Characteristics and Properties
Rational numbers include all integers (since any integer 'n' can be written as n/1). When expressed as a decimal, a rational number either terminates (e.g., 1/4 = 0.25) or repeats a sequence of digits indefinitely (e.g., 1/3 = 0.333...). They can be positive, negative, or zero.
Practical Examples of Rational Numbers
Common examples of rational numbers include 5 (which is 5/1), -7 (which is -7/1), 1/2 (0.5), 3/4 (0.75), -2/3 (-0.666...), and 0.23 (which is 23/100). Any fraction you can write with integers in the numerator and denominator (with a non-zero denominator) is a rational number.
Significance in Mathematics
Rational numbers form a foundational subset of the real numbers. They are closed under addition, subtraction, multiplication, and division (except by zero), meaning that performing these operations on two rational numbers always yields another rational number. This property makes them crucial in algebra, arithmetic, and understanding numerical systems.