The difference between rational and irrational numbers fundamentally lies in their definitions and characteristics.
Rational Numbers: These are numbers that can be expressed as a fraction or quotient of two integers, where the denominator is not zero. In other words, if a number can be written in the form a/b, where a and b are integers and b is not equal to zero, it is considered a rational number. For example, the numbers 1/2, 3, -4, and 0.75 are all rational because they can be represented as fractions.
Moreover, rational numbers can be represented in decimal form as either terminating or repeating decimals. For instance, 0.5 is a terminating decimal, while 0.333… (which represents 1/3) is a repeating decimal.
Irrational Numbers: In contrast, irrational numbers cannot be accurately expressed as fractions of integers. They are non-terminating and non-repeating when represented as decimals. Examples of irrational numbers include π (pi), the square root of 2 (√2), and the mathematical constant e. For these numbers, there is no fraction of the form a/b that can exactly represent their value.
In summary, the key difference is that rational numbers can be expressed as a ratio of integers, while irrational numbers cannot. This distinction is crucial in various fields of mathematics, influencing calculations, proofs, and the understanding of the number line.