Rational numbers and irrational numbers are both categorized under the set of real numbers because they represent all possible values on the number line. Real numbers encompass any value that can be found along this line, including whole numbers, fractions, and decimals.
Rational numbers are those that can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero. For example, numbers like 1/2, -3, and 7.5 fall into this category. Since they can be represented in a fractional form, they are considered ‘rational.’
On the other hand, irrational numbers cannot be expressed as simple fractions. These numbers have non-repeating, non-terminating decimal expansions. Common examples include numbers like π (pi) and √2. Although they cannot be expressed as fractions, they still correspond to a specific point on the number line, which allows them to qualify as real numbers.
In summary, both rational and irrational numbers are essential components of the real number system because they provide a complete coverage of all possible magnitudes and measures we can encounter, making them integral to mathematics and our understanding of quantity.